Question

A thin uniform spherical shell and a uniform solid cylinder of the same mass and radius are allowed to roll down a fixed incline without slipping, starting from rest. The ratio of times take by them to roll down the same distance is

Answer 1

Assume each object has the same mass and the same outside diameter.

ΣTA=IAαΣTA=IAα

I will assume clockwise rotation = positive,

∴mg(sinθ)R=(IA)α∴mg(sinθ)R=(IA)α

but α=aRα=aR

∴mg(sinθ)R=(IA)aR∴mg(sinθ)R=(IA)aR

or

a=mg(sinθ)R2IAa=mg(sinθ)R2IA

but the parallel axis theorm states IA=IC+mR2IA=IC+mR2

a=mg(sinθ)R2IC+mR2a=mg(sinθ)R2IC+mR2

The object with the smallest mass moment of inertia (IC)(IC) will have the largest acceleration down the plane.

Solid Sphere: IC=25mR2IC=25mR2

Solid Cylinder: IC=12mR2IC=12mR2

Thin walled Hollow Sphere: IC=23mR2IC=23mR2

The solid sphere wins the race.