Physics homework

Question

In an experiment of rolling motion, similar to the demo video done in class, two objects are rolled down a slope starting at rest and the same position on the slope. Both objects roll without slipping.

The first object is a hoop of radius R_h and mass M_h, with all the mass at the outside of the hoop. The moment of inertia for a hoop is:

I = M_h R_h²

The second object is a sphere of radius R_s and mass M_s, with the mass distributed in some unknown way, but with a moment of inertia:

I = 0.65 M_s R_s²

If v_s is the (center of mass) velocity of the sphere at the bottom of the slope and v_h is the (center of mass) velocity of the hoop at the bottom of the slope, what is the ratio of the velocities?

What is v_s/v_h ?

Answer 1

Start by writing the change in height Δz for each object in terms of its change of P.E., then in terms of its change of translational plus rotational K.E. Note that the angular rotation rate is v/R for either object. After equating the expressions for Δz in terms of K.E. you have a simple equation to solve for the v ratio.