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

f_sR=I\dot\omega,\quad a=R\dot\omega

problems.proof.analysis

Static friction at the contact produces torque $f_sR=I\dot\omega$ . The no-slip constraint $a=R\dot\omega$ links rotation to translation.

problems.proof.pitfall. Pitfall: treating static friction as dissipative drag. In rolling without slipping on a stationary surface, the contact point is instantaneously at rest, so static friction does no work.