Question

A body of mass m and negligible size starts from rest and slides down the surface of a frictionless solid sphere of radius R. Prove that the body leaves the sphere when 0 = cos(2/3).

Answer 1

Answer:

$\theta=\cos^{-1}\dfrac{2}{3}$

Explanation:

Given that,

Mass = m

Radius = r

Angle $\theta=\cos\dfrac{2}{3}$

Prove that,

The body leaves the sphere when $\theta=\cos^{-1}\dfrac{2}{3}$

When the particle was contact

$mg\cos\theta=\dfrac{mv^2}{R}$

$Rg\cos\theta=v^2$.....(I)

Using conservation of energy

$mg(R-R\cos\theta)=\dfrac{1}{2}mv^2$

$v^2=2gR(1-\cos\theta)$....(II)

From equation (I) and (II)

$Rg\cos\theta=2gR(1-\cos\theta)$

$\cos\theta=2-2\cos\theta$

$\theta=\cos^{-1}\dfrac{2}{3}$

The body leaves the sphere when $\theta=\cos^{-1}\dfrac{2}{3}$

Hence proved.