Lab #2: Newton s Second Law

Transcription

1 Physics 144 Chowdary How Things Work Spring 2006 Name: Partners Name(s): Lab #2: Newton s Second Law Introduction In today s exploration, we will investigate the consequences of what is one of the single most important developments in the history of science: Newton's 2 nd Law: F net = m a, where net is the (vector) sum of all the forces acting on the object, m is the object s mass, and a r the object s (vector) acceleration. The development of this law, which forms the foundation of classical mechanics, planted the seeds for modern physics and technology and is the heart of much of our understanding of how things work. In particular, we will explore the relationship between acceleration, force, and mass, and explore how the motion of an object depends on the direction as well as the amount of each quantity. We ll also explicitly study the vector nature of force and develop some ideas regarding vectors. There are several goals of this lab: in particular, after finishing this lab, you should a) have further developed your ideas about position, velocity, and acceleration, and see how they relate to each other; b) gain some personal experience in how an applied force results in acceleration; c) make some measurements to verify Newton s 2 nd Law quantitatively; and d) get some experience with the vector nature of force by balancing forces in two dimensions. This lab will also give you more practice with the graphing concepts we went over last week. Specifically, we ll explore the following question: r r Does Does F = m a? r r F = m a? How do vectors add? Equipment For part of this lab, you ll use the low friction track along with the Motion Detector connected to the LabPro interface as in Lab #1. This time, you ll use a cart with a fan attached. The fan should serve as a constant force. You ll notice that you can change the angle at which the fan blows, so we can adjust the amount of force that is parallel to the track. For the other part of this lab, you ll use a force table that consists of a central ring with strings tied to it. The strings run over pulleys which can be placed at any angle on the edge of the table and to which weights can be attached. When the hanging weights are in equilibrium, the tension in the string is equal to the weight, we can have the tensions in the various strings pull on the central ring in any direction we want (along the plane of the table). If the string forces balance, the central ring will remain stationary. If the forces don t balance, then when we pull out the central retaining pin, the central ring will accelerate in the direction of the net unbalanced force.

2 Constant Force Fan? (1) Examine your fan cart. At the front of the room is a scale; determine the mass of your fan cart and write that mass down. Convert to kg. Mass of fan cart: g = kg (2) If you turn on the fan, the cart will experience a force that is intended to be constant in time. What measurements could we make that would show if the fan exerts a constant force? Discuss this in your lab group, and design an experiment that will enable you to determine if the net force acting on the fan cart is in fact constant. Write down your ideas and experiment. Discuss your experiment idea with your instructor. (3) Make sure the fan is lined up at 0 o. Put the fan cart on the track and aim the cart so that it would accelerate towards the detector. Turn on the motion detector and move the cart back and forth with your hand to find the useful detection range of your detector. Write down the minimum and maximum range of your detector. Convert to m. Minimum range: cm Maximum range: cm m m (4) Place the cart near the minimum range distance and turn on the fan. Gently and briefly push the cart away from the detector. Don t use the motion detector here. In words, describe the motion you observe.

3 (5) If the acceleration of the cart is indeed constant, what would a graph of its position vs. time look like? What would a graph of its velocity vs. time look like? (Hint: consider another constant acceleration situation you are familiar with: throwing a ball straight up in the air). On the axes below, sketch qualitative (no numbers needed) graphs of position vs. time and velocity vs. time. (6) Now, make some measurements using the motion detector. Turn on the fan cart, give it a brief and gentle push away from the detector, and show a plot of position (distance) vs. time and velocity vs. time. Did your measured graphs look like your predicted curves from step (5)? Discuss. (7) How would you obtain acceleration from a velocity vs. time graph? Using the same analysis you did last week, determine the acceleration of the fan cart. Make sure you only analyze the region where the cart is headed towards the detector (i.e. on its way back). If you don t have a good graph, try again until you do. Briefly describe your method. Write down your experimentally obtained acceleration. Convert to m/s 2. acceleration: cm/s 2 = m/s 2 (8) What is the net force acting on the fan cart. Briefly describe your reasoning and show any calculations. Net force: N Discuss your results with your instructor before moving on.

4 Does F = m a? (1) In the previous step, you determined the net force acting on the cart. For the remainder of this lab, we ll assume that this force remains constant in magnitude (strength) If the mass of the cart were increased by 200 g, would the acceleration of the cart increase, decrease, or remain the same? What would be the acceleration of the cart? Explain your reasoning/show your calculations below. Predicted acceleration of fan cart when mass increased by 200 g: m/s 2 (2) Repeat your calculation, but this time increase the mass by another 200 g (in other words, the fan cart g). Predicted acceleration of fan cart when mass increased by 400 g: m/s 2 (3) Now, do the experiment. Adding 200 g isn t too hard; it can be placed in the front of the fan cart. Adding the other 200 g might be hard; you can probably balance it on the back or on top of the batteries without too much difficulty. Measure the accelerations for the two cases, and fill out the table below: Experiment Force (N) (assumed constant, so fill in same value) mass (kg) predicted acceleration (m/s 2 ) measured acceleration (m/s 2 ) comments? Fan Cart no prediction Fan Cart g Fan Cart g (4) Do you find compelling evidence to believe in F = m a? Describe how your experiment supports or contradicts this statement of Newton s Second Law.

5 r How about F r = m a? In the previous step, you checked on Newton s Second Law in the form F = m a. But we know that forces and r r F = m a. We can accelerations are actually vectors, so that Newton s Second Law is more correctly given by check on the vector nature of force by adjusting the angle of the fan. Since the cart can only move along the track due to the grooves on the track, if we change the angle of the fan, we are changing the amount of force in the direction of motion. The acceleration should change by the same amount. (1) How can we relate the direction of the fan to the amount of force along the track? We ll need to use some trigonometry. Here are some examples of a force vector at a certain angle to a track. In each case, determine the amount of force that points along the track. Keep track of positive and negative. Write the amount of force pointing along the track next to the picture. If your group can t figure out what to do, consult your instructor. Check your results with your instructor before moving on. track direction track direction track direction track direction 30 o 60o (2) We call the amount of force that points in a particular direction the component of the force in that direction, or the force component. You will adjust the angle of the fan to 20 o and 40 o. In the space below, calculate the force component for those two angles, using the force you determined for the fan previously. Using just the mass of the fan cart, also calculate the acceleration of the fan cart. (3) Now, perform the experiment and measure the acceleration of the fan cart. Don t forget to remove the extra mass from the fan cart. Fill in the table below. Experiment Force (N) (assumed constant, so fill in same value) Force Component (N) predicted acceleration (m/s 2 ) measured acceleration (m/s 2 ) comments? 0 o no prediction 20 o 40 o

6 The Vector Nature of Forces So far, all the motion you have measured has been one dimensional (i.e. the motion took place along one line). You began with the force pointing along that line, but then saw some evidence for the vector nature of force by aiming the force at an angle to the allowed motion. In general, forces act in many directions and it is necessary to consider their full vector nature. For simplicity, we will deal with the F v net = 0 case, in other words, with no acceleration. You will be working with a force table, which is the large heavy object that looks like a modern-art coffee table. You can attach different masses to strings, which are each also attached (via pulleys) to the central ring. When you look down on the force table from above, you will see that the four strings can be adjusted to different angles. Each partner should take turns climbing up onto the table to look down on the force table. The retaining pin is the straight metal pin which runs through the central ring; it makes sure that even when the forces aren t balanced, nothing moves. When removing the retaining pin, make sure your partners are holding the masses and slowly release them. If you haven t balanced the forces successfully, the masses will rapidly drop, which could be quite startling. If you have questions about this apparatus, please come find me. Note that each hangar has a mass of 50 g. So you ll need to include that mass when bringing the total mass up to some specified value: for example, if you are to have 100 g hanging from a string, then you only need to add 50 g to the 50 g hangar to reach your target mass. An appropriate way to represent forces is in terms of vectors, which we draw using arrows: the length of the arrow represents the magnitude (strength) of the vector, and the direction the arrow is aimed in represents the direction of the vector. We can also use trigonometry to break vectors up into their vector components (the part of the vector that lies along a particular direction, and the part of the vector that lies perpendicular to that direction). You dealt with the component of the force in the direction of the track in the previous part of this problem. Here, you will have to combine the effects of two or more vectors, and there are two equivalent ways to do this. Your instructor will discuss graphical addition of vectors on the board with the class; you can continue along until that is done. (1) Set up the force table with two of the pulleys at 0 o and 180 o. Make sure any other pulleys are weightless (no mass hangers on those strings). Make sure that the string at 0 o and the string at 180 o each have one mass hangar (50 g) apiece. Place an additional 50 g on the hangar at 0 o, bringing its mass up to 100 g total. Convert g to kg and determine the weight of the 100 g mass in Newtons (assume g = 10 N/kg = 10 m/s 2 for convenience). weight of 100 g mass: N (2) When that mass is balanced, what must be the tension in the string attached to that weight? Briefly explain. tension in string connected to 100 g mass, when balanced: N (3) Since the pulley only acts to change the direction of the string, that means that the end of the string tied to the central ring is exerting the same magnitude of force, but in a direction lying along the plane of the table and given by the angle you set the string at. If you want to make sure the central ring is balanced, what force should you pull at 180 o? How much (total) mass should you hang on that string to accomplish this?

7 (4) Place your predicted mass on the 180 o string. Recall that the hangar itself has mass. Pull out the retaining ring to verify that your central ring remains stationary. Was your prediction correct? Discuss. (5) On the provided graph paper, using pencil, carefully draw a vector diagram. The vector diagram should be accurate in both direction and also should be scaled. My suggestion is to use a scale so that 1 Newton is 10 boxes (a box is 5 mm). Draw both tail-to-tail and head-to-tail. Note that the head-to-tail version shows more clearly that the vectors add up to zero. (6) Now, put the retaining pin back into the central ring and move the pulley at 180 o to 120 o (don t change any masses). Can the net force exerted on the central ring still be zero? Check whether the central ring is balanced by carefully removing the retaining ring. Was the central ring balanced? Describe your observations. (7) On the provided graph paper, using pencil, carefully draw the vector diagram for this situation. Draw a headto-tail vector diagram, to scale at correct angles. (8) Look at your vector diagram from the previous step. It s clear that the forces don t add up to zero, since the vectors don t close in. Carefully draw on the vector diagram a single vector that closes the vector diagram; put an arrow on the vector. Measure the length of that vector using your ruler and also measure the angle (with respect to 0 o ). Convert (using your scaling factor) the length of the vector to a magnitude in Newtons and also to mass. vector length: mm angle: (degrees, counterclockwise from 0 o ) Force: N mass: kg = g (9) Use your results from the previous step to set up a third string at the predicted angle holding the correct mass (again, don t forget that the hangar itself has mass 50 g). Are the forces balanced? Pull out the retaining pin to check. Was your prediction correct? Discuss. (10) Put a total of 100 g at 0 o and a total of 100 g at 90 o. On the provided graph paper, using pencil, carefully draw the vector diagram for this situation. Draw a head-to-tail vector diagram, to scale at correct angles. Draw the vector that closes the vector diagram. Use this to determine the mass and angle you should hang from the third string to balance the forces. What are your predicted mass and angle? predicted total mass: g predicted angle: degrees (measured counterclockwise from 0 o ).

8 (11) Test out your prediction. Were you correct? (12) Set up the following situation: 100 g total at 30o. 200 g total at 135o. Determine using a vector diagram what mass your should put at what angle to balance out these forces. After making your prediction, test it out. Were you correct?