# Physics 211R: The Work - Kinetic Energy Theorem

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1 Physics 11R: The Work - Kinetic Energy Theorem Reading Assignment: Chapter 7, Sections -8 Introduction: F/A-18E/F Super Hornet U.S. Navy photo by Photographer's Mate 3rd Class John Sullivan Aircraft take off from the deck of ships via a catapult system. Essentially, a large force is applied to the aircraft as it is displaced across the deck of the ship. The purpose of this is obvious. A plane needs to reach a particular speed before it can remain airborne and the catapult provides the means to do this. In the language of Kinematics, one would describe the motion of the plane as having acceleration. In the language of Newton s Laws, one would explain that the Net Force on the plane was responsible for causing the accelerated motion. However, there is another language used by Physicists to explain situations such as this aircraft example. This language, and approach to solving problems, is founded upon the concepts of work and energy. The purpose of this lab is to accelerate a cart using two different mock catapult systems and to analyze each of these systems from a work and energy perspective. Work is accomplished on an object any time a force acts over a particular displacement, d, such that the force, or some component of it, is parallel to the displacement. If the force is constant throughout the entire displacement of the object, then the following equation is valid: W sub = F sub d cos θ or W sub = F // sub d where θ is defined as the angle between the force and displacement vectors, sub refers to the descriptive subscript clarifying which force is doing the work, and F // sub = F sub cos θ. If the subscript is friction, for example, then the equation describes the work done on the object by friction. It the subscript is gravity, for example, then the equation describes the work done on the object by the force of gravity. If the subscript is net, for example, then the equation describes the net (or total) work done on the object. It is important to clarify exactly which force, and therefore, work, is being described, because most objects are acted upon by more than one force simultaneously. The total work done on an object describes the overall result of the transfer of energy caused by all of the forces combined. Work is considered to be positive, negative, or zero in value, depending on the value of θ. (Recall that cosθ = 1 if θ = 0 and cosθ = -1 if θ = 180.) In addition, it is important to keep in mind that the above equation is valid if and only if F // sub is constant. If the force is not constant (variable), then the work done cannot be determined via the equation above. Instead, the following integral (for the case of one-dimensional displacement along the x-axis in which θ = 0 ) must be evaluated:

2 W sub = x x i f F sub ( x) dx Essentially, the work done is found by determining the area under the Force vs. Position graph. See Section 7-8 of the text by Halliday, Resnick, and Walker for a three-dimensional analysis of determining the work done by a non-constant force. One example of a non-constant force is a spring force. Springs exert a force that varies in a predictable linear fashion described by Hooke s Law: F x F by the spring = - k x where x is the displacement of the spring from its equilibrium (at rest) position and k represents the force constant of the spring (N/m). Notice the negative sign and the subscript of the force, as it is important to understand the importance of these. Section 7-6 of the text explains the significance of this in detail. The work done by a spring force, due to its linear nature, is rather simple to calculate using the integral above. Computer programs that perform integration calculations do so using various numerical techniques. Data Studio has the ability to estimate the area under a particular plot of data using one of these methods. The Work-Kinetic Energy Theorem describes what happens when a particular force, such as the one supplied by the catapult, does work to cause only the kinetic energy of the object to change. It is written as follows: W by a particular force = K = K f K i This equation, then, would not be valid if this particular force caused another type of energy to change, such as gravitational potential energy or thermal energy. However, the Work-Kinetic Energy Theorem can be applied to all situations if one is very careful to define the work done as the Net (or Total) Work done on the object. This version of the Work-Kinetic Energy Theorem is more versatile: W total = K = K f K i where the Total Work is determined by the sum of the work done by each of the individual forces acting on the object, such as: W total =W by an applied force +W by friction +W by gravity +W by a spring +W by the normal force (etc.) Therefore, if the work done by a particular force appears not to be equal to the change in kinetic energy of the object, then the system should by analyzed for possible work (positive or negative) done by other forces. (Read Section 7-3 for a thorough explanation of the Work-Kinetic Energy Theorem.)

3 The Work - Kinetic Energy Theorem Goals: Determine the Work done by a constant & a non-constant force. Verify the Work-Kinetic Energy Theorem. Determine the Spring Constant, k, of a given spring and use it to calculate the work done by a spring. Equipment List: Data Studio 1. meter track with adjustable feet Dynamics cart with force sensor attached Ultrasonic motion sensor String Pulley Scale balance (for measuring the mass of the cart) Mass hanger and mass set Spring Computer & Equipment Set Up: There are many calculations to be performed in this Lab. Therefore, it will be more efficient to take the time to completely set up Data Studio before starting the lab activities. 1. Measure the total mass of the dynamics cart and the attached force sensor. Record this mass, m.. Set up Data Studio to read the data collected from the force sensor connected to the dynamics cart and the motion sensor located at the end of the 1. meter track. The motion sensor does not need to be calibrated but the force sensor does. Always remember to first remove all tension and then press the TARE button to re-zero the force sensor before data is taken for each trial. 3. Change Sampling Options so that Periodic Samples = 50 Hz. Change the motion detector s Trigger Rate so that it is also 50 Hz. 4. Open the Experiment Calculator (click on the calculate button) and define the calculation for Kinetic Energy, K= ½ mv. 5. Create a graph of Velocity vs. Time. Once this graph is displayed, drag the input icon for position data and drop it on the x-axis so that the graph plots Velocity vs. Position. Next, click and drag Graph1 (under Displays) and drop it on the calculator icon (the kinetic energy calculation under Data) in order to also graph Kinetic Energy vs. Position (remember to drag the position input icon to the x-axis for this graph too, otherwise time is displayed). Click the Statistics button to open the statistics area of the Kinetic Energy vs. Position graph. Set up this area to display the maximum and minimum values of your data.

4 6. Create a graph of Force vs. Time. Once this graph is displayed, click on the input icon for the x-axis data and change it to position so that the graph plots Force vs. Position. Click the Statistics button to open the Statistics area of the graph. Select Area from the Statistics Menu so that the area between the data and the x-axis will be calculated. Force Sensor Motion Detector Dynamics Cart Track 7. Set up the equipment as shown above, for Activity 1, and get ready to take data. Lab Activity 1: Work Done by a Constant Force 1. Press Record and gather data as the cart moves in one direction along the track while being pulled with a constant force by the hanging mass. Be sure that the cart is released from rest. Note its starting position and ending position relative to the motion detector. (Use the yellow measuring tape located on the track.). Note the region of the graphs over which this motion took place. Remember that the x-axis of each graph is Position, not Time. 3. Note the area calculated by the integration function on the Force vs. Position graph over the constant force interval. Answer the following questions: (Hint: Consider the direction of the Force and the Displacement) 1.) Why is this value negative?.) Is the work done on the cart by the Force (due to the hanging mass) positive or negative? Explain. (Realize that you will need to interpret the sign of this value correctly for all further analysis.) 4. Using the Maximum and Minimum information, determine the Change in the Kinetic Energy ( K) of the cart between these values. Record the K. 5. Highlight the region of the Force vs. Position graph over which the cart was being pulled along by the hanging mass. (In other words, remove extraneous data from the integration calculation). Determine and record the value of the Work done by the Force created by the hanging mass. Include the appropriate sign and units. 6. Copy each graph (including the statistics information) into the Word template by clicking on the Display menu and selecting Export Picture. 7. By what % does the Work done by the hanging mass differ from the K of the cart? (Show your calculation.) 8. Recall that the Work Kinetic Energy Theorem (W = K) implies that W refers to the total work done on the cart, not the work done by any particular individual force. Considering all of the forces acting on the cart, why is it reasonable to assume that the work done by the hanging mass is the total work done on the cart? Explain what might account for the % difference calculated above.

5 Lab Activity : Work Done by a Non-Constant Force (ex. a spring) 1. Unhook the hanging mass from the cart and put the string, hanger, and masses away. Carefully hook a spring to the force probe. Do NOT, at any time during the lab, over-stretch this spring!. Hold the cart at rest in front of the motion detector while carefully stretching the spring a short distance down the track away from the motion detector. Keep the far end of the spring stationary throughout the entire collection of data. 3. Press Record and gather data as the cart moves in one direction along the track while being pulled by the spring. Be sure that the cart is released from rest. Make note of its starting position and ending position relative to the motion detector. 4. Note the region of the graphs over which the cart was being pulled by the spring. Remember that the x-axis of each graph is Position, not Time. 5. Using the Maximum and Minimum information, determine the K of the cart over this region. Record the K. 6. Highlight the region of the Force vs. Position graph over which the cart was being pulled along by the spring. (In other words, remove extraneous data from the integration calculation.) Determine and record the value of the Work done by the spring. Include the appropriate sign and units. 7. Copy each graph (including the statistics information) into the Word template by using Paste Special. Paste each as if it were a picture. 8. By what % does the work done by the spring differ from the K of the cart? (Show your calculation.) 9. Recall that the Work Kinetic Energy Theorem (W = K) implies that W refers to the total work done on the cart, not the work done by any particular individual force. Why is it reasonable to assume that the work done by the spring is the total work done on the cart? Explain what might account for the % difference calculated above. Lab Activity 3: Analyzing the Work Done by a Spring using Hooke s Law 1. Using the graphs from Activity, determine the spring constant, k, of the spring used above. Record your value using SI units. Explain how you obtained your answer.. Hooke s Law defines a position variable, x, the stretch/compression of the spring as the position of the end of the spring measured from the spring s rest position. Explain how this variable differs from the position data taken by the motion detector. Explain how the position data taken by the motion detector could be altered to determine the position of the spring described by Hooke s Law. Write a simple formula for x in terms of the positions measured by the motion detector.

6 3. Using your answers to the previous two questions, calculate the work done by the spring force using equation 7-40: (Note the location of the position of the i and f subscripts.) W spring = 1 4. Because this method differs from the one used to calculate the work done by the spring in Activity, compare this value to the K of the cart by, again, determining the % by which the work done by the spring differs from the K of the cart? 5. In Activity, you calculated the work done by the spring using the area under the Force vs. Position graph. In Activity 3, you calculated the work done by the spring by determining k from the Force vs. Position graph, calculating x as defined in Hooke s Law, and using equation Both methods start with the same data and both results are compared to the K of the cart. Which method of calculating the work done by the spring gives a more correct result? Support your answer. kx i 1 kx f

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