AP Physics 1 Lesson 17 Momentum and Collisions Rotational Kinetic Energy Conservation of mechanical Energy Student Performance Outcomes Use conservation of momentum principles to solve problems with angular elastic and inelastic collisions. Use conservation of energy principles to solve problems with rotational elastic collision principles. Use conservation of energy to solve problems involving transfer of energy Name Date Period In the previous two lessons we have explored similarities between linear and angular motion. We will continue making more comparisons. Engage 1. Derive the equation to calculate angular momentum. Think about the variables in the equation for linear momentum and their angular equivalents. Watch the video linked below: https://www.youtube.com/watch?v=unsyqlfir6o (key words: pirouette from IceTokioHotelFan) 2. How could you explain that the angular velocity of the skater has increased without an apparent external torque being applied? Explain I Notes: Rotational or angular momentum L is the product of an object s moment of inertia and its angular velocity about the center of mass. Explore I 3. Using the rotational inertia apparatus from the last lesson collect data to uncover what happens to the angular momentum of a system when the mass (moment of inertia) is changed. This will be tricky as you will see shortly, but I am confident you will pull it off. ü Attach a mass to the end of the string and allow it to fall towards the ground. ü When the mass hits the ground start your timer to record the time for one revolution. You can also simply give the apparatus a spin, in which case each trial would be like a different set-up. ü As the apparatus is still spinning, drop a steel ball into the cup on each end. Make sure they are dropped simultaneously ü Record the time for one revolution. ü Repeat 5 more times for a total of 6 trials. ü Record your measurements in the data table below. Trial 1 Trial 2 Trial 3 Trial 4 Trial 5 Trial 6 Rotational Inertia Time before drop velocity before Momentum before Rotational Inertia with steel balls Time after drop velocity after Momentum after 4. How does the angular momentum before you dropped the additional steel ball compare to the angular momentum after the drop? 1
5. Are the values reasonably close? What uncertainties can account for any differences? 6. If the angular momentum before and after are equal then the experiment would support what law that now can be extended to rotational motion? Just as we have used the law of conservation of momentum in collisions involving linear motion, we can apply it when objects have angular motion. Watch the following video: https://www.youtube.com/watch?v=kc24c8w0wo0 (keyword: sticks and disc from Luc Dunn) 7. What is the total angular momentum immediately before the collision? (a simple zero or not-zero will do) 8. What is the total angular momentum immediately after the collision? (a simple zero or not-zero will do) 9. Since conservation of angular momentum applies as you have shown in the earlier experiment, do you feel you need to change your answer to qu. 7? Explain II This is where it is getting crazy, any moving object also has angular momentum since we can arbitrarily assign a point (axis of rotation) that this object is rotating about. The graphic below shows a falling object P and it shows its radius from the intersection of the x and y- axis. In this case we can manipulate our equations to show how to calculate the angular momentum of the falling object. L = I ω and since I = m r 2 and ω = v/r we can get L = m v r or simply L = p r 2
Explore II This might be the single most important activity we have done all year. It is very comprehensive as it touches on several important concepts. I can t stress enough how important it is for everybody in your group to completely understand what you are doing and why you are doing it. 10. Watch the video linked here (https://www.youtube.com/watch?v=qamidhwf8u0 keywords: Direct Measurement by Peter Bohacek)) and use the information from the video to demonstrate conservation of angular momentum in this collision. Steps to follow: ü Find the angular momentum of the marble immediately before impact. You need its mass, linear velocity and for radius the distance from the center of the wood block to the impact point. ü Find the angular momentum of the wood block before the collision using its moment of inertia (I) and its angular velocity. ü Find the angular momentum of the marble after the collision. ü Find the angular momentum of the wood block after the collision. ü Compare the sum of the momenta before with the sum of the momenta after the collision. Show your work here: 11. Was angular momentum conserved? 12. Was linear momentum conserved? Show your calculations. 3
Notes: Explain III Finally, the last concept we have to tackle. We are looking at the equivalent of kinetic energy when the object is rotating. We have conveniently neglected this type of energy every time we looked at a rolling object as a ball rolling down an incline. A rolling object doesn t just have linear kinetic energy, but also rotational kinetic energy. Rotational kinetic energy can be calculated using the analog to the translational kinetic energy formula -- all you have to do is replace inertial mass with moment of inertia, and translational velocity with angular velocity! If an object exhibits both translational motion and rotational motion, the total kinetic energy of the object can be found by adding the translational kinetic energy and the rotational kinetic energy: Explore III Roller Derby Lab Set up your table at an incline. Yes, set it up; don t just hold it up. A. Choose a metal ball and bocci ball and release them at the same time from the high end of the table. Use qualitative observations to determine the winner of the race. Repeat several times until you have consistent results and you are confident in your decision. 13. Which ball got to the bottom first? Only consider significant differences to answer this question (why?). 14. From this observation does the mass of the ball or its size (radius) affect its rotational velocity as it rolls down the table? B. Choose the two discs and place the 4 metal balls on the inside in the compartments closest to the center in one disc and in the outer compartments of the second disc. Release them at the same time from the high end of the table. Make qualitative observations. Repeat several times until you have consistent results. 15. Which disc made it to the bottom first? 16. By placing the steel balls in different compartments, what is now different about the discs and what stayed the same? 17. Combining the results from the two set-ups, what affects the rotational velocity of an object that rolls down an incline and what doesn t? 4
Explain IV Notes: Since conservation of energy applies to the rolling object, we can determine the velocity of a rolling object based on its starting gravitational potential energy. This approach is similar to finding the velocity of a falling object using the height it was dropped from. Start with an object initially at rest at the top of the ramp, calculate the final linear velocity at the bottom of the ramp. We will write the moment of inertia in a generalized form for convenience later on: Where A is 1 for a hoop, 1/2 for a cylinder or disk, 3/5 for a hollow sphere and 2/5 for a solid sphere. Now we solve by energy conservation. At the top of the ramp, the object is at rest, so: At the bottom of the ramp, the object is rolling at a final velocity having fallen the full height of the ramp so and is rolling without slipping so: Setting energy at the top and bottom of the ramp equal (assuming no frictional losses) we can solve for final velocity: For a falling object that is not rotating the moment of inertia is zero (A=0) and therefore the denominator is this equation is 1. The equation looks like the one we have been using many times before we considered rotation. 5
Explore IV Purpose: The purpose of this lab is to apply energy conservation to objects that rotate and to review projectile motion> Procedure: 1. Set up a ramp to roll a bocci ball down. The end of the ramp should be flush with the edge of the lab table. The angle the ramp makes with the horizontal should be between 15 and 30 degrees. 2. Record the ball s mass (m) and radius (r), the height of the top of the ramp above the table (h), the angle the ramp makes with the horizontal (θ) (use tan ylx), and the table height (y) in the table below. m (kg) r (m) h (m) Θ (deg) y (m) 3. Derive an equation for the ball s speed at the end of the ramp using the fact that energy is conserved. The potential energy at the top of the ramp will equal the sum of the translational (KT) and rotational kinetic energy (KR) at the bottom of the ramp (assume U=O at bottom of ramp). Use I=2/5 mr 2 for the moment of inertia of a solid sphere. Use your equation to calculate the speed of the ball as it leaves the ramp (Vo). Show your work below. 4. Draw a free body diagram for the ball as it is rolling down the ramp below. 5. Which force is responsible for the rotation of the ball? 6. Since the ball has angular acceleration rolling down the ramp, there must be a net. 6
7. Using your knowledge of projectile motion, predict how far the ball will land from the end of the ramp (x). Hint: since the ball is rolling down the ramp you have to consider both the horizontal and vertical components of its resultant velocity. It does have already an initial vertical velocity. 8.. Place a target on the floor so that the bull s eye is where you predict the ball will land. Roll the ball and mark the target where the ball hits it. Measure the x distance from this mark to the edge of the ramp and record it. Calculate the percent error where your prediction is the theoretical value. 9..Calculate the KT of the ball and KR of the ball at the bottom of the ramp. What percent of the total energy is KR? Explain V 10. How would the target s position change if we used a) a hollow sphere b) empty can c) solid cylinder? Verify your predictions. Rank the four different objects in terms of the distance they traveled horizontally (x) from highest to lowest. Explain this order. Explain VI 1. Revisiting the activity with the marble and the stick. Was this an elastic or inelastic collision? Support your choice with appropriate calculations. 7