Identical Hollow and Solid Spheres

[cassie123]

Homework Statement

Two spheres look identical and have the same mass. However, one is hollow and the other is solid. Describe an experiment to determine which is which.

Homework Equations

mgh= ½ m v^2 + ½ I ω^2
where I= 2/3 mr2 for a hollow sphere
I=2/5 mr2 for a solid sphere

The Attempt at a Solution

You could allow the two spheres to roll an identical incline from rest. For both spheres, the gravitational potential energy will be transformed to both rotational kinetic energy and translational potential energy when they reach the base.

Since a solid sphere has a smaller moment of inertia, it is less resistant to rotation. More of the original gravitational potential energy will be converted into rotational potential energy for the solid sphere than for the hollow sphere. Thus, the hollow sphere must have more translational kinetic energy and will reach the bottom at a greater translational velocity than the solid sphere will.

Logically I believe that the solid sphere should go faster.. so I am not confident in my logic above.

Could you also argue that the at the moment released from rest the solid sphere will begin to rotate to fall down the incline before the hollow sphere due to the differences in inertia?

 

[andrewkirk]

Since a solid sphere has a smaller moment of inertia, it is less resistant to rotation. More of the original gravitational potential energy will be converted into rotational potential energy for the solid sphere than for the hollow sphere.

Think over this again.

If the solid sphere has a smaller moment of inertia, will its rotational energy be higher or lower than that of the hollow sphere, if they are rolling at the same rate?

Say sphere A will have a lower rotational energy than sphere B when rolling at the same rate, and both have the same mass. Given the same energy input to both, what can we then say about which one must be rolling faster?

 

[cassie123]

Think over this again.

If the solid sphere has a smaller moment of inertia, will its rotational energy be higher or lower than that of the hollow sphere, if they are rolling at the same rate?

Say sphere A will have a lower rotational energy than sphere B when rolling at the same rate, and both have the same mass. Given the same energy input to both, what can we then say about which one must be rolling faster?

Based on the equation for the conservation of energy: if a solid sphere has a smaller moment of inertia it will then have a lower rotational energy than a hollow sphere. So, the solid sphere must have a higher translational energy and reach the bottom at a higher velocity.
Better?

 

[andrewkirk]

Yep!

 

[jbriggs444]

Based on the equation for the conservation of energy: if a solid sphere has a smaller moment of inertia it will then have a lower rotational energy than a hollow sphere. So, the solid sphere must have a higher translational energy and reach the bottom at a higher velocity.
Better?

If the spheres have the same velocity then the one with the higher moment of inertia will have the higher rotational kinetic energy, right?

But they do not have the same velocity. The proposed experiment only works if their velocities are different. Instead, something else is being held constant.