The apparatus shown at right consists of two identical spools connected by a mas

Question

The apparatus shown at right consists of two identical spools connected by a massless, inextensible thread that runs over an ideal pulley. The thread is wrapped around spool A many times, but it is attached to a fixed point on spool B, so that spool B will not rotate.

The spools are released from rest from the same height at the same instant.

a. Based on your answers to part a:

? Is the magnitude of the center-of-mass acceleration of spool A (while it is falling) greater than, less than, or equal to that of spool B? Explain.

Is the translational kinetic energy of spool A just before it hits the ground greater than, less than, or equal to that of spool B? Explain.

Is the total kinetic energy of spool A just before it hits the ground greater than, less than, or equal to that of spool B? Explain.

Explanation / Answer

The magnitude of the center of mass acceleration is equal for both spool A and spool B. This can be determined with a simple free body diagram; the net force on both spools is equal, so therefore the acceleration of the center of mass must be equal.

This answer may seem counter intuitive if you think about conservation of energy but I will attempt to clarify.

Due to the net force on both objects being equal, and therefore the acceleration of both objects is equivalent, both A and B will have the same speed when they hit the ground because they accelerated at the same value over the same time interval. Therefore the translational kinetic energy of A and B are equivalent.

Extrapolating from this, spool A must have greater total kinetic energy because in addition to the translational kinetic energy, it also has rotational kinetic energy.

If this doesn't make sense, think about which way the pulley will rotate after the spools are dropped. Is the net external work done on both spools the same?

The short answer being that there is more work done on spool A, and that is why spool A has a greater total energy than spool B; I'll let you think about why though.

For me it is easier to think about net force first, and therefore acceleration, then translational energy, then total energy; but it can be done with work first.

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