PHYSICS UNION MATHEMATICS. Physics II. Dynamics. Student Edition. Supported by the National Science Foundation (DRL ) and Science Demo, Ltd.

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1 PHYSICS UNION MATHEMATICS Physics II Dynamics Student Edition Supported by the National Science Foundation (DRL ) and Science Demo, Ltd.

2 PUM Physics II Dynamics Most of the module activities were adapted from: A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, Used with permission. Some activities area based on FMCE (Thornton and Sokoloff) and on Ranking Task Exercises in Physics (O Kuma, Maloney, and Hieggelke). Contributions of: E. Etkina, T. Bartiromo, A. Boudreaux, S. Brahmia, J. Chia, C. D Amato, J. Flakker, J. Finley, S. Kanim, R. Newman, J. Santonacita, E. Siebenmann, R. Therkorn, K. Thomas, M. Trinh. This material is based upon work supported by the National Science Foundation under Grant DRL Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF). 2 PUM Dynamics II

3 Table of Contents LESSON 1: FORCE AS AN INTERACTION 4 LESSON 2: EXPERIMENTAL DESIGN AND ASSESSMENT 17 LESSON 3: MOTION DIAGRAMS & FORCE DIAGRAMS 19 LESSON 4: WHAT S A FORCE TO DO? 24 LESSON 5: INERTIAL AND NON-INERTIAL REFERENCE FRAMES 29 LESSON 6: NEWTON S SECOND LAW: QUALITATIVE 33 LESSON 7: NEWTON S SECOND LAW: QUANTITATIVE 38 LESSON 8: DESIGN AN EXPERIMENT 44 LESSON 9: APPLYING NEWTON S SECOND LAW 48 LESSON 10: NEWTON S THIRD LAW: QUALITATIVE 52 LESSON 11: NEWTON S THIRD LAW: QUANTITATIVE 54 LESSON 12: TWO-BODY PROBLEMS 59 LESSON 13: DESIGN AN EXPERIMENT 62 LESSON 14: FRICTION 66 LESSON 15: PUTTING IT ALL TOGETHER 72 LESSON 16: COMPONENTS 74 LESSON 17: FINDING THE COEFFICIENT 81 LESSONS 18: PRACTICE 83 LESSON 19: REVIEW 88 PUM Dynamics II 3

4 1.1 Observe and Represent Lesson 1: Force as an Interaction a) Pick up a tennis ball and hold it in your hand. Now pick up a medicine ball and hold it. Do you feel the difference? How can you describe what you feel in simple words? b) Think of how we represented the motion of objects in the last module. What are some possible ways of representing the interaction between your hand and the tennis ball? c) Let s choose the ball as our object of interest. Represent the medicine ball with a dot and label the dot with Ball Draw an arrow to show how your hand pushes the ball. Connect the tail of the arrow to the dot. This arrow represents the force that your hand exerts on the ball. Did You Know? The word force is used in physics for a physical quantity that characterizes the interaction of two objects. A single object does not have a force by default, as the force is defined through the interaction of two objects. Remember that all physical quantities are measured in units. The unit of force is called the newton (N), where 1 N = (1 kg)(1 m/s 2 ). d) How could you label this force arrow to show that it is the force your hand exerts on the ball? Add this label to your representation. Here s An Idea! To show that the force arrow represents the push that the hand exerts on the ball, we can use a symbol F with two little words at the bottom on the right. These are called subscripts. For example: If we look at the interaction of a golf ball and a golf club while the club is hitting the ball. Then if we choose the golf ball as the object of interest, the golf club exerts a force on the golf ball. As a label for an arrow on a force diagram, this would be written as F club on ball. e) What do you think would happen to the ball if your hand were the only object interacting with it? What does this tell you about other objects interacting with the ball? f) What other objects are interacting with the ball? List each object and the direction of the push or pull. 4 PUM Dynamics II Lesson 1: Force as an Interaction

5 1.2 Test Your Reasoning a) In the previous activity, did you say that air interacts with the ball for part f? How do you think air interacts with the ball? b) What experiment can you perform to test your idea about whether the air pushes up or down on the ball? c) Use the video experiment on the website or CD if your class does not have the equipment. Before watching the video or performing the experiment, write a prediction of what should happen to the bottle based on your hypothesis. d) Watch the video or perform the experiment: (Or on the CD it is on the List of Videos, Bottle in a vacuum). e) Summarize what effect the air has on the ball. 1.3 Represent and Reason a) In activity 1.1, did you say that gravity interacts with the ball? Gravity is not an object; you cannot hold or touch it. So when we use the word gravity to note the pull down on all objects on Earth, what is the object that exerts this downward pull? b) Add another arrow on your diagram in 1.1 (c). Label the arrow with the appropriate subscripts. c) What do you notice about the length of the arrows in your diagram? What do you think would happen if the arrow representing the interaction with your hand were longer than the arrow due to the interaction with Earth? If it were the other way around? d) Now draw a diagram for the heavy ball. How are the force arrows different from the arrows on the diagram for the tennis ball? Did You Know? The diagrams you created in activity 1.1 through 1.3 are called force diagrams. Force diagrams are used to represent the forces exerted on an object of interest (system) by other objects. A system is an object or group of objects that we are interested in analyzing. Everything outside the system is called the environment and consists of objects that might interact with and affect the system object s motion. These are external interactions. When we draw force diagrams, we only consider the forces exerted on the system object(s). PUM Dynamics II Lesson 1: Force as an Interaction 5

6 1.4 Represent and Reason a) Think of a word to describe the force arrows in each force diagram. Did You Know? When the forces exerted on an object of interest are balanced, we say that the object is in EQUILIBRIUM (equilibrium does not necessarily mean rest). b) How might we represent our force diagrams with a mathematical representation or math statement? Write a math statement for the medicine ball. Need Some Help? Imagine putting an axis next to the force diagram with the origin at the dot. You can use + for the up direction and for the downward direction. For example: Let s take the situation of a puppy curled up in your lap. Then we can write the total force exerted on the puppy by your legs and Earth as: F legs on dog + F Earth on dog = 0. c) For your math statement, does it matter whether you chose up as positive or down as positive? How would this affect the math statement you wrote? What happens to the total force exerted on the ball if we switched the axis? Did You Know? Notice that depending on the orientation of the axis, either F Hand on Ball or F Earth on Ball has a negative value, thus the sum of a positive and a negative number can be zero. How do we know which force is positive and which one is negative? If the force arrow points in the positive direction of the chosen axis, we consider the force positive. If the y axis points down, for example, then F Earth on Ball >0 and F Hand on Ball <0. d) Look at your force diagrams for the tennis ball and medicine ball? What is the same about the diagrams? What is different? e) Write an expression for the forces exerted on the tennis ball similar to the expression you wrote for the medicine ball. Is the tennis ball in equilibrium? Explain. 1.5 Observe and Explain 6 PUM Dynamics II Lesson 1: Force as an Interaction

7 a) Perform the experiments described in the first column. Then record your data and fill in the empty cells. Remember that the scale, as a measuring instrument, has an uncertainty of measurement associated with it. Experiment Draw a picture of the apparatus. List objects interacting with the object of interest. Draw a force diagram for the object. Discuss what objects exert forces balancing the force that Earth exerts on the object. What is/are the direction of the balancing force/forces? Write a mathematical expression for the forces exerted on the object. Specify your axis. (a) Hang an object from a spring scale. Record reading of the scale here (b) Lower the object onto a platform scale so it touches the scale. Record new reading of the spring scale Record the reading platform scale (c) Remove the spring scale and leave the object on the platform scale. Record new reading platform scale (d) You place the object on a horizontal meter stick whose ends rest on two blocks. Record what happens (e) You place the object on a thick, foam cushion. Record what happens (f) You place the object on a tabletop. Record what happens (g) You place the block on the platform scale and then tilt the scale at a PUM Dynamics II Lesson 1: Force as an Interaction 7

8 small angle. Record what happens a) Some people think that only alive (animate) objects can exert forces. The table is not alive. How can a table push on an object? b) A book rests on top of a table. Jim says that the force exerted by the table on the book is always the same in magnitude as the force exerted by Earth on the book. Why would Jim say this? Do you agree or disagree with Jim? If you disagree, how can you argue your case? 1.6 Reason a) Summarize in what direction the force is exerted on an object of interest by the supporting object. b) Is this force always equal in magnitude and direction to the force that Earth exerts on the object? Provide experimental evidence and reasoning to support your opinion. c) Look at the force diagram shown in the Did You Know? below. How would the force diagram change if instead of dragging the box on a smooth floor, you dragged it on the carpet? Did You Know? The diagrams we constructed above are force diagrams. A force diagram is a physical representation used to analyze and evaluate processes involving forces. In order to create a force diagram, follow the 6 steps below. 1. Sketch the situation SKETCH FORCE DIAGRAM y 2. Circle the object of interest 3. Draw a dot representing the box F Floor on Box 6. Label the forces F Rope on Box 4. Identify interactions between the system and other objects. Here: Earth, floor, rope and surface F Earth on Box Check for understanding: What does the length of an arrow on the diagram mean? 8 PUM Dynamics II Lesson 1: Force as an Interaction 5. Draw forces to represent interactions, watch the length of arrows

9 Did You Know? System: A system is the object of interest that we choose to analyze. Make a sketch of the process that you are analyzing. Then circle the object of interest your system. Everything outside that system is called the environment and consists of objects that might interact with and affect the system object s motion. These are external interactions. Force: A force that one object exerts on another characterizes an interaction between the two objects. The force causes some effect or influence of the one object on the second object. Forces are represented by a symbol with an arrow above it to show that the force has direction and with two subscripts indicating the two objects. For example, if Earth pulls on a! ball, we note the force exerted by Earth on the ball as: F Earth on Ball. The arrow above force indicates that force is the physical quantity that both has magnitude and direction. The symbol also indicated that in this case our system is the ball and Earth is the external object. If we are interested in the force that the ball exerts on Earth, we will write it as! F Ball on Earth. 1.7 Reason Describe a situation in which a surface exerts ONLY a horizontal force on the object. Draw a picture of the situation. Then draw a force diagram. 1.8 Represent and Reason A person pushes a box across a very smooth floor. a) Examine the force diagram to the right. Do the forces in the vertical direction balance? Do the forces in the horizontal direction balance? b) Draw an arrow to indicate the direction of the unbalanced force, if there is one. Discuss whether the result is reasonable. 1.9 Represent and Reason Read each of the scenarios and then draw a force diagram for the selected object of interest. PUM Dynamics II Lesson 1: Force as an Interaction 9

10 1. You are throwing a tennis ball upward. Consider the moment right before the ball leaves your hand. The ball is the object of interest. 3. The ball is at the top of the flight. The ball is the object of interest 2. The ball is flying up. The ball is the object of interest. 4. The ball is being caught by you. Consider the moment when your hands are stopping the ball. The ball is the object of interest. Now use the rubric below to self-assess your force diagrams. Do not worry about the length of force arrows yet. Missing An attempt Needs improvement Acceptable No force diagram is constructed. Force diagram is constructed but contains major errors: missing or extra forces (not matching with the interacting objects), incorrect directions of arrows, or incorrect relative length of force arrows. Force diagram contains no errors in force arrows but lacks a key feature such as labels of forces with two subscripts or forces are not drawn from single point. The diagram contains forces for each interaction and each force is labeled so that one can clearly understand what each force represents. Relative lengths of force arrows are correct. 10 PUM Dynamics II Lesson 1: Force as an Interaction

11 Homework 1.10 Represent and Reason a) Draw force diagrams and use them to determine the direction of the unbalanced force exerted on the following objects of interest: i. A hockey puck moving on ice slows to a stop. The puck is the object of interest. ii. A box is sliding down an inclined plane. The box is the object of interest. iii. You start lifting up a heavy suitcase; the suitcase is the object of interest. iv. A boat floats in the ocean; the boat is the object of interest. v. You are pulling a sled on fresh snow at constant speed; the sled is the object of interest. vi. You are pushing a lawnmower; the lawnmower is the object of interest. b) Examine the unlabeled force diagrams and come up with a real life situation that they might describe. Then label each force with the appropriate subscripts Reason You are sitting on a chair reading this problem. What are the objects that exert forces on the chair? Choose the right answer: (a) no forces are exerted on the chair; (b) you are exerting a force; (c) you, Earth and gravity; (d) Earth, floor, and you; (e) floor, you and gravity Represent and reason You are holding a medicine ball steady. Choose the force diagram that represents the forces exerted on the ball best: a) b) c) d) e) F H on B F H on B F E on B F E on B F E on B F E on B F H on B 1.13 Estimate uncertainty F gravity Rob and Tina collected data using a scale that had divisions every newton (N): 0 N, 1 N, 2 N, 3 N, etc. When Tina hung her bag on the spring scale, she wrote the reading of the scale as 2.2 N. Rob repeated the experiment and wrote the reading as 2.3 N. They used the same bag. Why are their numbers different? Who do you think is correct? Based on your answer, decide how precisely you can measure the force with this scale Estimate uncertainty PUM Dynamics II Lesson 1: Force as an Interaction 11

12 Find three measuring devices in your house (each one needs to show the quantity that it is measuring, the units of measurement, and a scale). a) Write down the experimental uncertainty for each instrument. You may need to recall how we determined experimental uncertainty in the kinematics module. b) Now take a measurement with each instrument and write the result so that it incorporates the experimental uncertainty Estimate Uncertainty You drop a tennis ball from the second floor window three times. Your friend Tina is using a stopwatch to record the time of flight. Here are the results of her measurements: 1.11 s; 1.14 s; and 1.12 s. What is the average time Tina measured? What is the random uncertainty of the measurement? Write the result in the form of t = t average ±!t Significant Figures Two friends are discussing their cash allowance. Jake says that he gets $15month. Robin says that she gets $15.00 a month. Who is more likely to get more per month? Reflect: What did you learn about forces in this lesson? You heard the expression May the force be with you. Explain why it is not a possible thing for a force in physics. 12 PUM Dynamics II Lesson 1: Force as an Interaction

13 Lesson 1 A (Supplement): Vectors and Their Components 1a.1 Represent and Reason (a) In activity 1.5 you used a spring scale and a platform scale. You lowered the object onto a platform scale so it touched the scale. You recorded the new reading of the spring scale and the reading of the platform scale. Represent the situation with a force diagram. Assign magnitudes to the forces. What is the total force exerted on the object in this situation? (b) Obtain a block (a textbook will work too) and attach two spring scales to it, so that they are parallel to the desk surface, make a 60 0 angle with each other and show the same reading. Observe what happens to the box. Represent the situation with the force diagram. What is the total force exerted on the book? Did You Know? There are two general types of physical quantities force is an example of one type and temperature is an example of the other type. As you know, forces can be exerted in different directions (direction is very important if you are trying to hammer a nail into the wall), while there is no direction associated with your body temperature. Physical quantities that contain information about their magnitude and direction are called vector quantities and are represented!! by symbols with an arrow on top ( F, v, etc. ). For example, force is a vector quantity. When you push a door, your push can be represented with a force arrow; the stronger you push, the longer that arrow must be. The direction of the push is represented by the direction of the arrow on the force diagram. Physical quantities that do not contain information about direction are called scalar quantities and are written using italic symbols ( m, T, etc. ). Mass is a scalar quantity as is temperature. To manipulate scalar quantities, you use standard arithmetic and algebra rules - addition, subtraction, multiplication, division, etc. You add, subtract, multiply, and divide scalars as though they were ordinary numbers. Vectors are more complicated. For example, you just pulled on the box exerting equal strength pulls at an angle of 60 0 with respect to each other. Was the resultant pull twice as large as the pull exerted by each person, or more or less? In what direction was the resultant pull? To answer these questions, we need to learn how to perform mathematical operations on vectors. PUM Dynamics II Lesson 1: Force as an Interaction 13

14 a) Addition! A +! C To add two vectors we will use the following graphical! technique! to illustrate. Suppose we want to add the two vectors A and C!. To add them graphically, we redraw A! A and place the tail of C! at the head of A!!, keeping the length and the C orientation of both vectors the same as before (we can move vectors parallel to each other; therefore moving vectors A!! and C from their original location will not change them. While we can move vectors from one place to another for addition, we! cannot change the magnitude or direction of a vector while moving it). Having moved vector C!, we draw another vector, R!! C A +, from the tail of A! to the head of C! as in the figure at the right. This vector R!! represents the result of the addition of two vectors A and C!! R = A! + B!. We can write the resultant vector as a mathematical equation: R! = A! + C!. This vector has a magnitude and direction. As you see in the Figure, the magnitude of the vector is not equal to the sum of the magnitudes of A! and C!.! A b) Subtraction! A!! C You are familiar with subtraction a little bit already this is what we do when we find the!! v vector on motion diagrams. Basically, to find the vector! P, which is equal to! A!! C, you need to find the vector that you need to add to! C to get! A. c) Practice Label each vector in the pairs of vectors below and! A -! C! C! P =! A!! C!C +! P =! A find their sum and their difference. d) Make up 10 examples of vector addition, in 3 of them the vector s should be in the same direction, in 3 in the opposite and in 4 at an angle. Add the vectors and then act out every example. You can use any props available, including people. 14 PUM Dynamics II Lesson 1: Force as an Interaction

15 Check your Tip to Tail Vector Diagrams: Scale present (ex. 1 cm = 1 m/s) Orientation of the diagram remains consistent Vectors are labeled with magnitude, units and direction Vectors are done in proper format (tip to tail or coordinate) Vectors are drawn to scale Protractors, rulers and straight edge is used when drawing vector diagram Vector components are solid lines and resultant is dotted (or vice versa) Supplemental Material for those who can use Phet simulations Vector Analysis- Tip to Tail Method Tip to Tail: One vector tail starts from the origin and each vector added on after is added to the arrow tip of the previous vector. The vector tips become the new origin for each subsequent vector arrow. It is very important that you keep your orientation (0º, 90º, 180º, 270º or E, N, W & S) consistent in each diagram to be as accurate as possible with the direction of the vector. The magnitude of the vector is represented in the length of the arrow and should be to scale (ex. 1 cm = 1 m/s or 5 cm = 100 N). You can only add similar vectors with the same units (ex. Velocity and velocity, acceleration and acceleration, m/s² and m/s²). PhET Simulation Go to phet.colorado.edu and click on Play with Sims button. From the side menu bar select Math and then select Vector Addition Simulation. Select the Run Now button and a new window should pop up. Begin making your tip-to-tail diagrams by selecting a vector arrow from the bucket. Drag and drop the vector into the space provided. PUM Dynamics II Lesson 1: Force as an Interaction 15

16 Changing The Vector Arrow You can change the length and direction of the arrow by clicking on the tip and dragging it to the direction you choose or extend or compress it in size. Notice the box on the top has symbols and numbers. These will change when you change the length or direction of your arrow. Click on another arrow and it will display its values. Align the vector arrows so they are tip to tail. To get the resultant, click on Show Sum and a green arrow will appear in the space. Drag the arrow over to your diagram so that the Resultant (green) arrow s tail is at the origin and the tip of the Resultant is at the last arrow tip. You have just made a tip-to-tail vector diagram. NOTE: When doing this by hand, you will label the vector s magnitude (size) and direction next to the vector itself. The Resultant will be represented by a dotted line instead of a green line and the components will be solid lines. Answer the following questions using the simulation. 1. You take a walk and travel 20 meters in the north direction (90º). a. Could this arrow represent another type of vector, like 20 m/s North or 200 N 290º? Explain. b. Next, you turn left and walk 10 meters to the west (180º). Click show sum to find the resultant of these vectors. Arrange the vectors to display a tip-to-tail vector diagram. How far are you from your initial position? In which direction is the resultant? In what direction would you have to travel to get from your end point back to your starting point? c. Your friend says that you will get different answers if you set up your vectors in the wrong order, so you need to be extra careful. Do you agree with your friend? Support your answer with evidence. 2. On the weekend, you decide to run a couple of errands for your parents (the holidays will be here in no time!) on your way to your friend s house. You travel East on Rt. 537 driving 15 m/s, then travel North on Rt. 9 moving 21 m/s, and then take the back roads traveling 11 m/s at 328º to your friends house. What was the resultant velocity for your trip? For this next problem, select Style 2 so that you can see the x and y components of a vector at an angle and use a scale to convert the units associated with the arrow s length to the force in your force diagram. 3. A boy pulls his sled across newly fallen snow. The weight of the sled is 150 N and the boy pulls the sled with a constant velocity with a rope that makes a 45º angle with the ground. The ground exerts an upward force of 100 N and a horizontal force (friction) of 40 N. Draw a force diagram to represent this situation. What is the force the rope exerts on the sled? 16 PUM Dynamics II Lesson 1: Force as an Interaction

17 Lesson 2: Experimental Design and Assessment 2.1 Observe and Find a Pattern You have the following equipment: a spring scale, a plastic bottle filled with sand, a container of water, and a ruler (measuring tape). a) Examine the equipment that you are given. Think of possible questions you can answer using the equipment. Focus your questions on the magnitudes and directions of forces that the water exerts on the bottle. b) As a group, decide what question(s) you are going to investigate and the experiment you are going to conduct to answer your question(s). Draw a picture of the apparatus. c) What physical quantities will you measure? What do you think are the dependent and independent variables? How are you going to record and represent the data? d) Perform the experiment and collect the data. Represent the interactions of the bottle with the scale, Earth, and water using a force diagram. e) What patterns did you find? Summarize your findings for the question(s) you posed. f) Based on the experiment you performed, does the water push up or down on the object? Does the push depend on how deeply the object is submerged? If you cannot answer these questions from you experiment, conduct a second experiment to answer them. 2.2 Communicate Homework Write a report about your investigation so that a person who did not see your experiment can repeat it and obtain similar results. Use the rubrics on the next page to help write your report. Be sure to include what you learned from performing the experiment(s)? 2.3 Equation Jeopardy The following equations are mathematical descriptions of several situations that you might have encountered in activity 2.1. Try to visualize the equations and describe what situations they could describe. Choose the upward direction as positive. a) 10.0 N + (-10.0 N) = 0 b) 7.0 N N + (-10.0 N) = 0 c) 4.0 N N + (-10.0 N) = 0 For all three situations draw force vectors and show the total force using vector addition. PUM Dynamics II /Lesson 2: Experimental Design and Assessment 17

18 Scientific Ability Missing An attempt Needs some improvement Acceptable 1 Is able to identify the question to be investigated The question is not mentioned. An attempt is made to formulate the question but it is described in a confusing manner, or is not the question of interest. The question is posed but there are minor omissions or vague detail. The question to be investigated is clearly stated. 2 Is able to decide what is to be measured and identify independent and dependent variables The chosen measurements will not produce data that can be used to achieve the goals of the experiment. The chosen measurements will produce data that can be used at best to partially achieve the goals of the experiment. The chosen measurements will produce data that can be used to achieve the goals of the experiment. However, independent and dependent variables are not clearly distinguished. The chosen measurements will produce data that can be used to achieve the goals of the experiment. Independent and dependent variables are clearly distinguished. 3 Is able to use available equipment to make measurements At least one of the chosen measurements cannot be made with the available equipment. All chosen measurements can be made, but no details are given about how it is done. All chosen measurements can be made, but the details of how it is done are vague or incomplete. All chosen measurements can be made and all details of how it is done are clearly provided. 4 Is able to describe what is observed in words and with a picture of the experimental setup. No description is mentioned. A description is mentioned but it is incomplete. No picture is present. A description exists, but it is mixed up with explanations or other elements of the experiment. A labeled picture is present. Clearly describes what happens in the experiment both verbally and by means of a labeled picture. 5 Is able to identify sources of experimental uncertainty No attempt is made to identify experimental uncertainties. An attempt is made to identify experimental uncertainties, but most are missing, described vaguely, or incorrect. Most experimental uncertainties are correctly identified. Experimental uncertainties due to all instruments are correctly identified. Random uncertainty is considered. 6 Is able to evaluate specifically how identified experimental uncertainties may affect the data No attempt is made to evaluate experimental uncertainties. An attempt is made to evaluate experimental uncertainties, but most are missing, described vaguely, or incorrect. Or only absolute uncertainties are mentioned. The final result does take the identified uncertainties into account but is not correctly evaluated. The final result is written with the uncertainty. 18 PUM Dynamics II /Lesson 2: Experimental Design and Assessment

19 Lesson 3: Motion diagrams & Force diagrams 3.1 Observe and Represent Consider the following experiment: You have a bowling ball and a board (or anything that rolls easily, a billiard ball or a low friction cart on a track). You place the ball on the floor and push it with the board continuously trying to exert a constant force. a) Sketch the situation. b) Perform the experiment; then describe the motion of the ball in words. c) List all of the objects interacting with the bowling ball while it is being pushed. d) Draw a motion diagram for the ball. Indicate the direction of the! v! arrow. e) Draw three force diagrams for the ball as it is rolling on the floor. Place these diagrams under the motion diagram showing for what clock readings you created them. Does your picture make sense to you? a) Observe experiment in What is happening to Eugenia s speed? Draw a motion diagram and then 3 consecutive force diagrams for every instant when she drops a beanbag. Then draw a force diagram for Eugenia. Compare the direction of the!! v arrow on the motion diagram to the direction of the unbalanced force on the force diagrams 3.2 Represent and Reason a) Look at the force diagrams you drew in 3.1 (in for both experiments). Are the force arrows changing as the ball rolls on the floor? Are there any forces that are balanced? If so, please indicate which and explain why you think so. b) Indicate if there is an unbalanced force exerted on the ball on each diagram. Indicate the direction of the unbalanced force with an arrow. c) Indicate the direction of the velocity change arrow (! v! ) on the motion diagram for the instances matching the force diagrams. 3.3 Observe and Represent Consider this new experiment: You push the ball to start it moving. Once it is already rolling, you lightly push the front of the moving bowling ball continuously with a board in the direction opposite to the direction of motion. Try to exert a constant force. Make sure you do not push so hard that the ball stops very quickly. The slower the process occurs, the easier it will be for you to analyze it. a) Sketch the situation. b) Perform the experiment and describe the motion of the ball in words. PUM Dynamics II Lesson 3: Motion diagrams & Force diagrams 19

20 c) List all of the objects interacting with the bowling ball while it is being pushed in the direction opposite to its motion. d) Draw a motion diagram for the entire time that the ball is in motion. e) Draw three force diagrams for the ball as it is in motion; place those diagrams under the motion diagram for the relevant clock readings. f) Examine your force diagrams. Do they change as the ball slows down? Indicate which forces are balanced and which forces are unbalanced. How do you know? Draw an arrow to show the direction of the unbalanced force. g) Indicate the direction of the change in velocity arrow on the motion diagram for each of the clock readings for which you drew force diagrams. a) Observe the experiment in Draw a motion diagram for the ball, as it is moving upward and the force diagram for the three instances on the motion diagram. Compare the direction of the!! v arrow on the motion diagram to the direction of the unbalanced force on the force diagram. 3.4 Represent and Reason Place a bowling ball on a tilted surface and let it roll down on a very smooth floor. Observe the ball rolling on the floor. Does its velocity change? If the floor is smooth the ball should not slow down. a) Draw a motion diagram for the ball. What is the direction of the velocity change? b) Draw force diagrams for the ball for three different clock readings. Are the diagrams the same or different? What is the direction of the unbalanced force? c) If the floor is infinitely long, how long will the ball move before it stops? Should it ever stop? a) Use the outcome in the experiment in to compare Eugenia s motion to the motion of the bowling ball. Are they similar? Different? How do you know? 3.5 Find a Pattern Consider the experiments you performed in activities Examine the force and motion diagrams for each experiment. a) Is there a pattern in the directions of the unbalanced forces that other objects exert on the ball and in the directions of the! v arrows on the motion diagrams for the ball? b) Is there a pattern in the directions of the unbalanced forces that other objects exert on the ball and the directions of the! v! arrows in the motion diagrams? 20 PUM Dynamics II Lesson 3: Motion diagrams & Force diagrams

21 c) Use the pattern that you found to formulate a statement relating the force diagram to the motion diagram. d) How do you understand the difference between the words motion and change in motion? Give an example. e) Do you think the net force exerted on an object causes motion or change in motion? f) Who was the observer recording the velocity changes for the object in the above activities? Would there be observers for whom the statement relating the force diagram to the motion diagram would not be true? 3.6 Test the Pattern a) Design 2 different experiments whose outcome you can predict using the statement you formulated in activity 3.5 (c). Need Some Help? The statement you are testing is the rule or pattern you noticed between the direction of the unbalanced force on the force diagram and the direction of the! v! arrow on the motion diagram for the same object. The experiment you design should try to rule out the pattern, not to prove it. b) Write predictions for the outcome of each experiment based on the pattern you noticed. c) Perform the experiment and record the outcome. How did your prediction compare to the outcome? Did you succeed in the disproving the pattern? Explain your judgment. 3.7 Test the Pattern Homework You have a medicine ball. When you place it on a bathroom scale, the scale reads 6 pounds (a unit of force in a the British system). Imagine that a friend drops a medicine ball, and it falls straight down on a bathroom scale. a) Draw a force diagram for the ball when it sits on the scale at rest. Draw a motion diagram for the ball. b) Draw a motion diagram for the ball when it just touches the scale but is not yet stopped. c) Draw a force diagram to match the motion diagram. Assume that the scale reads the force that the scale exerts on the ball. Make a prediction about the reading of the scale as it stops the falling ball using the pattern between the motion diagram and the force diagram you formulated and tested during the lesson. PUM Dynamics II Lesson 3: Motion diagrams & Force diagrams 21

22 d) After you made the prediction, watch the videos. Make sure that in the second clip, you move frame by frame. then e) What judgment can you make about the pattern you formulated? 3.8 Represent and Reason a) Draw a motion diagram for a book sliding on a table coming to a stop. Draw a force diagram for the book. Are the force diagram and motion diagram consistent with each other? Explain. b) You are holding a birthday balloon filled with helium. Draw motion and force diagrams for the balloon. Are the force diagram and motion diagram consistent with each other? Explain. c) You are holding a birthday balloon filled with helium and then let it go. Draw motion and force diagrams for the balloon the moment you let it go. Are the diagrams consistent with each other? Explain. d) The balloon reaches the ceiling. Draw motion and force diagrams for the balloon the moment the top of it touches the ceiling. Check the consistency of your representations. Can you represent the balloon as a particle in this case? Explain. e) A matchbox slides down a steep incline. Draw a motion diagram and a force diagram for the matchbox as it slides down the incline. Check the consistency of your representations. 3.9 Pose a Problem Consider the scenario: You are playing ice hockey. Pose a problem similar to the two activities above. Then solve your problem Reason Below you have descriptions of processes, unlabeled motion diagrams, and unlabeled force diagrams. Label the force diagrams and motion diagrams and match the diagrams so all three describe the same motion. Words: (a) a book sliding on a table top to a stop; (b) a book dragged on the tabletop at constant speed; (c) an object thrown upward on its way up; (d) and object thrown upward at the top of its flight; (d) a car skidding to a stop. Notice that some force diagrams are extra they do not match any word descriptions. 22 PUM Dynamics II Lesson 3: Motion diagrams & Force diagrams

23 Force diagrams a) b) c) d) e) f) Motion diagrams a) b) c) e) PUM Dynamics II Lesson 3: Motion diagrams & Force diagrams 23

24 Lesson 4: What s a Force to Do? 4.1 Test an Idea Aaron has a hypothesis that says objects always move in the direction of the unbalanced force exerted on them by other objects. a) Design an experiment whose outcome might be consistent with Aaron s hypothesis. Carefully describe what you are going to do. Draw a force diagram for the system object for the experiment you described. b) Make a prediction about the object s motion based on Aaron s hypothesis. Use the rubrics below to help in your reasoning. Need Some Help? A Hypothesis is a general belief, pattern, model, or rule developed from observations. Hypotheses can be used to develop predictions. A Prediction is a statement that describes the outcome of a specific experiment (prior to conducting the experiment). Predictions must be based on the hypothesis being tested. We can use our hypothesis and prediction to write an H-D statement, which uses the logic of hypothetico-deductive reasoning: If blah-blah is correct (describe my idea/hypothesis/explanation) and I do such and such (describe your testing experiment), then such and such should happen (describe your predicted outcome). But it did not happen (describe the outcome of the experiment), therefore I need to revisit my hypothesis, examine the experiment, etc. OR And it did happen (describe the outcome of the experiment), therefore my hypothesis has not been disproved yet. REMEMBER! Your predicted outcome must be based on your hypothesis. c) Now perform the experiment and record the outcome. How did the outcome compare to the prediction? Can you say that you proved Aaron s hypothesis? 4.2 Test an Idea a) Think of an experiment you can perform in which you can exert a force on an already moving object in a direction that is different from the direction of its motion. Describe the experiment carefully and draw a force diagram. b) Use Aaron s idea to make a prediction about the object s motion. Use the rubrics below to help in your reasoning. 24 PUM Dynamics II Lesson 4: What s a Force to Do?

25 c) Perform the experiment and record the outcome. Compare the outcome to your prediction. What judgment can you make about Aaron s HYPOTHESIS? Scientific Ability Is able to distinguish between a hypothesis and a prediction Is able to make a reasonable prediction based on a hypothesis Is able to make a reasonable judgment about the hypothesis Missing No prediction is made. The experiment is not treated as a testing experiment. No attempt to make a prediction is made. No judgment is made about the hypothesis. An attempt A prediction is made but it is identical to the hypothesis. A prediction is made that is distinct from the hypothesis but is not based on it. A judgment is made but is not consistent with the outcome of the experiment. Needs some improvement A prediction is made and is distinct from the hypothesis but does not describe the outcome of the designed experiment. A prediction is made that follows from the hypothesis but does not have an if-and-then structure. A judgment is made and is consistent with the outcome of the experiment but assumptions are not taken into account. Acceptable A prediction is made, is distinct from the hypothesis, and describes the outcome of the designed experiment. A prediction is made that is based on the hypothesis and has an if-and-then structure. A reasonable judgment is made and assumptions are taken into account. 4.3 Hypothesize Based on the experiments above and the experiments you performed in lesson 3, formulate the relationship between the unbalanced force (sum of the forces) exerted by other objects on the object of interest and the motion (or change in motion) of the object of interest. 4.4 Test an Idea James thinks that when the sum of the forces exerted by other objects on the object of interest is zero, the object is at rest. a) Design two experiments to test James s hypothesis. b) Write a prediction for each experiment based on James s hypothesis. c) Perform the experiments and record the outcomes. Compare the outcomes to your predictions. d) What judgment can you make about James s hypothesis now? Why would James have such an idea? Did You Know? Relationship between force and motion: If external objects exert an unbalanced force on the system object of interest, its motion changes so that the!! v arrow in a motion diagram describing the motion of this object is in the same direction as the unbalanced force exerted on the object. PUM Dynamics II Lesson 4: What s a Force to Do? 25

26 Homework 4.5 Reason For each part below, identify the system, draw a force diagram, draw a motion diagram, and determine if the force and motion diagram are consistent. a) Give an example for an object moving in the direction of the unbalanced force exerted on it by other objects. b) Give an example for an object moving in the direction opposite to the unbalanced force exerted on it by other objects. c) Give an example for an object moving at an angle with the unbalanced force exerted on it by other objects. d) You throw a small ball upward. 4.6 Represent and Reason Draw a motion diagram and three force diagrams for each of the following objects (the object of interest is underlined) once the object is in motion. Make sure that the force diagram and motion diagram are consistent with each other. Situation Motion diagram Force diagrams for three instances How do you know if they are consistent? A ball is dropped and is falling down. A ball is thrown upwards after it leaves your hand and before it reached the top of the flight A ball is thrown down. 26 PUM Dynamics II Lesson 4: What s a Force to Do?

27 A football is landing on a cushion and the cushion is being compressed. A rabbit sits in its cage. A matchbox slides down a smooth book cover. An air hockey puck slides across an air table. 4.7 Reason Read the statements below and classify each into one of three groups: experimental evidence, hypothesis, prediction. a) As the plants grow their mass increases. b) The mass of the plants increases because you water them. c) The increase in the mass of the growing plant should be exactly equal to the decrease in the mass of the potting soil in a pot with a plant. Measure when the soil is dry. d) The mass of the plants increases because they absorb carbon from the air. e) The mass of the plants increases because they absorb nutrients from the soil. f) The increase in the mass of the growing plant should be exactly equal to the mass of water used to water the plant. PUM Dynamics II Lesson 4: What s a Force to Do? 27

28 g) Explain how you understand the difference between the terms experimental evidence, hypothesis, and prediction. Provide your own example for each. Reflect: Is the following statement true or false: Objects on Earth always move in the direction of the total force exerted on them by other objects. If you disagree, how can you convince others in your opinion? 28 PUM Dynamics II Lesson 4: What s a Force to Do?

29 Lesson 5: Inertial and Non-inertial Reference Frames 5.1 Observe and Analyze You are sitting on a train and place a ping-pong ball on a tray table in front of you. The pingpong ball is at rest. All of a sudden, the ball starts rolling towards you. At the same time, your friend who was waiting for your train to depart, saw the train starting to move in the direction in which you were facing, but she saw the ball stationary and the train leaving from under it. a) Describe the motion of the ball when it starts rolling using a motion diagram for each observer: you on the train and your friend on the platform. b) Explain the behavior of the ball when it starts rolling using a force diagram for each observer: you on the train and your friend on the platform. Observer Motion diagram for the ball (description of motion) You on a train that is starting to move Your friend on the platform Force diagram for the ball (explanation of motion) Are the diagrams consistent? Explain. c) What can you say about the relationship between the unbalanced force and change in motion for the observer on the train that starts to move? d) How will you rewrite the relationship between force and change in motion to include the role of the observer? e) Why do you think your friend on the platform did not see the ball starting to move (i.e. change its motion) when the train started to move? Did You Know? Inertial reference frame: Inertia is the phenomenon when an object continues moving at constant velocity if no other objects interact with it or if the sum of all these interactions is zero. Reference frames in which we can observe this phenomenon are called inertial reference frames. If the sum of all forces exerted on the object is zero, then in an inertial reference frame, the object s velocity remains constant. PUM Dynamics II Lesson 5: Inertial and Non- inertial Reference Frames 29

30 Newton s first law of motion: We choose a particular object as the object of interest the system. If no other objects interact with the system object or if the sum of all the external forces exerted on the system object is zero (forces in the y direction are balanced and forces in the x direction are balanced), then the system object continues moving at constant velocity (including remaining at rest) as seen by observers in the inertial reference frames. 5.2 Reason Consider the following idea: The relationship between the unbalanced force and the change in motion depends on the observer. Apply this idea using all of the videos at: For some observers, THE RELATIONSHIP between the direction of the unbalanced force and the direction of the!! v arrow DOES NOT WORK. Identify such observers in every experiment. Need Some Help? Recall in the kinematics module that motion is relative. For each video, describe the motion of the object from multiple reference frames. Recall that when we draw force diagrams, we only consider the forces exerted on the system object(s). 5.3 Reason a) You are a passenger in a car. All of a sudden, your head jerks backwards. Explain this experiment from the reference frame of the car. Explain this experiment from the reference frame of the pavement. b) Describe what an observer on the ground sees when you stumble on a rock or slip on a banana peel (focus on the motion of your feet and your head, assuming that the head is only loosely attached to the body). Then describe what you observe. c) Use Newton s first law to explain the observations of the earth-based observer and your observations for the situation described in b. Who is in the inertial reference frame? How do you know? d) Imagine that you have an infinitely long, smooth table covered in sand. A bowling ball is hit once so that it starts rolling on the table but stops after 2 m. After removing some of the sand and repeating the experiment, the ball stops rolling after 5 m. How far do you think the ball will roll if ALL the sand is removed? e) If you take a ball whose mass is half of the mass of the bowling ball, how will the outcome of the last experiment change? 30 PUM Dynamics II Lesson 5: Inertial and Non- inertial Reference Frames

31 5.4 Explain Homework a) Provide two examples when the forces exerted on an object of interest are balanced. b) Choose an observer who will see this object moving with a constant velocity. c) Now choose an observer who will see this object moving with a changing velocity. d) Draw pictures of the same situation as seen by the two different observers. What is different about the reference frames of these two observers? Need Some Help? Example: A pendulum with a pendulum bob is attached to the ceiling of a car. For an observer sitting in the car, the bob instantly starts moving towards him. For an observer on the ground, the bob remains at rest but the car accelerates forward. 5.5 Represent and Reason An elevator starts at rest on the ground floor of a building and stops at the top floor. The elevator then returns to the bottom floor. The elevator is our object of interest. Complete the table that follows to determine how the force the supporting cable exerts on the elevator compares to the force Earth exerts on the elevator. The motion diagram and the force diagram should be consistent with each other and with the rule relating motion and forces developed in lesson 3. Experiment Sketch a motion diagram. Draw three force diagrams for 3 consecutive clock readings on the motion diagram. Check the consistency of the diagrams. (a) Elevator hangs at rest at the ground floor. (b) Elevator starts moving upward with increasing speed. PUM Dynamics II Lesson 5: Inertial and Non- inertial Reference Frames 31

32 (c) Elevator moves at a constant upward speed. (d) Elevator slows as it approaches the top floor. (e) Elevator starts moving down with increasing speed. (f) Elevator moves down at constant speed. (g) Elevator slows to a stop on ground floor. Summarize what you learned about how the magnitude of the force that the supporting cable exerts on the elevator compares to the force that Earth exerts on the elevator when the elevator moves at constant speed and changing speed. Reflect: Explain to your friends/parents/siblings the meaning of the words inertial/non inertial reference frame and give 2 examples of inertial frames from your everyday life and 2 examples of non-inertial reference frames. 32 PUM Dynamics II Lesson 5: Inertial and Non- inertial Reference Frames

33 Lesson 6: Newton s Second Law: Qualitative 6.1 Observe and Find a Pattern Student A is on rollerblades and stands in front of a motion detector. The motion detector produces the velocity-versus-time graphs shown. Student B (not on rollerblades) stands behind Student A and pushes her forward. Student A starts moving. The surface is very smooth (linoleum floor). a) Describe any patterns you see on the graph. b) Draw a motion diagram and a force diagram for student A (1) when student B is pushing and (2) when she is not pushing. Are the diagrams consistent with each other for each time interval? c) What is the meaning of the slope of the graph? Write a mathematical function that describes each part of one line on the graph. Write a second mathematical function that describes each part of a second line on the graph. What is the difference in the functions? d) What can you say about the relationship between an unbalanced force exerted on an object and its acceleration? 6.2 Reason Refer to the activity 6.1. a) Why do you think student A does not stop moving when student B stops pushing? b) Break each motion down into two parts: when student B is pushing and when student B is not pushing. What is different about the motion of student A for the three cases? What is the same? c) Imagine that while student A is moving at a constant speed, student B starts pulling her in the direction opposite to her motion, exerting a constant force. Draw a motion diagram and a force diagram for student A and extend the existing velocity versus time graphs to represent the experiment. d) Repeat c but imagine that student B pulls even harder in the direction opposite to student A s motion. 6.3 Test the pattern Download on your computer PHET simulation Forces in one dimension. Turn off Friction. Click on graph Applied Force and Graph velocity. Use the file cabinet as your object. Predict the shapes of the graphs (sketch them) if you first exert a force of 200N on the cabinet for 6 seconds and then turn the force off but continue playing the simulation. After you sketched the PUM Dynamics II Lesson 6: Newton s Second Law: Qualitative 33

34 predicted graphs, play the simulation and reconcile your predictions with the simulation graphs. Repeat the same procedure for the force of 400 N. 6.4 Observe and Find a Pattern Student A is still on rollerblades, but this time she is wearing a backpack filled with textbooks. Student B pushes Student A several times; each time, student A adds three more books to the backpack. Student B pushes exerting the same force each time. a) Use the graph to find a qualitative pattern between the change in Student A s velocity and the amount of stuff in her backpack. Did You Know? Mass: Mass m characterizes the amount of matter in an object and the ability of the object to change velocity in response to interactions with other objects. The unit of mass is called a kilogram (kg). Mass is a scalar quantity, and masses add as scalars. b) What can you say about the velocity of Student A after Student B stops pushing her? c) Does the mass of an object affect the velocity of an object or the change of velocity when there is a force exerted on it? d) How does changing the mass of the object affect the acceleration of an object if the force exerted on it is the same? 6.5 Test the pattern (Phet simulations) Use the same simulation as before. This time predict the shapes of the applied force and velocity graphs you use the same magnitude force first exerted on the file cabinet (200 kg) and then on the refrigerator (400 kg). After you made the predictions, run the simulation. Did your prediction match the graphs on the simulation? If not, how can you reconcile? 6.6 Find a Relationship Summarize how the change in the velocity of the object depends on the forces exerted on it by other objects. Then summarize how the change in the velocity of the object depends on the mass. Use the words: more, less, and constant. Now combine these two relationships into one. 6.7 Test the Relationship a) Use the relationship you formulated in activity 6.4 to predict the shape of the velocity versus time graph for an object that is dropped b) Use the relationship you formulated in activity 6.4 to predict the shape of the velocity versus time graph for an object that is thrown downward. c) Explain the shape of the graphs in terms of the relationship you are testing. 34 PUM Dynamics II Lesson 6: Newton s Second Law: Qualitative

35 d) Conduct the experiment using a motion detector. If there is no motion detector in your classroom, use the graphs provided by your teacher to compare the prediction with the actual outcome. e) Revise the relationship if the prediction does not match the outcome. Did You Know? Two new words are used in physics to describe the processes and objects in activity 6.3. The first is inertia. It describes the motion of an object that does not interact with any other objects AND inertia also describes the motion of an object whose interactions with other objects are balanced. Only observers in inertial reference frames observe inertia. The second word is inertness. Inertness is a property of objects that describes how hard it is to change their velocity by exerting forces on them. 6.8 Reason Examine the graphs in activity 6.4. Discuss which parts of the graph relate to the word inertia and which parts relate to the word inertness. What is a familiar physical quantity that is a quantitative measure of inertness? 6.9 Represent and reason Homework Draw a velocity versus time graph for the following scenario: A bowling ball is rolling on the floor. Then a person starts pushing it very lightly in the direction opposite to the direction of motion and continues pushing for a while. Finally the person stops pushing the ball. How many possible graphs can you draw for this scenario? Examine the effects of assumptions on your answer Reason When you talk about the change in velocity of an object and the unbalanced force exerted on that object, what is the cause and what is the effect? How do you know? 6.11 Equation Jeopardy The following system of equations describes forces exerted on an object in the vertical and in the horizontal direction. Describe two different situations that this system can describe in which the object of interest is moving in two different directions: x direction: 12.0 N +(- 9.0 N) = 3.0 N y direction: 20.0 N +(-20.0 N) = 0 PUM Dynamics II Lesson 6: Newton s Second Law: Qualitative 35

36 6.12 Ranking Tasks Examine the forces exerted on each object and the mass of each object. Rank the magnitude of the accelerations of the objects from largest to smallest. Each arrow represents a force exerted by some other object on the object of interest. Be sure to explain the reasoning behind your ranking. A B C 1,000 g 400 g 200 g D E F 1,000 g 500 g 200 g 6.13 Reason You throw a ball up. Draw a motion diagram and two force diagrams for the ball on its way up and another set of diagrams (one motion and two force) for the ball on its way down: (a) first ignoring air resistance, and then (b) taking air resistance into account. When does the ball have the largest! v!? (c) Represent the motion of the ball for the case of no air resistance with the position versus time graph and velocity versus time graph Reason A string pulls horizontally on a cart so that it moves faster and faster along a smooth frictionless horizontal surface. When the cart is moving medium-fast, the pulling is stopped abruptly. (a) Describe carefully in words what happens to the cart s motion when the pulling stops. (b) Illustrate your description with motion diagrams, force diagrams and position and velocity-versus-time graphs. Indicate on the graphs when the pulling stopped. What assumptions did you make? 6.15 Reason Your friend solving the previous problem says that after the string stops pulling, the cart starts slowing down. (a) Give a reason why he would think this way. (b) Do you agree with his opinion? Explain your opinion. (c) Explain carefully how you can design an experiment to test his idea. 36 PUM Dynamics II Lesson 6: Newton s Second Law: Qualitative

37 Reflect: What did you learn about Newton s laws in this lesson? How did you learn it? If you were to explain in simple words to a person who never took physics what Newton s second law is about, what would you say? Why do you think Newton s second law is important? PUM Dynamics II Lesson 6: Newton s Second Law: Qualitative 37

38 Lesson 7: Newton s Second Law: Quantitative 7.1 Observe and Find a Pattern Imagine an experiment in which one or more identical springs pull one or more identical carts in the same direction on a smooth horizontal track. (The springs are stretched the same amount so that each spring exerts the same force on the cart.) Experiment number Number of springs Number of carts Acceleration of carts m/s m/s m/s m/s m/s m/s m/s m/s 2 a) Use the data table above. Draw pictures and force diagrams for the cart in Experiments 1 4. Picture and Force Diagram for Experiment 1 Picture and Force Diagram for Experiment 2 Force Diagram for Experiment 3 Force Diagram for Experiment 4 b) Use the data in the table above to devise a relationship that shows how the carts acceleration depends on the carts mass and on the sum of the forces exerted on the carts by the springs, Earth, and the track. Need Some Help? When doing such analysis, it is possible to devise a relationship between the dependent variable and each independent variable, one at a time. Then combine these relationships to get a final relationship. For example, use some of the data to see how the acceleration depends on the number of springs. Then use other parts of the data to see how the acceleration depends on the number of carts. You can also collect your own data using the video experiments if you prefer (optional): 38 PUM Dynamics II Lesson 7: Newton s Second Law: Quantitative

39 and Observe and Find a Pattern Imagine springs are attached to both ends of a cart. The springs can pull the cart left or right. Each spring pulls with the same strength, but the number of springs on either side of the cart can vary. a) Examine the data in the table that follows. Experiment Number of springs pulling to the right Number of springs pulling to the left Acceleration of the cart m/s m/s m/s m/s 2 b) Draw a force diagram for the cart in each experiment. Show the horizontal forces only; the upward force exerted by the surface on the cart s wheels and the downward force exerted by Earth on the cart balance. Force Diagram for Experiment 1 Force Diagram for Experiment 2 Force Diagram for Experiment 3 Force Diagram for Experiment 4 Force Diagram for Experiment 5 c) Explain why there are negative signs in the acceleration column of the data table. d) Use the data in the table above to devise a relationship between the cart s acceleration and the sum of the forces exerted by the springs, Earth, and the track on the cart. PUM Dynamics II Lesson 7: Newton s Second Law: Quantitative 39

40 e) Is this relationship consistent with the relationship you came up with in the previous activity? Explain. 7.3 Explain In the two previous activities, you analyzed experiments in which the motion of an object was affected by other objects. a) Mathematically represent the relationship between the object s acceleration, the unbalanced force exerted on it by other objects, and its mass. Make sure that you write the relationship as a cause-effect relationship. Need Some Help? Think of cause-effect relationships you encounter in everyday life. For example, you set your alarm early (cause), you get to use the bathroom first (effect). Another example would be, you stub your toe on a rock (cause), your toe starts aching (effect). Determine what variables in part a can be causes (independent variables) and what variable can be the effect (dependent variable). b) Represent the relationship graphically. What is your dependent variable and what are your independent variables? How many graphs do you need in order to represent the relationship? c) How does acceleration of an object depend on the unbalanced force exerted on it? How does the acceleration depend on the mass of the object? d) What variable(s) did you hold constant in answering the questions above? Why was this necessary? e) What does it mean if one quantity is directly proportional to another quantity? What does it mean if it is inversely proportional? Give integer examples and draw graphs to explain your reasoning. 7.4 Test Your Idea a) Design an experiment whose outcome you can predict using the idea that you invented in 7.3. b) Make a list of the equipment will you need. Describe the experiment in words and with a picture. 40 PUM Dynamics II Lesson 7: Newton s Second Law: Quantitative

41 Here s An Idea! Use the motion of objects that you are familiar with. One possibility would be a ball thrown upward. c) Make a prediction about the outcome of the experiment using the idea you invented in 7.3. d) Perform the experiment and record the outcome. How does your prediction compare to the outcome? What judgment can you make about your idea? 7.5 Test Your Idea You can now test your idea with the Phet simulations. Use Forces in 1 dimension again. Friction should be off. Turn on force and acceleration graphs. Choose the object you wish to push and the force you wish to exert. Do not play the simulation yet. Sketch the graphs of force and acceleration with the scales on them labeled carefully. Then play the simulation do you need to revise your predicted graphs? After you do this, sketch the graphs position and velocity versus time and then play the simulation again. If you have time, repeat for a different force, make sure you vary both magnitude and direction! Did You Know? In the previous activities, you have developed and tested Newton s Second Law of Motion. Newton s Second Law of Motion: We choose a particular object, or group of objects, as our system object. The acceleration a! of the system is directly proportional to the unbalanced (net) force F! net = F! 1 on S + F! 2 on S F! n on S =! F! n on S exerted by other objects on the system object and inversely proportional to the mass m of the system object:!! F a = net =!! F n on S m S Newton s second law is written for the vectors. You cannot use them to make calculations, thus you need to use it in component form. To do this, use the force diagram. If the forces are pointed in the positive direction along x or y axes, they enter y the equation with positive signs, if in the opposite direction with negative signs. For example: Two forces are exerted on an object in the vertical direction: a 20 N force downward and a 10 N force upward. The mass of the object is 2 kg. How do we write Newton s second law in component form for this situation? Consider the upward to be the positive direction. The force diagram for the situation is on the right. Newton s second law is a! =! F!! n on O F = 1 on O + F! 2 on O m O m O. Component of force 1 in the y direction is negative and the component of force 2 is positive. Newton s second law in component form for the situation becomes: m S F 2 on o F 1 on o PUM Dynamics II Lesson 7: Newton s Second Law: Quantitative 41

42 a y = F + F 1 on O y 2 on O y = (!F ) + (F ) 1 on O 2 on O (!20 N) + (10 N) = =!5 N/kg =!5 m/s 2. m O m O 2 kg Follow the steps in the above equation carefully to see how you can find the acceleration. The sign of acceleration is negative. It means that the acceleration points down. The object is either sloping down on its way up or speeding up on its way down. 7.6 Reason Homework When you studied kinematics, you learned that all objects fall with the same acceleration: 9.8 m/s 2. Use this observational evidence and Newton s Second Law to write a mathematical expression for the force that Earth exerts on any object. 7.7 Reason a) Two forces are exerted on an object in the vertical direction: a 20 N force downward and a 10 N force upward. The mass of the object is 25 kg. (1) What do you know about the motion of this object? (2) Represent the motion of the object with a force diagram and a motion diagram. (3) Write Newton s second law in component form. b) You pull a 20-kg sled, exerting an unbalanced, horizontal force of 30 N on it for 10 seconds. (1) What is the acceleration of the sled? (2) What is the speed of the sled after 3 seconds? (3) What force do you need to exert on the sled if you wish to keep it going at that constant velocity? c) You hang a picture using two ropes, each at an angle of 30 with the vertical. (1) Draw a sketch of the situation. (2) Draw a force diagram for the picture. (3) If the mass of the picture is 5 kg, what is the force that each rope must exert on the picture to keep it stable? (4) How can you use trigonometry to solve the problem? 7.8 Represent and Reason Complete the table that follows. The system object is underlined. Write a word description of the situation. An elevator pulled by a rope is slowing down on its way up. Sketch the situation and circle the system object. Draw a force diagram; do not forget the axes. Label the forces, if needed. Draw a motion diagram. Draw the direction of the acceleration and of the net force. Are they consistent? 42 PUM Dynamics II Lesson 7: Newton s Second Law: Quantitative Write Newton s Second Law in component form.

43 a y = T R on O + (!F E on O ) m!2.0 m 800 N+(-1000 N) = 2 s 10 kg 30 o Did You Know? Gravitational force: The magnitude of the gravitational force that Earth exerts on any object near its surface equals the product of the object s mass m and the gravitational constant g: F E on O = m g where g = 9.8 m/s 2 = 9.8 N/kg on or near the earth s surface. The force points toward the center of Earth. 7.9 Reason Two small balls of the same material, one of mass m and the other of mass 2m, are dropped simultaneously from the Leaning Tower of Pisa. Earth exerts a bigger force: (a) On the 2 m ball. (b) On the m ball. (c) Earth exerts same force on both balls because they fall with the same acceleration. Reflect: What did you learn about Newton s laws in this lesson? How did you learn it? If you were to explain in simple words to a person who never took physics what Newton s second law is about, what would you say? Why do you think Newton s second law is important? PUM Dynamics II 43

44 8.1 Design an Experiment Lesson 8: Design an Experiment You have a spring and a set of objects of known masses. Your goal is to create a mathematical model for the spring constant, which relates the forces exerted by the spring to the stretch of the spring. a) Design an experiment to determine the relationship between how much the spring stretches from the original unstretched position as it pulls on an object and the force that it exerts on that object. b) Sketch the setup for your experiment and outline the plan. What instruments are you going to use? c) What are you going to measure? What are the dependent and independent variables? d) Perform the experiment, collect the data, and decide how you can best represent it both graphically and mathematically. e) Represent the data and decide how you will represent the uncertainties in the measurements on your graph. f) What pattern do you notice? How can you express the pattern mathematically? Examine the units of the slope on the graph. What information do the units give you about the spring? Did You Know? A spring constant is a quantity that determines the force that the spring must exert on an object in order to stretch it by 1 m from its original length. The unit of the spring constant, k, is 1 N/m. g) What is the spring constant of your spring? How certain are you in your answer? h) Finish your lab report. Make sure that data are easy to understand and the experiments is described clearly enough for anyone to repeat it and obtain similar results. 8.2 Reason A 0.5 kg object attached to a 0.5 m spring stretches it by 0.1 m. Draw a force diagram for this situation and determine the spring constant of the spring Design an Experiment Attach a 100 g object to a spring scale and lift the scale slowly. Notice the reading on the scale. Can you get it to read 1 N when the object is moving? Can you get it to read more? Can you get it to read less? How were you able to achieve this? 44 PUM Dynamics II Lesson 8: Design an Experiment

45 8.4 Test an Idea Ritesh says: If you hang an object on a spring scale, the scale always reads how much force Earth exerts on the object. Erin says: The scale always reads the unbalanced force exerted on the object. a) Think of what you can do to test their opinions. Describe the experiment. b) Make a prediction based on each of the hypotheses c) Perform the experiment or watch the video and record the outcome: d) Make a judgment about each person s hypothesis. What can you say the scale measures? Did You Know? Elastic force: An object that stretches or compresses like a spring exerts an elastic force on some other object that is causing it to stretch or compress. The elastic force exerted by the spring on that object points in the direction opposite to the stretch (or compression). The magnitude of the force is directly proportional to the distance x that the elastic object (spring) is stretched or compressed from its equilibrium position (at x = 0): F S on O = k x The force constant k in units N/m is a measure of the stiffness of the spring (the ratio of the force in N needed to stretch the spring 1 m). 8.5 Reflect Homework Make a list of things you learned from the lab activity above with the springs. How did you learn each? 8.6 Reason Calculate the relative uncertainties of your scale measurements. When was your knowledge about the force more accurate? Need Some Help? Recall from kinematics that when you measure a quantity and the measurement is small, the uncertainty is a big percentage of the measurement. When the measurement is large, the uncertainty is a smaller percentage of the measurement. The relative uncertainty is the fraction of the measurement due to the uncertainty. PUM Dynamics II Lesson 8: Design an Experiment 45

46 For example, when the measurement is 2 N and the uncertainty is 0.5 N, the relative uncertainty becomes 0.5/2 = 0.4 or 40%. When the measurement is 15 N and the uncertainty is 0.5 N, the relative uncertainty is 0.5/15 = 0.3 or 30%. Therefore for the accuracy of measurement with a particular instrument, we want to measure quantities that are large. 8.7 Reason What is the uncertainty in your parents car speedometer? When is the measurement of the speed more accurate: driving in the city or on a highway? Explain. 8.8 Reason If you are to stand on a scale in a very powerful elevator in a very tall building, what might happen to the scale reading as the elevator takes you from the 1 st floor to the 42 nd floor? Use force diagrams, motion diagrams, and Newton s Second Law to explain your answer. 8.9 Regular problem You hang a 5-pound bag of groceries on a spring scale that has a visible spring. The scale stretches from being 20 cm long to being 26 cm long. What is the spring constant? (Hint: make sure you use SI units for all quantities. If you do not know how to convert pounds to Newton s use the textbook or the internet for help.) Regular problem You probably know from experience that door springs are difficult to stretch. (a) What maximum 4 force do you need to exert on a relaxed spring with a 1.2! 10 -N/m spring constant to stretch it 6.0 cm from its equilibrium position? (b) What average force do you need to exert? Reason Why is it harder to stretch the spring that is already stretched than an unstretched spring? Explain your answer using force diagrams, graphs, etc Reason A person stands on a bathroom scale in a spaceship that is drifting at constant velocity in outer space far from the Solar system. What is the reading of the bathroom scale compared to its reading when on Earth? (a) The same as on Earth since the mass of the person did not change. (b) Zero since the person is not pressing on the scale at all. (c) About half the reading on Earth Reason A man stands on a scale and holds a heavy object in his hands. What happens to the scale reading if the man quickly lifts the object upward and then stops lifting it? (a) The reading increases, returns briefly to the reading when standing stationary, then decreases. (b) The reading decreases, returns briefly to the reading when standing stationary, then increases. (c) Nothing since the mass of the person with the book remains the same. Thus the reading does not change. 46 PUM Dynamics II Lesson 8: Design an Experiment

47 PUM Dynamics II Lesson 8: Design an Experiment 47

48 Lesson 9: Applying Newton s Second Law Problem-Solving Strategy for Dynamics Problems Sketch and Translate: Read the text of the problem at least three times to make sure you understand what the problem is saying. Visualize the problem, make sure you see what is happening. Sketch the situation described in the problem; include all known information. Choose a system object and make a list of objects that interact with the system. Indicate the direction of acceleration, if you know it. Simplify and Diagram: Consider the system as a particle. Decide if you can ignore any interactions of the environment with the system object. Draw a force diagram for the system. Label the forces with two subscripts. Make sure the diagram is consistent with the acceleration of the system object (if known). Include perpendicular x- and y-coordinate axes. Draw a motion diagram and make sure the force and the motion diagrams match. Represent Mathematically: Apply Newton s Second Law in component form to the situation you represented in the force diagram. Add kinematics equations if necessary. Solve and Evaluate: Solve the equations for an unknown quantity and evaluate the results to see if they are reasonable (the magnitude and the sign of the answer, its units, and how the result changes if one of the quantities becomes zero it the result what your common sense tells you? Make sure you go back to the force and motion diagrams to make sure your answer is consistent with both. Need Some Help? Here is an example that applies the strategy shown above: A 5-kg object (Earth exerts a 50 N force on it) is lifted by a cable that exerts a 70 N force on it. Calculate the acceleration of the object. Translate: The object is our system; Earth and the cable interact with the object. The acceleration is up. y F C on O F E on O Simplify and Diagram: In this case there are two forces exerted on the object one exerted by the cable and one exerted by Earth. We choose the positive axis to be down. Represent Mathematically: Newton s second law in component form: a O y = F C on O y + F E on O y m O Solve and Evaluate: The component of the force exerted by the cable is negative as the force points in the negative direction, the component of the force exerted by Earth is positive. Thus (!70 N) + 50 N a O y = = (!4 N/kg) = (-4 m/s 2 ). The negative sign of acceleration means that it 5 kg is pointed upward the elevator is accelerating in the upward direct (not necessarily moving in 48 PUM Dynamics II Lesson 9: Applying Newton s Second Law

49 the upward direction, it can be slowing down). 9.1 Represent and Reason Description of the object of interest is underlined 1) A 2.2 kg bucket of clams sits at rest on a desk. A Sketch the situation. Circle the object of interest. Draw the direction of the acceleration, if known. B Translate the givens into physical quantities. Given in description: a = 0 (sits ) m = 2.2kg C Draw a force diagram for the object of interest. F table on bucket Bucket D Can you evaluate any of the forces in the force diagram? Which are negative and which are positive? F E on B y = -mg=-22 N E Write Newton s Second Law in component form. Fill in anything you know and solve for anything you do not know. a y = F Earth on Bucket y + F Table on Bucket y m 0 = (!22N) + F Table on Bucket 2.2kg F Earth on 2) A 5kg bucket of clams hangs motionless from a spring that stretches 40 cm. bucket 3) A man pulls a 40kg refrigerator up an elevator shaft with a rope at a constant speed. Come up with your own for an object in equilibrium PUM Dynamics II Lesson 9: Applying Newton s Second Law 49

50 with 3 or more other objects interacting with it. 9.2 Regular Problem In a grocery store, you push a 14.5 kg shopping cart. It is initially rolling at a constant speed of 2.0 m/s. You push on it in the direction opposite to its motion exerting a force of 12.0 N. a) Draw a force diagram and a motion diagram for the cart when you start pushing in the direction opposite to its motion. b) Write Newton s second law in component form for the process. c) Assuming you push the cart exerting constant force for a while, how far will it travel in 3 seconds? (Ignore friction for all parts of this problem.) Use the problem-solving strategy steps illustrated above. 9.3 Regular problem An astronaut, while pushing a beam into place on the International Space Station, exerted a 150-N force on the beam. The beam accelerates at 0.15 m/s 2. Determine the mass of the beam. 9.4 Design an Experiment a) Describe an experiment that you can design to test Newton s Second Law. Decide on what aspect of the law you can test. b) Brainstorm different possibilities and decide which ones are the best. Think of the criteria for choosing the best experiment. d) What equipment will you need? What data will you collect? a) What is your prediction about the relationships in the data based on Newton s Second Law? b) What is the difference between Newton s Second Law and the predictions? 9.5 Reason An elevator is pulled upward so it moves with increasing upward speed the force exerted by the cable on the elevator is constant and greater than the downward gravitational force exerted by Earth. When the elevator is moving up medium fast, the force exerted by the cable on the elevator changes abruptly to just balance the downward gravitational force of Earth the sum of the forces that the cable and Earth exert on the elevator is now zero. Now what happens to the elevator? Explain. Represent your answer with a position and velocity-versus-time graphs. What assumptions did you make? 50 PUM Dynamics II Lesson 9: Applying Newton s Second Law

51 Homework 9.6 Practice Fill in the table below. The system object is underlined. Description of the object of interest is underlined 1) A 72 kg crate on a freight elevator accelerates upwards at a rate of 0.2 m/s 2 while moving down. A Sketch the situation. Circle the object of interest. Draw a motion diagram and the direction of the acceleration, if known B Translate the givens into physical quantities. C Draw a force diagram for the object of interest. D Can you evaluate any of the force components in the force diagram? Which are negative and which are positive? What if you changed the direction of the axes? E Write Newton s Second Law in component form. What can you determine using the information in the problem? 2) A kg crate on a freight elevator accelerates downwards at a rate of 0.4 m/s 2 while moving up. 3) A physics teacher of mass m is holding onto a rope attached to a hot air balloon and is accelerating upwards at a m/s 2. PUM Dynamics II Lesson 9: Applying Newton s Second Law 51

52 10.1 Observe and Explain Lesson 10: Newton s Third Law: Qualitative Student A and Student B both wear rollerblades or are on chairs with wheels. Student B pushes Student A abruptly. If you are not observing the real-life experiment, watch the video experiments: and a) Observe what happens during the instant of the push to both students, and describe your observations in words. b) Draw motion diagrams and force diagrams for each student for the instant when B pushes A. Use the diagrams to explain the observations. c) Why didn t Student B s velocity change in activities 6.1 and 6.3? Explain using force diagrams Test your Idea Use Newton s Second Law and the explanation that you devised in the previous activity to predict what will happen if Students A and B, both on rollerblades, start throwing a heavy medicine ball back and forth to each other. If you have the equipment, perform the experiment and then check whether your prediction matches the outcome. You can also watch the video of the experiment at: a) What was your hypothesis? b) Where did the hypothesis come from? c) What was your prediction? d) Did the outcome of the experiment prove the hypothesis to be right or fail to disprove it? Apply Examine a fan cart on your desk. Turn on the fan and observe its motion. Draw a motion diagram for the cart and decide what object exerts the force to accelerate the cart. Use the sail on your desk to test your answer. Record all your observations and the testing experiment very carefully. 52 PUM Dynamics II Lesson 10: Newton s Third Law: Qualitative

53 Homework 10.4 Represent and Reason Fill in the empty spaces and draw pictures representing the situations. Place force arrows on the pictures; remember to think about the lengths of the arrows. a) When Earth exerts a force on the book, the book exerts a force on b) When a table exerts a force on the book, the book exerts a force on c) When a tennis racket exerts a force on the ball, the ball exerts a force on d) When car tires push back on Earth s surface, Earth s surface 10.5 Relate List 5 everyday experiences that support the idea that when object B exerts a force on object A, object A will simultaneously exert a force on object B. Discuss whether you can always observe the effects of these forces that the interacting objects exert on each other. List possible reasons. PUM Dynamics II Lesson 10: Newton s Third Law: Qualitative 53

54 Lesson 11: Newton s Third Law: Quantitative 11.1 Observe and Find a Pattern The goal of this experiment is to determine a mathematical relationship between the force that object A exerts on object B and the force that object B exerts on object A when they are interacting with each other. Available Equipment: Force probe sensors with hooks on the end and computers. Note on equipment: You will be using a new sensor for this experiment; it is called a force probe. A force probe is a sensor that sends a signal to the computer indicating the force exerted on its tip. The software interface helps you plot force as a function of time for two different force probes. a) Take one of the probes, connect it to the computer, and gently pull or push on it. Examine the graph on the screen and make sure it makes sense to you. Need Some Help? The force probe is an object. You exerted a force on this object, and this force was recorded as a function of time. b) Design and perform enough experiments to find a pattern in the readings of the two probes when they record forces that two interacting objects exert on each other. Here s An Idea! Place one force probe on the table and tap it gently with the second probe. The probes are very delicate, and if you use them to tap each other, you must do it lightly. Even a light tap will register on the graph. c) Write a short report, summarizing the experiments and your findings. For your report, be sure to include the following: 1) For each experiment, describe the setup in words and sketch the graphs that you see on the computer. 2) Find a pattern in the pairs of graphs representing the force-versus-time functions recorded by each probe during an interaction. 3) Formulate a tentative rule relating the force that object A exerts on object B to the force that the object B exerts on the object A. 4) Use the observational experiment rubric and data analysis rubric to write your report about this experiment. 54 PUM Dynamics II Lesson 11: Newton s Third Law: Quantitative

55 11.2 Test Your Hypothesis Available Equipment: Force probe sensors with hooks on the end, a track, carts, objects of different masses to put on the carts, cushions, elastic bands, and computers. a) State the rule you are going to test which you devised in b) Brainstorm a list of possible experiments whose outcome can be predicted with the help of the rule. Choose two experiments from your list. c) Briefly describe your chosen design. Include a labeled sketch. d) Use the tentative rule to make a prediction about the outcome of each experiment. e) Perform each experiment. Record the outcome. f) Does the outcome support the prediction? Explain. g) Based on the prediction and the outcome of the experiments, what is your judgment about the rule? h) Think of the assumptions that you used to make the predictions for the testing experiments. How could these assumptions have affected your judgment? i) Talk to your classmates in other lab groups and find out about their results. Are they consistent with yours? j) Use the testing experiment rubric to write your report Reason Use the rule that you devised and tested to decide who exerts a larger force for the following situations: 1. A mosquito on a car s windshield or a car s windshield when a mosquito smashes into it; 2. A reflex hammer (Taylor hammer) on your knee or your knee on the hammer when a doctor taps your knee with it. Reconcile your answers with your observations of these phenomena. Did You Know? Newton s Third Law of Motion: When two objects interact, object 1 exerts a force on object 2. Object 2 in turn exerts an equal-magnitude, oppositely directed force on object 1:! F 1 on 2 =!! F 2 on 1. Each force above is caused by one object and is exerted on another object. Since these two forces are exerted on two different objects, they cannot be added to find a net force. PUM Dynamics II Lesson 11: Newton s Third Law: Quantitative 55

56 11.4 Reason A horse is pulling a carriage. According to Newton s third law the force exerted by the horse on the carriage is the same in magnitude but points in the opposite direction of the force exerted by the carriage on the horse. How are the horse and carriage able to move forward? (a) The horse is stronger. (b) The total force exerted on the carriage is the sum of the forces exerted by the horse and the ground s surface in the horizontal direction. (c) The net force on the horse is the sum of the ground s static friction force on its hooves and the force exerted by the carriage. (d) Both b and c are correct Reason Homework a) A book sits on the tabletop. What is the Newton s Third Law pair for the force that Earth exerts on the book? b) If Earth exerts a 5 N force on the book, what is the force that the book exerts on Earth? c) What is the acceleration of the book if Earth is the only object exerting a force on it? d) Why does the book fall onto Earth but Earth does not fall onto the book? The mass of Earth is 6.00 x kg. Need Some Help? Use Earth s mass to calculate Earth s acceleration. What does this tell you about the motion of Earth? e) The Sun s mass is 2.00 x kg. It pulls on Earth, exerting a force of about N. What is the force that Earth exerts on the Sun? 11.6 Reason Two students sit on office chairs with wheels. Student A pushes student B away from him. Student B does nothing. Does student B exert the force on A? How do you know? 11.7 Reason a) You hit a stationary puck with a hockey stick. The stick exerts a 100 N horizontal force on the puck. What is the force exerted by the puck on the stick? How do you know? b) A truck rear ends a small sports car that is moving in the same direction as the truck. The collision makes the truck slow down and the sports car is propelled forward. What object 56 PUM Dynamics II Lesson 11: Newton s Third Law: Quantitative

57 exerts a larger force on the other object: the truck on the car or the car on the truck. Explain how your answer reconciles with Newton s third law and with the fact that the sports car is damaged more than the truck. c) Earth pulls on an apple exerting a 1.0 N force on it. What is the force that the apple exerts on Earth? Why does the apple fall towards Earth but Earth does not move towards the apple? d) The tree branch exerts a 1.0 N force holding the apple. What is the force that the apple exerts on the tree branch? 11.8 Reason Use Newton s third law to predict what will happen if you try to open a door wearing rollerblades. Draw a force diagram for yourself to help make the prediction Represent and Reason Your friend says that if Newton s third law is correct, no object would ever start moving. Here is his argument: You pull a sled exerting a 50 N force on it. According to Newton s third law the sled exerts the force of 50 N on you in the opposite direction. The total force is zero, thus the sled should never start moving. But it does. Thus Newton s third law is wrong. What is your opinion about this answer? How can you convince your friend of your opinion? Reason The Moon orbits Earth because Earth exerts a force on it. The Moon, therefore, has to exert a force on Earth. What is the visible result of this force? Regular Problem A person of mass m is standing on the floor of an elevator that starts from the first floor and reaches the 21 st floor. a) Make two kinematics models for the motion of the elevator. Describe them in words. What is the same about the two models? What is different? b) Now describe the same models with motions diagrams, with position, velocity and acceleration versus time graphs, and with algebraic functions. c) Choose one of the models and draw force diagrams for the person for three different parts of the trip. d) Write a mathematical expression that will help you determine the magnitude of the force that the person exerts on the floor when the acceleration of the elevator is upward and again when the acceleration is downward. What is a reasonable magnitude for the elevator s acceleration? PUM Dynamics II Lesson 11: Newton s Third Law: Quantitative 57

58 e) Who is pushing harder the elevator s floor on the person or the person on the floor? Earth on the person or the floor on the person? Reason Your young sister jumps off a chair holding a 0.50-kg container of ice cream. What is the force with which the container pushes on her hand when she is in the air? Explain. (a) Almost 5 N. (b) The same as she exerts on the container. (c) 0 N. (d) Both b and c are correct Reason A spaceship moves in outer space. What happens to its motion if there are no external forces exerted on it? If there is a constant force exerted on it in the direction of its motion? If something exerts a force opposite its motion? (a) It just keeps moving; it speeds up with constant acceleration; it slows down with constant acceleration. (b) It slows down; it moves with constant velocity; it slows down. (c) It slows down; moves with constant velocity; it stops instantly Reason You stand on a bathroom scale in a moving elevator. What happens to the scale reading if all of a sudden the cable holding the elevator breaks? (a) The reading will increase. (b) The reading will not change. (c) The reading will decrease a little. (d) The reading will drop to 0 instantly Estimate Imagine that all people on Earth decided to jump simultaneously. What will happen to Earth? Reflect: What did you learn about Newton s third law so far? How is it different from the second law? Give 3 examples of Newton s third law from everyday life. 58 PUM Dynamics II Lesson 11: Newton s Third Law: Quantitative

59 12.1 Represent and Reason Lesson 12: Two-Body Problems 1 2 You push two crates that have different masses on a smooth surface. Fill in the table that follows for the two situations. Use the rule that relates the forces that two objects exert on each other while interacting (devised in lesson 11). Situation 1: You push crate 1, which pushes against the smaller crate 2. Situation 2: You now reverse the positions of the crates and push crate 2, which pushes on larger crate 1. (a) You push crate 1. Show the force that 2 exerts on 1. Show the force that 1 exerts on 2. 1 (b) You push crate 2. Show the force that 1 exerts on 2. Show the force that 2 exerts on c) Based on the diagrams in (a) and (b), should it be easier to push the crates in one situation than the other? Explain. d) Is your answer to (c) consistent with Newton s Third Law? e) Calculate the sum of the forces from part (a), the force crate 1 exerts on crate 2 and the force crate 2 exerts on crate 1. How does this compare to the sum of these forces for part (b)? What does this imply about the magnitude of the force of one crate on the other and vice versa? 12.2 Represent and Reason This time, instead of pushing two crates, you connect them with a rope and attach another rope to crate 1. You pull this second rope horizontally, exerting a force F you on crate1. The masses of the crates are m 1 and m 2. a) Find the acceleration of the crates. Decide what assumptions you need to make to solve this problem. Follow the problem-solving strategy. b) Find the force that the rope connecting the two crates exerts on crate 2. Again, be sure to follow the problem-solving strategy. PUM Dynamics II Lesson 12: Two- Body Problems 59

60 12.3. Test your ideas a) Examine the experimental setup in the video experiment at: b) Before watching the video predict how much each object in the video will move in 1 second. (Based on Newton s 2 nd Law) Your prediction should contain an uncertainty value with it. Make sure you follow the problem-solving strategy closely and list all of your assumptions. c) How might the result be different from your prediction if the assumptions are not valid? 12.3 Continued Homework Perform the experiment (watch the video) and record the outcome. Clearly describe of how you found whether the prediction matched the outcome of the experiment. What can you say about Newton s Laws based on this experiment? 12.4 Represent and reason A person pulls on a rope, which in turn pulls a crate across a horizontal smooth surface, as shown below. Three motion diagrams for the crate are shown below (with v arrows only). In the table that follows, construct a force diagram for the crate. Check the consistency of your motion diagrams and force diagrams. (a) (b) (c) (a) (b) (c) 60 PUM Dynamics II Lesson 12: Two- Body Problems

61 12.5 Represent and Reason Examine the system on the right. Jon says that the force the rope exerts on the cart is always equal to m 1 g. Why would Jon have such an opinion? Do you agree or disagree? Explain your answer. m Reason You use the setup described in activity You first hold the cart with your hand so the system is at rest. Then you abruptly push the cart to the left and let it go. Describe the motion of the cart in words after you let it go. Explain the motion using force diagrams for both the cart and the hanging object. Then sketch the acceleration versus time graph for the cart Regular problem You pull a rope oriented horizontally. The other end of the rope is attached to the front of first of two wagons that have the same 20-kg mass. The rope exerts a force of magnitude T 1 on the first wagon. The wagons are connected together by a second horizontal rope that exerts a force of magnitude T 2 on the second wagon. Determine the magnitudes of T 1 and T 2 if the acceleration of the wagons is 2.0 m/s Regular problem Rope 1 pulls horizontally, exerting a force of 45 N on an 18-kg wagon attached by a second horizontal rope to a second 12-kg wagon. Make a list of physical quantities you can determine using this information, and solve for three of them (one has to be a kinematics quantity) Regular problem A person holds a 250-g block that is connected to a 300-g block with a string going over a light pulley with no friction in the bearing (similar to a system in the video experiment). After the person releases the 200-g block, it starts moving upward and the heavier block descends. (a) Choose each block as a system and draw force and motion diagrams for them. (b) What is the acceleration of each block? (c) What is the force that the string exerts on each block? (d) How long will it take each block to traverse 1.0 m? Reason In the pulley-2 block system in the problem above, a 300-g block was replaced with the block of the unknown mass. The first block is hanging 0.8 m above the second block. The system is at rest. Nobody is holding any of the block. What is the mass of the unknown block? PUM Dynamics II Lesson 12: Two- Body Problems 61

62 13.1 Design an Experiment Lesson 13: Design an Experiment You have helium balloons and air balloons at your disposal. You can attach different objects to the balloons. a) Decide what questions you can investigate using one or both balloons. Brainstorm a list of questions. b) Decide which one you and your group can perform with the equipment available. c) Decide what you need to know more about in order to pursue the question. d) Make a sketch of the experimental design and list the physical quantities that you plan to measure. e) Are you going to conduct an observational experiment or a testing experiment? If it is a testing experiment, make sure that you outline the hypothesis being tested and the prediction of the outcome of the experiment based on the hypothesis. If you cannot make a prediction, you should consider whether you are actually doing a testing experiment. f) Perform the experiment, record your data, represent the data, and perform necessary calculations. Do not forget about the uncertainties when you are presenting the final results. g) Write a report about your investigation. Include sketches of the equipment, motion diagrams and force diagrams if necessary, tables of data, graphs, and your findings, including the uncertainties. Make sure that the report is clear enough that a person who has not seen the experiment can repeat it. h) Use the rubrics to improve your report. 62 PUM Dynamics II Lesson 13: Design an Experiment

63 Ability Absent An attempt Needs some improvement Acceptable Is able to formulate the question to be investigated The question to be investigated is not mentioned. The question is posed but it is not clear. The question is posed but it involves more than one variable. The question is posed and it involves only one variable. Is able to design an experiment to answer the question The experiment does not answer the question. The experiment is related to the question but will not help answer it. The experiment investigates the question but might not produce the data to find a pattern. The experiment investigates the question and might produce the data to find a pattern. Is able to decide what is to be measured and identify independent and dependent variables It is not clear what will be measured. It is clear what will be measured, but independent and dependent variables are not identified. It is clear what will be measured, and independent and dependent variables are identified but the choice is not explained. It is clear what will be measured and independent and dependent variables are identified and the choice is explained. Is able to use available equipment to make measurements At least one of the chosen measurements cannot be made with the available equipment. All chosen measurements can be made, but no details are given about how it is done. All chosen measurements can be made, but the details of how it is done are vague or incomplete. All chosen measurements can be made and all details of how it is done are clearly provided. Is able to describe what is observed in words, pictures and diagrams. There is no description of what was observed. A description is mentioned but it is incomplete. No picture is present. A description exists, but it is mixed up with explanations or other elements of the experiment. A labeled picture is present. Clearly describes what happens in the experiments both verbally and by means of a labeled picture. Is able to construct a mathematical (if applicable) relationship that represents a trend in data No attempt is made to construct a relationship that represents a trend in the data. An attempt is made, but the relationship does not represent the trend. The relationship represents the trend but no analysis of how well it agrees with the data is included (if applicable), or some features of the relationship are missing. The relationship represents the trend accurately and completely and an analysis of how well it agrees with the data is included (if applicable). PUM Dynamics II 63

64 Homework 13.2 Finish Your Report In the next class, you will need to evaluate one of your classmate s report and your classmate will evaluate yours. Make a list of necessary elements that you will look for in your classmate s report Reason (adapted from FMCE) Questions below refer to a toy car which can move to the right or left on a horizontal surface along a straight line (the + position axis). The positive direction is to the right. 0 + Different motions of the car are described below. Choose the letter (A to G) of the accelerationtime graph which corresponds to the motion of the car described in each statement. You may use a choice more than once or not at all. If you think that none is correct, answer choice J The car moves toward the right (away from the origin), speeding up at a constant acceleration. The car moves toward the right, slowing down at a constant acceleration. The car moves toward the left (toward the origin) at a constant velocity. The car moves toward the left, speeding up at a constant acceleration. 64 PUM Dynamics II

65 5. The car moves toward the right at a constant velocity. Questions below refer to a coin which is tossed straight up. After it is released it moves upward, reaches its highest point and falls back down again. Use one of the following choices (A through G) to indicate the acceleration of the coin during each of the stages of the coin's motion described below. Take up to be the positive direction. Answer choice J if you think that none is correct. A. The acceleration is in the negative direction and constant. B. The acceleration is in the negative direction and increasing C. The acceleration is in the negative direction and decreasing D. The acceleration is zero. E. The acceleration is in the positive direction and constant. F. The acceleration is in the positive direction and increasing G. The acceleration is in the positive direction and decreasing The coin is moving upward after it is released. The coin is at its highest point. The coin is moving downward. Reflect: Among the multiple choice questions find those that relate to forces changing in time. So far we mostly studied constant forces. What happens to Newton s second law if the sum of the forces exerted on the object changes with time? PUM Dynamics II 65

66 Lesson 14: Friction 14.1 Observe and Find a Pattern Perform the following experiment: Rest a wooden block (or some other object, like your shoe) on a table. Attach a large spring scale to a string attached to the front of the block. Pull the scale harder and harder. Notice what happens to the scale reading while the block does not move. Notice the reading right before the block starts moving and right after. Keep the block moving but not accelerating. a) Fill in the table that follows by constructing a force diagram for the block (the system) for these five situations. The block sits on the table with no scale pulling it. The spring pulls on the block, which does not start moving. The spring pulls harder, but the block still does not move. The spring pulls on the block, and the block is just about to start moving. The spring pulls the block at a slow, constant velocity. b) Describe in words how the magnitude of the force that the table s surface exerts on the block varies with the force exerted by the spring pulling on the block. c) Compare the magnitude of the force just before the block starts moving to the magnitude when it is moving at a constant velocity. What do you observe? d) What object is exerting this friction force for the scenarios given above? e) Summarize your findings for the friction force exerted on an object at rest and on the same object moving at a constant velocity. Did You Know? The friction force is a resistive force exerted by the surface on an object. There are two kinds of friction forces you observed in the experiments above. The static friction force is variable. As you saw, once the maximum static friction force is overcome, the object will start to move. The kinetic friction force is the resistive force exerted on a moving object. 66 PUM Dynamics II Lesson 14: Friction

67 14.2 Observe and Find a Pattern Instead of the block in the previous activity, you have rectangular blocks with different surface areas and different types of surfaces on which the block slides horizontally. The force that the string exerts on the block (as measured by the spring scale reading) when the block just starts to slide is recorded in the table that follows. This force is equal in magnitude to the maximum static friction force (as we discovered in the previous activity). Examine the data in the table that follows. Mass of the block Surface area Quality of Maximum surfaces static friction force 1 kg 0.1 m 2 Medium smooth 3.1 N 1 kg 0.2 m 2 Medium smooth 3.0 N 1 kg 0.3 m 2 Medium smooth 3.1 N 1 kg 0.1 m 2 A little rougher 4.2 N 1 kg 0.1 m 2 Even rougher 5.1 N 1 kg 0.1 m 2 Roughest 7.0 N Now decide how the maximum static friction force that the surface exerts on the block depends on the surface area of the block and on the roughness of the two surfaces Observe and Find a Pattern A spring scale pulls a 1 kg block over a medium smooth surface. The reading of the scale can be used to determine the magnitude of the maximum static friction force in this instance, the force when the block starts to slide. In some experiments, a compressible spring also pushes vertically down on the block (see the second block). Use the data in the table to draw a graph of the maximum static friction force versus the normal force the surface exerts on the block. Extra downward force exerted on the 1-kg block Normal force exerted by the board on the block 0 N 10 N 3 N 5 N 15 N 4.5 N 10 N 20 N 6 N 20 N 30 N 9 N Maximum static friction force PUM Dynamics II Lesson 14: Friction 67

68 b) Express mathematically a relationship between the normal force and the maximum static friction force Test your idea Design an experiment to test that the magnitude of the maximum static friction force is equal to f s surface on object = µn surface on object. Describe what you will do, what data you will collect and what the predicted outcome should be if the expression is correct. Then perform the experiment and make a judgment about the hypothesis Reason a) Take a textbook and drag it with your pinky finger. Repeat but this time have your neighbor push down lightly on the book. Repeat 3 more times with your neighbor pushing down successively harder. Draw a force diagram for each case. What can you say about the maximum static friction force? b) Consider the previous activity. Why would we consider the normal force exerted on the object rather than the force of Earth exerted on the object? c) A person is holding a book against a vertical wall, pushing on it horizontally. The book is at rest. Draw a force diagram for the book. Check if all forces balance. Which force prevents the book from falling down? Why, if you do not push on the book hard enough, does the book start falling? Did You Know? When two objects touch each other, they exert a normal force on each other. In physics it is customary to break the force that each exerts on the other into two forces a force perpendicular to the surfaces and the force parallel to the surfaces. Normal force: This is the perpendicular component of the total force that one object exerts on the other object. it points perpendicular to the surface of contact. Often one symbol N is used to denote this force (do not confuse with the Newton, N). There is no equation for calculating the normal force. Its magnitude must be determined for each situation by some other method. Static friction force: The parallel component of the force that two objects exert on each other is called a friction force. The friction force of the one object on the other object points parallel to the surfaces of contact. If the objects are not moving with respect to each other, the friction force that they exert on each other is static. The static friction force between two surfaces opposes the tendency of one surface to move across the other and provides flexible resistance (as much as is needed) to prevent motion up to some maximum value. This maximum static friction force depends on the relative roughness of the surfaces (on the coefficient of static friction µ s between the surfaces) and on the magnitude of the normal force N between the surfaces. The magnitude of the static friction force is always less than or equal to the product of these two quantities: F s surface on object = µ s N 68 PUM Dynamics II Lesson 14: Friction

69 Kinetic friction force: The kinetic friction force between two surfaces is exerted parallel to the surfaces and opposes the motion of one surface relative to the other surface. The kinetic friction force depends on the relative roughness of the surfaces (on the coefficient of kinetic friction µ k ) and on the magnitude of the normal force N between the surfaces: F k surface on object = µ k N 14.6 Design an Experiment Use Phet simulations apply your understanding of friction forces. Go to Forces in 1 Dimension. This time do not turn off Friction. Choose three objects and design experiments that will allow you to determine what coefficient of static and kinetic friction the programmers who write the simulation used. Write a report in which you will describe the experiments and explain how you used the results to determine the coefficients. Do your results make sense? Did the programmers use the knowledge of Newton s laws and friction forces to write the program? 14.7 Observe and Represent Homework Imagine that you could watch yourself walk in slow motion. Analyze your steps in terms of the force of friction that the floor exerts on your foot and in terms of Newton s Second and Third Laws. In order to do this, break the step into two parts: (1) when you put the foot down to finish up the previous step, and (2) when you are pushing off the floor to start a new step. Draw force diagrams to represent your reasoning Evaluate Jamie says that the force of friction is something that we should reduce in order to make the cars go faster. What friction force could she mean? Do you agree or disagree with her opinion? If you agree, how would you argue for it? If you disagree, how would you argue against it? 14.9 Represent and Reason Some students are trying to move a heavy desk across the room. Diana pushes it across the floor at the same time that Omar and Jeff pull on it. Omar pulls on the desk, exerting a (-150) N force, and Jeff pulls exerting a (-125) N force. There is also a (-200) N friction force exerted by the floor on the desk. The net force exerted on the desk is 27 N. a) Make a sketch of the situation. b) Draw a force diagram for the desk. Draw a motion diagram. c) Write an algebraic statement that describes the force diagram you drew. d) How hard is Diana pushing? e) Is the desk moving with a constant velocity or is it speeding up? How do you know? PUM Dynamics II Lesson 14: Friction 69

70 f) What would happen if, after a few seconds, the boys stopped pulling? Regular problem According to Auto Week magazine, a Chevrolet Blazer traveling at 60 mph (97 km/h) can stop in 48 m on a level road. Determine the coefficient of friction between the tires and the road. Do you think this is kinetic or static friction? Explain Regular Problem A 50-kg box rests on the floor. The coefficients of static and kinetic friction between the bottom of the box and the floor are 0.70 and 0.50, respectively. (a) What is the minimum force a person needs to exert on it to start the box sliding? (b) After the box starts sliding, the person continues to push it exerting the same force. What is the acceleration of the box? Reason A car is moving to the right increasingly faster. A leaf is on the back vertical side of the car and does not slide down. Explain how this can be Regular problem The Ford P2000 fuel cell car has a mass of 1520-kg. While it is traveling at 20 m/s, the driver applies the brakes to stop the car on a wet surface with a 0.40 coefficient of friction. (a) How far does the car travel before stopping? (b) If a different car with the mass 1.5 times as much as the mass of the Ford P2000 is on the road traveling at the same speed and the coefficient of friction between the road and the tires is the same, what will its stopping distance be? Does the answer make sense to you? Reason Compare the ease of pulling a lawn mower and pushing it. In particular, in which case is the friction force that the grass exerts on the mower greater? (a) The same. (b) Pulling is easier. (c) Pushing is easier. (d) Not enough information to answer. 70 PUM Dynamics II Lesson 14: Friction

71 PUM Dynamics II Lesson 14: Friction 71

72 Lesson 15: Putting It All Together 15.1 Reason You stand on a bathroom scale that reads 712 N (160 lb) (You can use your own weight in newtons in this problem if you wish). You place the scale on an elevator floor and stand on the scale. a) What does it read at the beginning of the ride when the elevator accelerates up at 2.0 m/s 2? b) What does the scale read when the elevator continues to move up at a constant speed of 4.0 m/s? c) What does it read at the end of the ride when the elevator slows down at a rate of magnitude 2.0 m/s 2? 15.2 Reason Describe in words a problem for which the following equation is a solution and draw the force diagram that is consistent with the equation (specify the direction of the axis): (30 kg)( 1.0 m/s 2 ) = 100 N f S on O 15.3 Reason Explain the whiplash phenomenon from the point of view of an observer on the ground and an observer in the car Reason You push a bowling ball along a bowling alley. Draw force diagrams for the ball: (a) just before you let it go; (b) when the ball is rolling along the alley; (c) as the ball is hitting a pin. (d) For each force exerted on the ball in parts (a)-(c), draw to the side the Newton s third law pair force and indicate the object on which these third law forces are exerted Reason James Steward, 2002 Motocross/Supercross Rookie of the Year, is leading the race when he runs out of gas near the finish line. He is moving at 16 m/s when he enters a section of the course covered with sand where the effective coefficient of friction is Will he be able to coast through this 15-m long section to the finish line at the end? If yes, what is his speed at the finish line? What assumptions did you have to make to solve this problem? 15.6 Reason According to Auto Week magazine, a Chevrolet Blazer traveling at 60 mph (97 km/h) can stop in 48 m on a level road. Determine the coefficient of friction between the tires and the road. Do you think this is coefficient of kinetic or static friction? Explain Reason A 50-kg box rests on the floor. The coefficients of static and kinetic friction between the bottom of the box and the floor are 0.70 and 0.50, respectively. (a) What is the minimum force a person needs to exert on it to start the box sliding? (b) After the box starts sliding, the person continues 72 PUM Dynamics II Lesson 15: Putting It All Together

73 to push it exerting the same force. What will happen to the box? Answer this question quantitatively Reason A wagon is moving to the right faster and faster. A book is pressed against the back vertical side of the wagon and does not slide down. Explain how this can be Regular Problem A Navy Seal of mass 80 kg parachuted into an enemy harbor. At one point while falling, the resistive force of air exerted on him was 520 N. What can you determine about his motion? Regular Problem Karen Nyberg, 60-kg astronaut, sits on a bathroom scale in a rocket that is taking off vertically with an acceleration of 3 g. What does the scale read? Regular Problem A 0.10-kg apple falls off a tree branch that is 2.0 m above the grass below. The apple sinks m into the grass while stopping. Determine the force that the grass exerts on the apple while stopping it. Indicate any assumptions you made. [This is a twopart problem.] Regular Problem An 80-kg fireman slides 5.0 m down a fire pole. He holds the pole and exerts a 500-N steady force that resists his downward slide so that he is not going too fast when he reaches the bottom. He stops in 0.40 m by bending his knees. What can you determine using this information? [This is a two-part problem.] Reason Earth exerts a 1.0-N gravitational force on a tennis ball as it falls toward the ground. (a) What force does the ball exert on Earth? (b) Compare their accelerations due to these forces. The mass of the ball is about 100 g and the mass of Earth is about 6! kg Reason You push a bowling ball along a bowling alley. Draw force diagrams for the ball: (a) just before you let it go; (b) when the ball is rolling along the alley (for two clock readings); (c) as the ball is hitting a bowling pin. (d) For each force exerted on the ball in parts (a) (c), draw to the side the Newton s third law force, and indicate the object on which these third law forces are exerted. PUM Dynamics II Lesson 15: Putting It All Together 73

74 Lesson 16: Components Reason The x- and y-components of several unknown forces are listed next ( F x, F y ). For each force, draw on an x, y coordinate system the components of the force vectors and determine the magnitude and direction of each force: (a) (+100 N, 100 N), (b) ( 300 N, 400 N), and (c) ( 400 N, +300 N) Reason The x- and y-components of several unknown forces are listed next ( F x, F y ). For each force, draw on an x, y coordinate system the components of the force vectors, and determine the magnitude and direction of each force: (a) ( 200 N, +100 N), (b) (+300 N, +400 N), and (c) (+400 N, 300 N). Need Some Help? Scalar components of a vector When working with well-defined x - and y -axes, we do not y need to use the vector components of F!, but can instead provide F = 5 N the same information about the vector by specifying what are called the scalar components F x and F y of the force F!. For example, in N x the Figure on the right, we see that F! is 5 N and points 37 0 above the negative x -axis. The x vector component of force F! is 4 N long and points in the negative x direction. Thus, the scalar x component of F! is F x = 4 N. Similarly, the y vector component of F! is 3 N in the positive y direction. Thus, we can specify the y effect of F! as F y = +3 N. Thus, the x and y components F x = 4 N and F y = +3 N tell us everything we need to know about the force F!. Note that since the scalar components of a vector are scalars, they are not written with a vector symbol above them. Finding the magnitude and direction of a force from its scalar components The scalar components of a vector contain all the information needed to reconstruct the original vector. To do this, first place the x -vector component of the vector so that its tail is at the origin (Figure on the next page). The x -scalar component indicates the direction and magnitude of the x -vector component along the x -axis. Then place the y -vector component so it its tail is at the head of the x -vector component. We also know the direction and magnitude of 74 PUM Dynamics II /Lesson 16: Components

75 the y -vector component relative to the y -axis from the sign and magnitude of the y -scalar component. The force vector points from the tail of the x -vector component to the head of the y -vector component. Together, the original vector and its two vector components form a right triangle, and the lengths of the sides of the triangle are the magnitudes of the respective vectors. Since the magnitudes of the vector components are equal to the magnitudes of the scalar components, we can use the Pythagorean theorem to determine the magnitude F of the force (how strong the force is, in newtons): F F + F. (3.1) 2 2 = x y We can also use this triangle, along with some trigonometry to determine the angle! ( 90 0 or less) that the force makes relative to the positive or negative x -axis: Fy tan! =, F or! = tan 1 Fy F. Thus, if we know the components F x and F y, we can determine the magnitude and direction of the force F!. Finally, if we know the magnitude of F!, we can find its scalar components: F x = F cos!; F y = F sin!. x x 16.3 Reason Finding components of a vector a) Represent each vector below as the sum of two vectors, one that is parallel to the x-axis and one that is parallel to the y-axis. Assign some integer values to the length of the vector! a and the angle ϑ and find the length of the components in terms of your chosen values. Show your work. b) Now express the length of each component in terms of the length of the vector! a and the angle ϑ. Do not forget the sign! PUM Dynamics II /Lesson 16: Components 75

76 y x a θ y x θ a Represent and reason You are pulling a 20-kg crate on a rug exerting a 300-N force at a 30 0 angle with the horizontal. The maximum coefficient of static friction between the rug and the crate is 0.5. Represent the situation with a force diagram, motion diagram and mathematically. Make sure your vertical forces and force components balance Represent and reason Inclined planes Draw a force diagram and show the direction of the acceleration for an object sliding down an inclined plane tilted at an angle ϑ with the horizontal. 76 PUM Dynamics II /Lesson 16: Components

77 On the force diagram, consider the direction of the following forces exerted on the object described above. a) Consider the frictional force (which is parallel to the surface) exerted by the surface of the incline on the object and the normal force exerted by the incline on the object (which points perpendicular to the surface). How do they compare to each other? b) Consider the force exerted by Earth on the object. How does the direction of the force that Earth exerts on the object compare to those mentioned in part (a)? How does the magnitude of this force compare to the magnitude of the normal force? c) Which forces are perpendicular to each other? Which of the forces that are exerted on the object are not perpendicular? 16.6 Evaluate Examine the two diagrams below, which represent an object of mass m accelerating down an inclined plane. The incline has a rough surface. One diagram has an axis that is upright. The other diagram has an axis that is tilted. For each situation, draw a force diagram and write " Newton s Second Law in variable/component form a x =!F x % " # $ m & ' and a y =!F y % # $ m & ' for each situation. Which coordinate axis would you prefer to use for evaluating objects on inclines and why? Scenario 1: Upright Axis! a θ y x PUM Dynamics II /Lesson 16: Components 77

78 Scenario 2: Tilted Axis y x a θ Did You Know? Components of a force: The force that one object exerts on another has the same effect on that object as the perpendicular components F x and F y of the force. The values of the components are: F x = ± F cos! F y = ± F sin! where F is the magnitude of the force and q is the angle (90 o or less) between the positive or negative x-axis and the direction of the force. The component is positive if it projects in the positive direction of the x- or y-axis and negative if it projects in the negative direction of the axes. Newton s Second Law in component form: a x = F 1 on S x + F 2 on S x F n on S x m S a y = F 1 on S y + F 2 on S y F n on S y m S = F net x m S = F net y m S 16.7 Regular Problem You are trying to pull a sled with two children on it up a hill that makes a 20 angle with the horizontal. The combined mass of the sled and children is 80 kg. The coefficient of kinetic friction is 0.2. How hard should you pull parallel to the hill s surface if you: a) want the sled to move at a constant speed? b) want the sled to accelerate at 1 m/s 2? c) If you let the sled sit on the hill, will it slide down or stay in place? The maximum coefficient of static friction is PUM Dynamics II /Lesson 16: Components

79 d) If you wish to use a pulley system to pull the kids up, what should be the mass of the object attached to the other side of the pulley to pull the sled up at constant speed? Reflection: How can you explain to someone who does not study physics what a component of a force is? Why is it important to understand components? Give an example from everyday life when understanding force components helps you move heavy objects. PUM Dynamics II /Lesson 16: Components 79

80 Homework 16.8 Represent and Reason Complete the table that follows. Write a description of the situation in words. 1. An elevator is slowing down on its way up. Sketch the situation and circle the system object. Draw a force diagram with perpendicular axes. Label the forces if needed. y R on OT! Draw the direction of acceleration and of the net force. Are they consistent? Write Newton s Second Law in component form. x E on OF! 2. y!! N T R on O S on O o o x! F E on O 3. y! N S on O 30 o 30 o 60 o! F E on O x 4.! T! R on 1 y T R on y x! F E on 1! F E on 2 x 80 PUM Dynamics II /Lesson 16: Components

81 17.1 Design an Experiment Lesson 17: Finding the coefficient You are to determine the maximum coefficient of static friction between a shoe and a board in two different ways. You have the following equipment: the shoe, a spring scale, the board, a meter stick, a pulley, objects of different masses, and a protractor (you do ot need o use all fo them). a) Devise a method using the spring scale. b) Describe the experiment you will perform and the mathematical procedure that you will use to solve the problem. c) What quantities will you measure and what quantities will you calculate? d) What are you assuming to be true in your procedure? e) Perform the experiment and calculate the coefficient of static friction. Do not forget that you cannot obtain an exact value. How do you know if the result is reasonable? f) Devise a second method using the block and board or a pulley and objects of different masses but not using the spring scale. Repeat steps (b) (e). g) Compare the outcome of the two methods. Do they agree within expected uncertainties? Explain. h) Write a complete report about your experiment. Use the rubrics to improve your report. Ability to design and conduct an application experiment Missing An attempt Scientific Ability Needs some improvement Acceptable 1 Is able to identify the problem to be solved No mention is made of the problem to be solved. An attempt is made to identify the problem to be solved but it is described in a confusing manner. The problem to be solved is described but there are minor omissions or vague details. The problem to be solved is clearly stated. 2 Is able to design a reliable experiment that solves the problem The experiment does not solve the problem. The experiment attempts to solve the problem but due to the nature of the design the data will not lead to a reliable solution. The experiment attempts to solve the problem but due to the nature of the design there is a moderate chance the data will not lead to a reliable solution. The experiment solves the problem and has a high likelihood of producing data that will lead to a reliable solution. PUM Dynamics II Lesson 17: Finding the coefficient 81

82 3 Is able to use available equipment to make measurements At least one of the chosen measurements cannot be made with the available equipment. All of the chosen measurements can be made, but no details are given about how it is done. All of the chosen measurements can be made, but the details about how they are done are vague or incomplete. All of the chosen measurements can be made and all details about how they are done are provided and clear. 4 Is able to evaluate the results by means of an independent method No attempt is made to evaluate the consistency of the result using an independent method. A second independent method is used to evaluate the results. However there is little or no discussion about the differences in the results due to the two methods. A second independent method is used to evaluate the results. The results of the two methods are compared using experimental uncertainties. But there is little or no discussion of the possible reasons for the differences when the results are different. A second independent method is used to evaluate the results and the evaluation is done with the experimental uncertainties. The discrepancy between the results of the two methods, and possible reasons are discussed. 5 Is able to identify the shortcomings in an experimental design and suggest specific improvements No attempt is made to identify any shortcomings of the experimental design. An attempt is made to identify shortcomings, but they are described vaguely and no specific suggestions for improvements are made. Some shortcomings are identified and some improvements are suggested, but not all aspects of the design are considered. All major shortcomings of the experiment are identified and specific suggestions for improvement are made. 6 Is able to choose a productive mathematical procedure for solving the experimental problem Mathematical procedure is either missing, or the equations written down are irrelevant to the design. A mathematical procedure is described, but is incorrect or incomplete, due to which the final answer cannot be calculated. Correct and complete mathematical procedure is described but an error is made in the calculations. Mathematical procedure is fully consistent with the design. All quantities are calculated correctly. Final answer is meaningful. 7 8 Is able to identify the assumptions made in using the mathematical procedure Is able to determine specifically the way in which assumptions might affect the results No attempt is made to identify any assumptions. No attempt is made to determine the effects of assumptions. An attempt is made to identify assumptions, but the assumptions are irrelevant or incorrect for the situation. The effects of assumptions are mentioned but are described vaguely. Relevant assumptions are identified but are not significant for solving the problem. The effects of assumptions are determined, but no attempt is made to validate them. All relevant assumptions are correctly identified. The effects of the assumptions are determined and the assumptions are validated. 82 PUM Dynamics II Lesson 17: Finding the coefficient

83 Lessons 18: Practice 18.1 Represent and Reason A book rests on a table. a) Draw a sketch of the situation and identify objects that interact with the book. b) Draw forces representing these interactions (a force diagram for the book). c) If the book is stationary, these forces are equal in magnitude and opposite in direction. Can we say that they represent Newton s Third Law pair forces? If not, why not? d) Draw the Newton s Third Law force pairs for each force shown in the force diagram from part (b). Identify the cause of each of these forces and the objects on which each of these forces is exerted Regular Problem A large plane with a mass of 3.5 x 10 5 kg lands on a runway at a speed of 27 m/s. If the frictional force exerted by the road and the air on the plane is 4.3 x 10 5 N a) How long does it take the plane to stop? b) How far does the plane travel in this time? c) What is the effective coefficient of friction? d) What is the force that the plane exerts on the runway? 18.3 Regular Problem The driver of a 1560-kg Toyota Avalon, traveling at 24 m/s on a level, paved road, hits the brakes to stop for a red light. Determine the distance needed to stop the car if the coefficient of kinetic friction between the car tires and the road is Sketch the situation described in the problem; provide all known information. Choose a system object in the sketch from the first cell of this table and make a list of objects that interact with the system object. Simplify objects and interactions if necessary. PUM Dynamics II Lessons 18: Practice 83

84 Draw a force diagram for the system object. Label the forces. Make sure the diagram is consistent with the motion of the system. Include perpendicular x and y axes. Apply Newton s Second Law in component form (x and y axes) to the situation shown in the force diagram. Combine the results from the above force analysis with kinematics to determine the unknown quantity. Evaluate the result to see if it is reasonable (unit, magnitude, and value for limiting situations) Regular Problem To give a 17 kg child a ride on a 3.4 kg sled, two teenagers pull at 35 angles to the direction of the sled s motion (see picture). The unpacked snow exerts a frictional force of 57 N. If both teenagers pull, each exerting a force of 55 N, what is the acceleration of the sled? 84 PUM Dynamics II Lessons 18: Practice

85 18.5 Regular Problem Two of your neighbor s children (40 kg together) sit on a sled. You push on the back child, exerting a 50 N force on him directed 37 o below the horizontal. The sled slides forward with a constant velocity. Complete the table below to answer the question: What is the coefficient of kinetic friction between the snow and the sled? Sketch the situation described in the problem; provide all known information. Choose a system object in the sketch from the first cell of this table and list objects that interact with the system. Draw a force diagram for the system object. Label the forces. Make sure the diagram is consistent with the motion of the system object. Include perpendicular x and y axes. Apply Newton s Second Law in component form (x and y axes) to the situation shown in the force diagram you drew. Solve the equations for the unknown quantities. Evaluate the results to see if they are reasonable (units, magnitudes, and values for limiting situations). PUM Dynamics II Lessons 18: Practice 85

86 18.6 Evaluate the Solution Identify any errors in the solution to the following problem and provide a corrected solution if there are errors. The problem: A 1000 kg elevator is moving down at 6.0 m/s. It slows to a stop in 3.0 m as it approaches the ground floor. Determine the force that the cable supporting the elevator exerts on the elevator as the elevator stops. Assume that g = 10 N/kg. Proposed solution: The elevator at the right is the object of interest. It is considered a particle, and the forces that other objects exert on the elevator are shown in the force diagram. The acceleration of the elevator is:, a = v 0 2 /2d = (6.0 m/s) 2 /2(3.0 m) = 6.0 m/s 2. The force of the cable on the elevator while stopping is: 18.7 Evaluate the solution T = ma = (1000 kg)(6.0 m/s 2 ) = 6000 N. Identify any errors in the solution to the following problem and provide a corrected solution if there are errors. The problem: You push a 20-kg lawn mower, exerting a 100-N force on it. You push 37 o below the horizontal. The effective coefficient of kinetic friction between the grass and mower is Determine the acceleration of the lawn mower. Assume that g = 10 m/s 2. Proposed solution: The situation is pictured at the right. The mower is the object of interest and is considered a particle. The forces that other objects exert on the mower are shown in the force diagram. The magnitude of the kinetic friction force is:,, f k = µ k N = 0.60(20 kg)(10 m/s 2 ) = 120 N. The acceleration of the mower is: a = (F f k )/m = (100 N 120 N)/(20 kg) = 1.0 m/s Design an Experiment Design a balloon racer. You are given 2 balloons, straws, paper, and tape. The racer should be designed using your understanding of forces. You will race this balloon racer against other students in the class. a) Design a method in which you can determine the time it takes to travel a given distance when relative uncertainty is taken into account. 86 PUM Dynamics II Lessons 18: Practice

87 b) Design a method to determine the average acceleration during this time when relative uncertainty is taken into account. c) Design a method in which to determine the average force the air pushing its way out of the balloon exerts on the balloon itself when relative uncertainty is taken into account. Be sure to include force diagrams d) What assumptions did you make when doing these calculations? How do your assumptions affect your calculated estimate? e) What is your relative uncertainty in each value? What could you have done to reduce uncertainty? v 18.9 Reason The graph for velocity of book being pushed by Jenny across a table shown on the right. Draw 5 force digraphs for the book for selected clock readings (marked on the graph). t 1 t 2 t 4 t Reason Go to the Phet simulation Forces in 1 Dimension. Turn friction off. You goal is to choose how to manipulate the force so the graph acceleration versus time looks as follows: Test your ideas Go to You will find several testing experiments listed below. For each experiment, follow instructions and answer all of the questions on the website. Submit your answers to your teacher. a) Cart on Track: Qualitative Testing Experiment b) Atwood's Machine with a spring scale Part 1: Qualitative Testing Experiment c) Atwood's Machine with Spring Scale Part 2: Qualitative Testing Experiment d) Spring scale up and down: Qualitative Testing Experiment Reflection: Of you could write a note to your past self about solving problems involving Newton s laws, what would you say? PUM Dynamics II Lessons 18: Practice 87

88 Lesson 19: Review 19.1 Equation Jeopardy Several mathematical statements are listed below. For each statement, describe a problem for which this statement could be a solution. Then represent the statement using a force diagram and a motion diagram. For one of the forces involved in the situation find Newton s third law pair. a) F unbalanced on object = (9.8 N/kg) x (3 kg) b) (-7 m/s) + (2 m/s) = (3 s) x a c) (-35 N) + (9.8 N) = (1 kg) x a d) F rope on sled F Jake on sled = (35 kg) x (0 m/s 2 ) e) a = m 1! m 2 m 1 + m 2 g f) (70 N) cos F floor on crate = (5 kg) x a 19.2 Diagram Jeopardy Six force diagrams are shown below. Describe a situation for each diagram; be sure each diagram can represent the situation created for it. For each situation, in what direction is the object moving? How many answers can you have? Draw a matching motion diagram and write Newton s Second Law in component form for each scenario Graph Jeopardy Three lines on the graph below describe three motions of an object. Tell a story about each motion. Draw a motion diagram and a force diagram. How many answers can you have? Determine the unbalanced force in each case if the mass of the object is 250 kg. Velocity (m/s) Time (s) 19.4 Reason Earth exerts a 5-N force on an apple, what is the force that the apple exerts on Earth? 88 PUM Dynamics II Lesson 19: Review

89 19.5 Reason You are pulling two boxes (10 kg and 15 kg) connected with a rope on a horizontal surface. You exert a 250 N force at an angle of 27 0 with the horizontal by pulling the second rope attached to a 15-kg box. Represent the situation with the force and motion diagram. Write Newton s second law in component form for each box. Consider two cases: the surface is a smooth floor and the boxes are made of laminated cardboard and the floor is a carpet and the boxes are made of regular cardboard Reason A horse is pulling a cart. According to Newton s third law the force that the cart exerts on the horse is always the same in magnitude and opposite in direction to the force that the horse exerts on the cart. How does the horse ever manage to get the cart moving? 19.7 Reason A woman pushes a 60 kg couch along a rough surface. The couch accelerates at a rate of 0.5 m/s 2. Coefficient of kinetic friction between the couch and the floor µ k = Make a list of physical quantities you can determine using this information and determine 2 of those quantities Reason A football player exerts a force of 1800 N to push a 40 kg blocking sled with an acceleration of 10 m/s 2 over a very rough surface. Make a list of physical quantities you can determine using this information and determine 2 of those quantities Reason A car locks its brakes and skids to a stop with an acceleration of 4 m/s 2. For tires on the road, µ k = Assume the car has a mass of 2000 kg Reason Mr. T. pulls a 400 kg walk-in refrigerator behind his car as he drives. The road exerts a 3000 N force on the car but the car does not accelerate. Explain why. Make a list of quantities you can determine using this information. Determine 2 of them Reason A football player exerts a force of 1800 N to push a 40 kg blocking sled on a rough surface. The µ k between the surface and sled is 0.5. Determine everything you can using this information Reason A car slows to a stop with an acceleration of 8 m/s 2. Assume the force of friction exerted by the air and the road on the car is N. Pose a question about this situation that you can answer and provide additional information if necessary. PUM Dynamics II Lesson 19: Review 89

90 19.13 Reason Two objects of masses m 1 and m 2 are connected with a light rope going over a light pulley. Draw a picture representing is situation. Then determine the accelerations of the object when the system is let go and the force that the rope exerts on both objects. How many different scenarios can you come up with? How are the acceleration and the force difference depending on the scenario? Reason Questions below refer to collisions between a car and trucks. For each description of a collision below, choose the one answer from the possibilities 1 though 6 that best describes the forces between the car and the truck. 1 The truck exerts a greater amount of force on the car than the car exerts on the truck. 2 The car exerts a greater amount of force on the truck than the truck exerts on the car. 3 Neither exerts a force on the other; the car gets smashed simply because it is in the way of the truck. 4 The truck exerts a force on the car but the car doesn't exert a force on the truck. 5 The truck exerts the same amount of force on the car as the car exerts on the truck. In questions a-d the truck is much more massive than the car. a. They are both moving at the same speed right before they collide. Which choice describes the forces? b. The truck is moving and the car is standing still right before they collide. Which choice describes the forces? c. The car is moving much faster than the truck right before they collide. Which choice describes the forces? d. The heavier truck is standing still when the car hits it. Which choice describes the forces? In questions e and f the truck is a small pickup and has the same mass as the 90 PUM Dynamics II Lesson 19: Review