Problem 1

A uniform solid sphere of mass

m

and radius

R

rolls without slipping from rest down an incline at angle

\alpha

. Its moment of inertia about its center is

I=\tfrac25mR^2

and gravity is

g

. Find the center-of-mass acceleration

a

and speed

v

after descending height

h

. (a) Combine translation, torque, and rolling constraint to find acceleration

a

and static-friction magnitude and direction. (b) Use energy conservation to find speed

v

and angular speed after vertical drop

h

, including rotational energy. (c) Check dependence on

R

and explain why static friction does not dissipate energy in pure rolling.

problems.proof.stepOf

\boxed{a=\tfrac57g\sin\alpha,\ f_s=\tfrac27mg\sin\alpha\text{ uphill},\ v=\sqrt{10gh/7},\ \omega=v/R,\ W_{f_s}=0}

problems.proof.analysis

This is the collected answer to all parts. It obeys the stated assumptions and conventions.