Two spools with equal radii and equal masses are wound onto opposite ends of a massless string,
the center of which is then hung over a massless, frictionless pulley, as in the diagram to the right.
The spool on the left is a solid cylinder, while the one on the right is hollow. The two spools are
released from rest at equal heights and are allowed to fall/unwind. The string does not slip across
any of the surfaces it contacts. A little while later, which of the following is observed? [Hint: Force
diagrams and applications of Newton’s 2nd law for both linear and rotational motion may help.]
a. The solid spool is at the same height as the hollow spool and the pulley is rotating clockwise.
b. The solid spool is at the same height as the hollow spool and the pulley is rotating counterclockwise.
c. The solid spool is lower than the hollow spool and the pulley is rotating clockwise.
d. The solid spool is lower than the hollow spool and the pulley is rotating counterclockwise.
e. The solid spool is lower than the hollow spool and the pulley is stationary
To determine the answer, we can analyze the forces and torques acting on the system.
Let's consider the forces acting on each spool. The weight of each spool acts downwards, while the tension in the string acts upwards. Since the spools are released from rest at equal heights, the net force on each spool must be zero according to Newton's second law.
Now, let's consider the torques acting on each spool. In order to rotate, the spools must experience a net torque. Since the spools are wound in opposite directions, the direction of the torque will depend on the rotation direction.
For the solid spool, when it unwinds, the string is unwinding from the high radius to the low radius. This causes the force at the unwinding point to be closer to the rotation axis. Therefore, the torque due to the tension in the string will be counterclockwise.
For the hollow spool, when it unwinds, the string is unwinding from the low radius to the high radius. This causes the force at the unwinding point to be farther from the rotation axis. Therefore, the torque due to the tension in the string will be clockwise.
Since the solid spool experiences a counterclockwise torque and the hollow spool experiences a clockwise torque, the pulley will rotate counterclockwise.
Therefore, the correct answer is:
b. The solid spool is at the same height as the hollow spool and the pulley is rotating counterclockwise.