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

problems.proof.prevStep problems.proof.nextStep

v=\sqrt{10gh/7},\quad\omega=v/R

problems.proof.analysis

Substituting the sphere’s inertia makes total kinetic energy $7mv^2/10$ , hence $v=\sqrt{10gh/7}$ . No slip then gives $\omega=v/R$ , with speed increasing downhill.