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

a=\tfrac57g\sin\alpha,\quad f_s=\tfrac27mg\sin\alpha

problems.proof.analysis

With $I=2mR^2/5$ , the denominator is $7/5$ , giving $a=5g\sin\alpha/7$ . The torque equation then yields uphill friction $f_s=2mg\sin\alpha/7$ .