Question

A sphere and cylinder with the same radius r and mass m are released at the same time to roll without slipping down the same inclined plane. Which one will reach the bottom first?

Answer 1

Answer:Sphere

Explanation:

Given

sphere and cylinder are released  are released at same time to roll without slipping

While rolling acceleration of an object is given by

$a=\dfrac{g\sin \theta }{1+\frac{I}{mr^2}}$

where $\theta$=inclination of Plane

I=moment of Inertia of body

m=mass of object

r=radius of object

Moment of inertia of cylinder is

$I=\frac{mr^2}{2}$

Moment of inertia of sphere is

$I=\frac{2}{5}mr^2$

Suppose they are released from a height h so time taken to reach bottom is given by

$t=\sqrt{\dfrac{2h}{a}}$

thus $t\propto \dfrac{1}{\sqrt{a}}$

acceleration of cylinder is less as compared to sphere because its MOI is high

thus time taken by cylinder is more compared to sphere

Therefore sphere will reach first at bottom