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

mgh=\tfrac12mv^2+\tfrac12I(v/R)^2

problems.proof.analysis

The contact point is instantaneously at rest, so static friction does no work and mechanical energy is conserved. After dropping by $h$ , $mgh=mv^2/2+I(v/R)^2/2$ .

problems.proof.pitfall. Pitfall: assuming friction points uphill simply because it always opposes motion. Here static friction supplies the torque; derive its direction from the rolling constraint.