Physics II P H Y S I C S U N I O N M A T H E M A T I C S. Dynamics. Student Edition

Transcription

1 P H Y S I C S U N I O N M A T H E M A T I C S Physics II Dynamics Student Edition Supported by the National Science Foundation (DRL ) and Science Demo, Ltd.

2 PUM Physics II Dynamics Adapted from: A. Van Heuvelen and E. Etkina, Active Learning Guide, Addison Wesley, San Francisco, Used with permission. 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

3 Table of Contents LESSON 1: FORCE AS AN INTERACTION 4 LESSON 2: EXPERIMENTAL DESIGN AND ASSESSMENT 12 LESSON 3: MOTION DIAGRAMS & FORCE DIAGRAMS 14 LESSON 4: WHAT S A FORCE TO DO? 18 LESSON 5: INERTIAL AND NON-INERTIAL REFERENCE FRAMES 24 LESSON 6: NEWTON S SECOND LAW: QUALITATIVE 28 LESSON 7: NEWTON S SECOND LAW: QUANTITATIVE 32 LESSON 8: DESIGN AN EXPERIMENT 37 LESSON 9: APPLYING NEWTON S SECOND LAW 41 LESSON 10: NEWTON S THIRD LAW: QUALITATIVE 45 LESSON 11: NEWTON S THIRD LAW: QUANTITATIVE 47 LESSON 12: TWO-BODY PROBLEMS 51 LESSON 13: DESIGN AN EXPERIMENT 54 LESSON 14: FRICTION 57 LESSON 15: PUTTING IT ALL TOGETHER 63 LESSON 16: COMPONENTS 65 LESSON 17: FINDING THE COEFFICIENT 73 LESSONS 18: PRACTICE 76 LESSON 19: REVIEW 81 PUM Dynamics 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. 1.2 Test Your Reasoning 4 PUM Dynamics Lesson 1: Force as an Interaction

5 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 the 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). 1.4 Represent and Reason PUM Dynamics Lesson 1: Force as an Interaction 5

6 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 the Earth as: Flegs on dog + FEarth 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 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. 6 PUM Dynamics Lesson 1: Force as an Interaction

7 Experiment Draw a picture of the apparatus. List objects interactin g with the object of interest. Draw a force diagram for the object. Discuss what objects exert forces balancing the force that the 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 small angle. Record what happens PUM Dynamics Lesson 1: Force as an Interaction 7

8 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 the 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 the 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? 5. Draw forces to represent interactions, watch the length of arrows 8 PUM Dynamics Lesson 1: Force as an Interaction

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 the Earth pulls on a ball, we note the force exerted by the Earth on the ball as: FEarth 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 the Earth is the external object. If we are interested in the force that the ball exerts on the Earth, we will write it as FBall 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. PUM Dynamics Lesson 1: Force as an Interaction 9

10 1.9 Represent and Reason Read each of the scenarios and then draw a force diagram for the selected object of interest. 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. 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 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 below and come up with a real life situation that they might describe. Then label each force with the appropriate subscripts Estimate uncertainty 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 haw precisely you can measure the force with this scale Estimate uncertainty 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. PUM Dynamics 11

12 Scientific Ability 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 Missing 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 An attempt Needs some improvement 12 PUM Dynamics Lesson 2: Experimental Design and Assessment Acceptable

13 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. PUM Dynamics Lesson 2: Experimental Design and Assessment 13

14 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 a force diagram for the ball. 3.2 Represent and Reason a) Look at the force diagram you drew in 3.1 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. Indicate the direction of the unbalanced force with an arrow. c) Indicate the direction of the velocity change arrow ( v ) on the motion diagram. 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. a) Sketch the situation. b) Perform the experiment and describe the motion of the ball in words. 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. e) Draw a force diagram for the ball. f) Examine your force diagram. 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. 14 PUM Dynamics Lesson 3: Motion diagrams & Force diagrams

15 3.4 Represent and Reason Imagine that you have a bowling ball that is moving on a very smooth floor (neglect all friction forces). While the ball is in motion, its velocity does not change. a) Draw a motion diagram for the ball. What is the direction of the velocity change? b) Draw a force diagram for the ball. 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? 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 r 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? 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 ball? 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. PUM Dynamics Lesson 3: Motion diagrams & Force diagrams 15

16 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. 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. 16 PUM Dynamics Lesson 3: Motion diagrams & Force diagrams

17 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. PUM Dynamics Lesson 3: Motion diagrams & Force diagrams 17

18 4.1 Test an Idea Lesson 4: What s a Force to Do? 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? Remember in our kinematics unit we spent some time talking about hypotheses, predictions, and H-D statements. Recall the difference between hypotheses and predictions. Also recall that the H-D statements are written in an If-And-Then statement. REMEMBER! Your predicted outcome always 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. 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. 18 PUM Dynamics Lesson 4: What s a Force to Do? 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.

19 4.3 Hypothesize Based on the experiments above and the experiments you performed in lesson 3, formulate the relationship between the unbalanced force 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 net force 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 r 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. 4.5 Reason Homework 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. PUM Dynamics Lesson 4: What s a Force to Do? 19

20 4.6 Represent and Reason Draw a motion diagram and a force diagram 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 diagram How do you know if they are consistent? A ball is dropped and is falling down. FE on O A ball is thrown down. A football is landing on a cushion and the cushion is being compressed. A rabbit sits in its cage. FS on R FE on R 20 PUM Dynamics Lesson 4: What s a Force to Do?

21 Situation Motion diagram Force diagram How do you know if they are consistent? A matchbox slides down a smooth book cover. FS on O FS on O FE on O 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. g) Explain how you understand the difference between the terms experimental evidence, hypothesis, and prediction. Provide your own example for each. PUM Dynamics Lesson 4: What s a Force to Do? 21

22 I. Complete the diagrams below. Additional Problems and Exercises Representing Horizontal Motion a) Which dot represents t = 0 in the motion diagram? Label t = 0. b) Where is the object at t = 0 in the motion diagram? Let that be the origin, and label it. c) What do the arrows represent in the motion diagrams? Label them. d) Indicate the direction of r v for the motion diagrams. e) Label the forces in the force diagrams. Motion Diagrams Force Diagrams A B. C. 22 PUM Dynamics Lesson 4: What s a Force to Do?

23 II. For each of the force diagrams, determine which motion diagrams from the previous activity can physically represent the force diagram. For each force diagram-motion diagram combination, tell a story that they could be representing. An example is done for you. Be creative! Force Diagram Motion Diagram # Story A. 7 Example: A turkey leg sits on a plate. The downward force is the F Earth on turkey and the upward force is F plate on turkey B. C. III. Pick one story for each of the three force diagrams and write a problem that includes numbers. Then solve the problem. PUM Dynamics Lesson 4: What s a Force to Do? 23

24 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. 24 PUM Dynamics Lesson 5: Inertial and Non-inertial Reference Frames

25 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 r 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? PUM Dynamics Lesson 5: Inertial and Non-inertial Reference Frames 25

26 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 the 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 a force diagram. Check the consistency of the diagrams. (a) Elevator hangs at rest at the ground floor. (b) Elevator starts moving upward with increasing speed. 26 PUM Dynamics Lesson 5: Inertial and Non-inertial Reference Frames

27 (c) Elevator reaches 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 the Earth exerts on the elevator when the elevator moves at constant speed and changing speed. PUM Dynamics Lesson 5: Inertial and Non-inertial Reference Frames 27

28 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 previous activity and graphs. 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. 28 PUM Dynamics Lesson 6: Newton s Second Law: Qualitative

29 6.3 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? 6.4 Find a Relationship Summarize how the change in the velocity of the object depends on the unbalanced force 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. 6.5 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. 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? PUM Dynamics Lesson 6: Newton s Second Law: Qualitative 29

30 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. 6.6 Reason Examine the graphs in activity 6.3. 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? 6.7 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. 6.8 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? 6.9 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) = Ranking Tasks 30 PUM Dynamics Lesson 6: Newton s Second Law: Qualitative

31 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 PUM Dynamics Lesson 6: Newton s Second Law: Qualitative 31

32 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, the 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): 32 PUM Dynamics Lesson 7: Newton s Second Law: Quantitative

33 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 the 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, the Earth, and the track on the cart. PUM Dynamics Lesson 7: Newton s Second Law: Quantitative 33

34 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. Here s An Idea! 34 PUM Dynamics Lesson 7: Newton s Second Law: Quantitative

35 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? Did You Know? In the previous lessons, 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 Fnet F r 1 on S F r 2 on S... F r n on S F r n on S exerted by other objects on the system object and inversely proportional to the mass m of the system object: r F a net r F n on S m S m S 7.5 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 the Earth exerts on any object. 7.6 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. 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? PUM Dynamics Lesson 7: Newton s Second Law: Quantitative 35

36 7.7 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? Write Newton s Second Law in component form. a y T ( F ) R on O E on O m 2 m 800 N+(-1000 N) 2 s 10 kg 30 o Did You Know? Gravitational force: The magnitude of the gravitational force that the 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 the Earth. 36 PUM Dynamics

37 Ability Missing An attempt Needs some improvement Acceptable 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) Use the rubrics below to improve your lab report. PUM Dynamics Lesson 8: Design an Experiment 37

38 Is able to record and represent data in a meaningful way Data are absent. Data are present but impossible to understand. Units are missing. Data are present and have units, but one needs to make a serious effort to understand the data. Data are present, organized, and recorded clearly. A table is made. Is able to analyze data using a graph No graph is present. A graph is present but the axes are not labeled. There is no scale on the axes The graph is present and axes are labeled, but the axes do not correspond to the independent and dependent variables or the scale is not accurate. The graph has correctly labeled axes. The independent variable is along the horizontal axis and the scale is accurate. Is able to construct a mathematical (if applicable) relationship that represents a trend in the 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). 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? 8.4 Test an Idea Ritesh says: If you hang an object on a spring scale, the scale always reads how much force the 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: 38 PUM Dynamics Lesson 8: Design an Experiment

39 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. 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. PUM Dynamics Lesson 8: Design an Experiment 39

40 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. 40 PUM Dynamics Lesson 8: Design an Experiment

41 Lesson 9: Applying Newton s Second Law Problem-Solving Strategy for Dynamics Problems Sketch and Translate: 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. 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 of the answer, its units, how the solution changes in limiting cases, and so forth). Need Some Help? Here is an example that applies the strategy shown above: A 5-kg object (the Earth exerts a 50 N force on it) is lifted by a cable that exerts a 100 N force on it. Calculated the acceleration of the object. Translate: The object is our system; the 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 the 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 the 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 is 5 kg pointed upward the elevator is accelerating in the upward direct (not necessarily moving in the upward direction, it can be slowing down). PUM Dynamics Lesson 9: Applying Newton s Second Law 41

42 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. 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 F Earth on Bucket y Table on Bucket y m 0 ( 22N) F Table on Bucket 2.2kg 2) A 5kg bucket of clams hangs motionless from a spring that stretches 40 cm. 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 with 3 or more other objects interacting with it. 42 PUM Dynamics Lesson 9: Applying Newton s Second Law

43 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 m/s. You push on it in the direction opposite to its motion exerting a force of 12 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) 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 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? PUM Dynamics Lesson 9: Applying Newton s Second Law 43

44 9.4 Practice Homework 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 172 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 PUM Dynamics Lesson 9: Applying Newton s Second Law

45 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. PUM Dynamics Lesson 10: Newton s Third Law: Qualitative 45

46 10.4 Represent and Reason Homework 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 the 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 the Earth s surface, the 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. 46 PUM Dynamics Lesson 10: Newton s Third Law: Qualitative

47 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. PUM Dynamics Lesson 11: Newton s Third Law: Quantitative 47

48 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 r 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. 48 PUM Dynamics Lesson 11: Newton s Third Law: Quantitative

49 11.4 Reason Homework a) A book sits on the tabletop. What is the Newton s Third Law pair for the force that the Earth exerts on the book? b) If the Earth exerts a 5 N force on the book, what is the force that the book exerts on the Earth? c) What is the acceleration of the book if the Earth is the only object exerting a force on it? d) Why does the book fall onto the Earth but the Earth does not fall onto the book? The mass of the Earth is 6.00 x kg. Need Some Help? Use the Earth s mass to calculate the Earth s acceleration. What does this tell you about the motion of the Earth? e) The Sun s mass is 2.00 x kg. It pulls on the Earth, exerting a force of about N. What is the force that the Earth exerts on the Sun? 11.5 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? 11.6 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 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) The Earth pulls on an apple exerting a 1.0 N force on it. What is the force that the apple exerts on the Earth? Why does the apple fall towards the Earth but the 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? 11.7 Reason PUM Dynamics Lesson 11: Newton s Third Law: Quantitative 49

50 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? 11.9 Reason The Moon orbits the Earth because the Earth exerts a force on it. The Moon, therefore, has to exert a force on the 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? e) Who is pushing harder the elevator s floor on the person or the person on the floor? The Earth on the person or the floor on the person? 50 PUM Dynamics Lesson 11: Newton s Third Law: Quantitative

51 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 (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 Lesson 12: Two-Body Problems 51

52 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 rough 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) 52 PUM Dynamics Lesson 12: Two-Body Problems

53 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. PUM Dynamics Lesson 12: Two-Body Problems 53

54 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. 54 PUM Dynamics Lesson 13: Design an Experiment

55 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. PUM Dynamics Lesson 13: Design an Experiment 55

56 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). 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. 56 PUM Dynamics Lesson 13: Design an Experiment

57 14.1 Observe and Find a Pattern Lesson 14: Friction 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. PUM Dynamics Lesson 14: Friction 57

58 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). Spring pushes 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 1 kg 1 kg 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 58 PUM Dynamics Lesson 14: Friction

59 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 N 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 the 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? Normal force: When two objects touch each other, they exert a normal force on each other. The force of the one object on the other object 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: When two objects touch each other, they exert a friction force on each other. 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 s N PUM Dynamics Lesson 14: Friction 59

60 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 k N 14.6 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? 14.8 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? f) What would happen if, after a few seconds, the boys stopped pulling? 60 PUM Dynamics Lesson 14: Friction

61 I. Complete the diagrams below. Additional Problems and Exercises Label the motion diagrams with the appropriate labels. Label the forces in the force diagrams. Motion Diagrams Force Diagrams A B. C. II. For each of the force diagrams identify the motion diagrams and force diagrams that describe the same situations. For each force diagram-motion diagram combination, tell a story about what that they could be representing. An example is done for you. Be creative! PUM Dynamics Lesson 14: Friction 61

62 Force Diagram Motion Diagram # Story A. 5 Example: Kids were being pulled by their cousin on a sled to the right in wet snow. The youngest one was scared and wanted to slow down, so the cousin now pulls less hard, letting the snow slow them down. The downward force is the F Earth on sled and the upward force is F ground on sled The force to the left is the F snow on sled and the force to the right is F cousin on sled B. C. III. Pick one story for each of the three force diagrams and write a problem that includes numbers. Then solve the problem. 62 PUM Dynamics

63 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. PUM Dynamics Lesson 15: Putting It All Together 63

64 15.7 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 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. 64 PUM Dynamics Lesson 15: Putting It All Together

65 16. 1 Reason Lesson 16: Components We learned that force, acceleration and velocity are physical quantities that have both magnitude and direction they are vector quantities. How do we do operations with vectors? a) Addition A r 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 C A +, we draw another vector, R, 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 r B r. We can write the resultant vector as a mathematical equation: R A r C r. 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 r C You are familiar with subtraction a little bit already this is what we do when we find the r v vector on motion diagrams. Basically, to find the vector P, which is equal to A r 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 r A r C rc r P r A find their sum and their difference. PUM Dynamics Lesson 16: Components 65

66 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. 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. 66 PUM Dynamics Lesson 16: Components

67 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? PUM Dynamics Lesson 16: Components 67

68 TO BE HANDED IN Name: Date: Period: The following vector problems should be done by hand, with ruler, protractor and graphing or computer paper and can be double checked by using the PhET simulation. 4. A duck waddles 2.5 m east and 6.0 m north. What is the duck s displacement? 5. While following directions on a treasure map, you walk 45.0 m south and then 7.50 m east to find the X marks the spot. Show two other ways you could have arrived at the same spot from your starting position. 6. A hiker walks 4.5 km northwest and then walks 4.5 km south. What is the hiker s displacement? What magnitude and direction should the hiker have to go to get straight back to her starting position? 7. A stoplight hangs from two wires that make a 30º angle with a line parallel to the ground. Each wire can exert a maximum of 560 N. What is the maximum weight the light can be so that the light does not fall to the ground? 8. You are flying a plane at 140 m/s northeast when a gale wind of 40 m/s heading north begins to blow. a. With what velocity would an observer on the ground see you flying? b. If you wanted to stay on the original course (140 m/s NE), what correction (vector component) would you need to make so that your original velocity became your resultant? 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) 68 PUM Dynamics Lesson 16: Components

69 16.2 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 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. Do not forget the sign! 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 with the horizontal. 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? PUM Dynamics Lesson 16: Components 69