This keeps getting rejected as an essay question but it is not an essay question. It is solving for the proofs listed below. I appreciate your help in solving it. Consider a strong but effectively massless cylindrical chamber (i.e. small enough chamber mass that we can ignore it for this problem) with its axis oriented vertically, as in the figure. Inside the chamber are two balls, the diameter of each matching the inner diameter of the chamber. The balls are bouncing up and down inside the chamber with equal speeds but opposite velocities such that they bounce off of one another every time they reach the center of the chamber. The mass of each ball is m and the speed of each is w as seen in the rest frame of the cylinder. Now, suppose that we accelerate the cylinder from rest to a speed v along a horizontal axis. The amount of work it will take to accomplish this acceleration is equal to the energy we are adding to the system. In this case we can write: W = K(v)− K(0) where K(v) is the kinetic energy of the system (that is, of the two bouncing balls) when the cylinder has reached a speed v. a. Prove that this work is given by: W = (γv − 1)Mc2 , where M is independent of v. b. For w=0 show that find that M=2m c. For w>0 show that M>2m. Hint: You know the velocities of the balls in the cylinder’s rest frame. Transform these velocities to the “lab frame” in which the cylinder is moving at a speed v using the velocity addition equations.
This keeps getting rejected as an essay question but it is not an essay question. It is solving for the proofs listed below. I appreciate your help in solving it.
Consider a strong but effectively massless cylindrical chamber (i.e. small enough chamber mass that we can ignore it for this problem) with its axis oriented vertically, as in the figure. Inside the chamber are two balls, the diameter of each matching the inner diameter of the chamber. The balls are bouncing up and down inside the chamber with equal speeds but opposite velocities such that they bounce off of one another every time they reach the center of the chamber. The mass of each ball is m and the speed of each is w as seen in the rest frame of the cylinder. Now, suppose that we accelerate the cylinder from rest to a speed v along a horizontal axis.
The amount of work it will take to accomplish this acceleration is equal to the energy we are adding to the system. In this case we can write:
W = K(v)− K(0)
where K(v) is the kinetic energy of the system (that is, of the two bouncing balls) when the cylinder has reached a speed v.
a. Prove that this work is given by: W = (γv − 1)Mc2 , where M is independent of v.
b. For w=0 show that find that M=2m
c. For w>0 show that M>2m.
Hint: You know the velocities of the balls in the cylinder’s rest frame. Transform these velocities to the “lab frame” in which the cylinder is moving at a speed v using the velocity addition equations.
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