We can use conservation of energy to compare the gravitational potential energy at the time of the hill to the rotational and kinetic energy at the bottom of the hill. 

The box would be the fastest as all of the gravitational potential energy would convert to translational energy.

The round objects would share the gravitational potential energy between translational and rotational kinetic energies.

The moment of inertia is equal to a numerical factor () times the mass and radius squared.  Since the mass is the same in each term, the speed does not depend on .

Additionally we can substitute angular speed for translational velocity using the equation

The radius cancels out and we are left with

Therefore the velocity is purely dependent on the numerical factor () in the moment of inertia and the height from which it was released.  Since all of these objects were released from the same height, we can examine the moment of inertia for each to determine which will be the fasters.

Hoop (wedding ring) = 

Hollow cylinder (empty can) = 

Solid cylinder (Battery) = 

Solid sphere (Marble) = 

From this we can see that the marble will reach the bottom at the fastest velocity as it has the smallest numerical factor.  This will be followed by the battery, the empty can and the wedding ring.

We can use the conservation of angular momentum in order to solve this problem.  The law of conservation of angular momentum states that the momentum before the collision must equal to the momentum after the collision.  Angular momentum is calculated with the equation

Before the collision we only have the potter’s wheel rotating.

We know that the moment of inertia of the wheel can be considered as a uniform disk.

We can convert the velocity of the wheel to rad/s

We can now calculate the momentum before the collision.

Now it is time to analyze the momentum after the collision.  At this point we have added a piece of clay which is now moving at the same angular velocity as the pottery.

We know that the moment of inertia of the clay can be considered as a uniform disk.

We know the angular momentum at the beginning equals the angular momentum at the end.

We can now solve for the angular velocity

A wheel can be looked at as a uniform disk.  We can then look up the equation for the moment of inertia of a solid cylinder.The equation is

We can now solve for the moment of inertia.

Since both spheres have the same radius and the same mass, we need to look at the equations for the moment of inertia of a solid sphere and a hollow sphere.

A solid sphere 

A hollow sphere 

If both of these have the same mass and radius, the only difference is the constant that is being multiplied by 

In this case the hollow sphere has a larger constant and therefore would have the larger moment of inertia.

This also conceptually makes sense since all the mass is distributed along the outside of the sphere meaning it all has a larger radius.  A solid sphere has mass that is both close to the center and farther away, meaning that it would have a reduced moment of inertia.