VIEWS: 6 PAGES: 7 POSTED ON: 5/22/2011 Public Domain
Lesson - Atwood's Pulley - Massive Name: _________________________ The applet Atwood simulates the motion of two masses connected by a massless, ideal string which passes over a massive pulley. Prerequisites Students should be familiar with the concepts of potential and kinetic energy. Learning Outcomes Students will become familiar with energy conservation and energy transformation. Instructions Students should know how the applet functions, as described in Help and ShowMe. Many of the step-by- step instructions in the following lesson are to be done in the applet. Contents Calculating Potential Energy and setting a zero-point for Potential Energy Calculating Potential Rotational and Kinetic Energy Total Mechanical Energy of a System and Conservation of Energy Potential Energy and defining a zero-point for Potential Energy In order to assign a numerical value to the potential energy of a body it is first necessary to define a "zero point" or reference level at which the potential energy is considered to be 0 J. You do this on the applet by repositioning the Ep Reference line (moving it up or down) On the applet, position the Ep Reference line at the same height as the center of the pulley as illustrated in Figure 1. Using the mass sliders, set m1 to 250 g m2 to 750 g pulley mass to 400 g Figure 1 Lesson – Atwood’s Pulley – Massive 1 of 7 Are the initial potential energies for mass 1 and mass 2 positive, negative or zero in this case? Hint: are the masses above or below the Ep Reference line? Explain your answer. Play the applet ( ), and produce two graphs showing the potential energy for each mass as a function of time (Do not reset the applet - if you do you will need to reset the Ep Reference line and masses). To view the graphs: 1. select the graph option ( ) after the applet is finished playing 2. select "time" for the horizontal axis 3. select "potential energy - 1" for the vertical axis (this graphs the energy of the mass on the left of the pulley) 4. fit the graph ( ) on the display and sketch it below 5. to display the energy graph for the second mass, change the vertical axis to "potential energy - 2" 6. fit the graph ( ) on the display and sketch it below Graph 1: Potential Energy - 1 vs. Time Graph 2: Potential Energy - 2 vs. Time Explain why the graphs look the way they do. Lesson – Atwood’s Pulley – Massive 2 of 7 Mass 1 climbs by 1.119 m and mass 2 drops by 1.119 m. Calculate the change in potential energy for each mass. Recall that , where Ep is the change in potential energy, m is the mass, g is the acceleration of gravity and h is the change in height. Can you determine the change in potential energy of each mass using only your graphs from Exercise 2? If so, how is this done? (Tip: use the "drag-and-zoom" button ( ) to zoom-in on points of interest on the graph. Alternately, generate a data table using the option menu ( ) and look up the potential energy at the beginning and end of the motion.) Calculate the initial potential energy of the pulley-mass system? Using your graphs from Exercise 2, add the initial potential energy of each mass (Ep1 + Ep2). This represents the initial potential energy of the pulley - mass system. Calculate the final potential energy of the pulley-mass system? Using your graphs from Exercise 2, add the final potential energy of each mass (Ep1 + Ep2). This represents the final potential energy of the pulley - mass system. Is the final potential energy of the system equal to the initial potential energy of the system? Lesson – Atwood’s Pulley – Massive 3 of 7 Calculating Potential, Rotational and Kinetic Energy In the previous example the initial potential energy of the system was greater than the final potential energy (by about 5.56 J). What happened to this energy? Answer: When you press play, the masses begin to accelerate. Mass 1 moves up, mass 2 moves down. A new energy form, kinetic energy, is now being created. Recall, kinetic energy is given by the expression , where m is the mass and v is the velocity. Re-run the applet using the same mass values used in the previous section. Produce kinetic energy - time graphs for each mass and sketch them below. Graph 3: Kinetic Energy - 1 vs. Time Graph 4: Kinetic Energy - 2 vs. Time Using the graphs above, determine the final kinetic energy of each mass. (Tip: use the "drag-and-zoom" button ( ) to zoom-in on points of interest on the graph. Alternately, generate a data table using the option menu ( ) and look up the final kinetic energy.) Ek1final = _________________ Ek2final = ___________________ Add the final kinetic energy of each mass. Has the missing 5.56 J been accounted for? Does the kinetic energy you just measured equal the missing potential energy? Lesson – Atwood’s Pulley – Massive 4 of 7 If you did all of your calculations carefully you will have discovered that there is still some energy missing! We forgot to account for the fact that the pulley has mass and that this mass is spinning. We have encountered a new form of energy - Rotational Energy. On the basis of your calculations, how much rotational energy must be stored in the pulley itself? A spinning mass has rotational energy. The rotational energy of the pulley equal to one half the product of the pulley's moment of inertia and the square of its angular speed. Expressed as an equation: Quantity Symbol SI Unit rotational energy E J moment of inertia I kg.m2 angular speed radian/s The moment of inertia (I) is a measure both of how much mass is spinning and how this mass is distributed around its rotational axis. It is assumed in this applet that the pulley is a uniform density cylinder, ( ), where Mp is the pulley mass and R is the radius of the pulley. Further, it is assumed that the string does not slip on the pulley and therefore, . Show by algebraic manipulation that the rotational energy can be re-written as . Using the applet, produce a rotational energy - time graph for the motion set-up in Exercise 1. Based on the graph or data tables produced by the applet what is the final rotational energy of the pulley? Does this verify your answer to Exercise 11? Lesson – Atwood’s Pulley – Massive 5 of 7 The total energy of the system is the sum of all the kinetic, rotational and potential energy terms at any instant. In Exercise 5, you calculated the initial energy of the system (it is all in the form of potential energy before the masses begin to move). Now, calculate the final energy of the system by adding the final potential energy (calculated in Exercise 6) and the final kinetic energy of each mass. final potential energy of mass 1 and 2 rotational energy of the pulley final kinetic energy of mass 1 final kinetic energy of mass 2 + _______________________ Final total energy Compare the initial and final total energies. What do you notice about these numbers? Total Mechanical Energy of a System and Conservation of Energy In the previous example, you should have noticed that the total energy of the system is constant. This is to say that the total energy before the masses move is identical to the total energy at any point during the movement. The applet Atwood assumes that there is no loss of energy from the system. This means that there is no frictional loss in the pulley and that air resistance on the moving masses can be ignored. It is assumed that energy is conserved. When this happens we can conclude that the total energy of the system is constant. This can be expressed in the following ways: The net change in energy in the system is zero. Energy is total energy = zero neither lost or created The total energy of the system before is equal to total Etotal initial = Etotal final energy of the system after any motion or change. The individual expressions for the energy can change but their sum must be zero. Increases in one term will be offset by decreases in other terms. These are just three ways of stating the Principle of Conservation of Mechanical Energy. Lesson – Atwood’s Pulley – Massive 6 of 7 Set up the applet with the following quantities. m1 = 200 g m2 = 800 g pulley mass = 600 g Place the reference line at the center of the pulley. Play the motion and use the graphing tool and data table option to collect the necessary data required to complete the following table and verify the Principle of Conservation of Mechanical Energy. Before Masses System When t = 0.500 s Released Ep1(J) Ep2(J) Ek1(J) Ek2(J) Erot (J) E total (J) Lesson – Atwood’s Pulley – Massive 7 of 7