Question

A solid disk and a steel ring have the same mass. When rolled down an incline, why does the disk have a greater translational velocity?

Answer 1

The solid disk have a greater translational velocity & will reach down first because it has a smaller rotational inertia. For the solid sphere, rotational inertia I = (2/5)*MR^2 & for horrow I = (2/3)*MR^2. Where M is mass and R is radius.

The kinetic energy is K = (1/2)*I* ω^2, where ω is the angular velocity, I is the inertia. K is the same for both sphere since they have the same mass and position on the plane and this gives the same potential energy. When the spheres roll, this potential energy is transfered to kinetic energy K.
From the equation of K, a larger "I" gives a smaller "ω", and 'ω' is higher for smaller "I" as K is same. so the solid sphere must reach down first as it has a smaller rotational inertia and this cause the sphere to roll faster than the steel ring as "ω" is higher.