Two small spheres of putty, A and B, of equal mass m, hang from the ceiling on massless strings of equal length. Sphere A is raised to a height h 0 as shown below and released. It collides with sphere B (which is initially at rest). The two spheres stick and swing together to a maximum height h f. (assume a perfectly inelastic collision, where there is no internal energy lost to deformation, heating, etc.) Find the height h f in terms of h 0. y x h0h0 hfhf A B B A Lowest point in path is the point of zero gravitational potential

Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7

1. Which of the following physics principles should we use to solve this problem. AA) Conservation of Total Mechanical Energy BB) Conservation of Momentum CC) Newton’s 2nd Law

Choice A Correct, but this is not the only principle that we need to apply. We will need to apply this principle to this problem twice in order to find relations between the initial height and the speed of ball A, and between the speed of the merged balls and the maximum height that the two balls reach together. We will also need to think about the conservation of momentum.

We will need to apply this principle to this problem in order to find a relation between the speed of ball A right before collision and the speed of the two balls together. We will also need to think about the conservation of energy. Choice B Correct, but this is not the only principle that we need to apply.

Newton’s 2nd Law will be of no use to us here. Remember, we are concerned with the speeds of the putty balls, and their initial and final heights. Newton’s 2nd Law will not lead us to useful relations. Choice C Incorrect

2. We want to find the speed at which putty ball A strikes putty ball B. We can find this using the law of conservation of energy. What types of mechanical energy does ball A have initially and just before colliding with ball B (final)? AA) Kinetic Gravitational potential BB) Gravitational potential Kinetic CC) Kinetic & Gravitational potential Gravitational potential Initial EnergyFinal Energy

Putty ball A starts from rest, so it will not have kinetic energy initially. Also, the lowest point of the ball’s motion is considered to be the point of zero gravitational potential (h=0). Choice A Incorrect

This is true because ball A starts from rest and reaches a point of zero gravitational potential with a speed v A. Choice B Correct

Choice C Incorrect Putty ball A starts from rest, so it will not have kinetic energy initially. Also, the lowest point of the balls’ motion is considered to be the point of zero gravitational potential (h=0).

3. Applying the law of conservation of energy to ball A gives which of the following expressions for the speed of ball A at the moment right before it collides with ball B? A)B)C)A)B)C) Initial Final

Choice A Correct Reasoning:

Choice B Incorrect kinetic energy gravitational potential energy

Choice C Incorrect kinetic energy gravitational potential energy

4. In order to find an expression for the speed of the merged balls (v AB ), in terms of the speed of ball A immediately before the collision (v A ), we need to use which conservation principle? AA) Conservation of Momentum BB) Conservation of Total Energy CC) Conservation of Angular Momentum

When we set the momentum before the collision equal to the momentum of the system after the collision, there will not be other variables in our relation between the speeds, because we know that m A =m B =m. Choice A Correct Note: In general, this problem can be solved for spheres with different masses. This problem is a more specific case of colliding pendulums. Note: In general, this problem can be solved for spheres with different masses. This problem is a more specific case of colliding pendulums.

We can relate the two speeds this way, but this expression will involve other variables. Choice B Incorrect

This does not help us with the problem at hand. Choice C Incorrect

5. Which of the following expressions correctly relates the speed of ball A immediately before the collision and the speed of both balls moving together? A)B)C)A)B)C)

Choice A Correct Reasoning: m A =m B =m

Choice B Incorrect P=total linear momentum 0,f subscripts represent before and after collision P=total linear momentum 0,f subscripts represent before and after collision Since m A =m B =m Since m A =m B =m

Choice C Incorrect Since m A =m B =m Since m A =m B =m P=total linear momentum 0,f subscripts represent before and after collision P=total linear momentum 0,f subscripts represent before and after collision

6. Once again, use the law of conservation of energy for the initial moment right before the collision to the final moment where balls A & B reach their maximum height. Which one of the following expressions is correct for h f ? A)B)C)A)B)C)

Choice A Incorrect m A =m B =m

Choice B Incorrect m A =m B =m

Choice C Correct Reasoning:

7. Use the relations that we found in previous questions to find an expression for the maximum height of the two balls together (h f ) in terms of the initial height of ball A (h 0 ). Which of the following expressions is correct? A)B)C)A)B)C)

Choice A Incorrect From our previous expressions: We see that:

Choice B Correct Reasoning: The two putty balls will reach a maximum height that is 1/4th of ball A’s initial height.

Choice C Incorrect From our previous expressions: We see that:

Reflection Questions: Answer this question twice more for balls of differing mass: –