Starting from rest at the top of a tall, steep ramp, a sphere rolls without slip

Question

Starting from rest at the top of a tall, steep ramp, a sphere rolls without slipping to the ground 6.9 m below. At the bottom, its speed is 9.0 m/s. Question The sphere is either solid or hollow: which is it? Physics Assume that air resistance is negligible, and that the sphere's total mechanical energy is conserved. The fact that it's rolling without slipping means it has two kinds of kinetic energy; be sure to account for both. It also means there is a definite relationship between its rotational and linear speeds. You don't need to know the sphere's mass or radius, but if you want to, you can use 2.0 kg and 10 cm.

Explanation / Answer

k*m*r^2 =moment of inertia for k=2/3 it is solid

for k=2/5 its hollow

mgh=0.5(mv^2 +I^2)

or,gh=0.5(v^2 +k (r)^2)

or,gh=0.5(81+81K)

or,6.9* 10=0.5(81+81k) (assuming g=10)

or,(138-81)/81=k

or,k=0.7 close to that of solid sphere

hance the sphere is solid.

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