Question

A solid disk with mass M (1.00 kg) and radius R (0.200 m) is sitting on a frictionless surface. We analyzed the situation at left below, where a force F (2.00 N) is applied for four seconds, by a string that has been wrapped around the outer surface of a cylindrical disk.

Two students are debating the following question. ‘The same force is applied for the same amount of time, but this time by a string attached to the edge of the disk (right Figure above). If both disks start at rest, which one travels 1 m first.” Which student do you agree with, and what would you say to the other student to point out the flaw in their reasoning?

Leo: They have the same acceleration since they have the same mass and same net forces being applied. They both start from rest so their linear motions should be identical as well and they’ll cross at the same time.

Jen: the first will experience a net force and torque , while the second has no torque. The first disk has both linear and rotational kinetic energy . Since the same amount of energy is given to both disks, and for the first disk it needs to be shared between two different forms of kinetic energy, its translational kinetic energy will be smaller. Therefore, its center of mass will be moving slower and it will take longer for this one to reach 1 m.

Answer 1

We agree with Leo, i.e both will travel 1 m at the same time We can say to Jen that force applied is constant for same amount of time in the two cases not the energy supplied. In left case the force will cause a linear acceleration and angular acceleration Since, the surface is frictionless the disk will continue to move with constant velocity and constant angular speed after the force application is completed

While in the right case, since force will cause linear translation only, it will again travel exactly as the previous disc but without any angular speed Also, a disc rolling from inclined plane will need friction to roll and there potential energy is split between kinetic and rotation al kinetic energy. Here, we have not said anything about the energy supplied, rather the force and time are given