Question

A uniform solid sphere of radius $R$ is placed on a smooth horizontal surface. It is pulled by a constant force $F$ acting along the tangent from the highest point. Calculate the distance traveled by the centre of mass of the solid sphere during the time it makes one full revolution

Answer 1

Given: A uniform solid sphere of radius R is placed on a smooth horizontal surface. It is pulled by a constant force F acting along the tangent from the highest point.

To find the distance traveled by the centre of mass of the solid sphere during the time it makes one full revolution

Solution:

Suppose the radius of the sphere be$R$ and the force by which it is pulled be $F$

Now, torque

$\tau =R\times F⟹\tau =I\alpha$

So angular acceleration,

$\alpha =\frac{\tau }{I}=\frac{R\times F}{\frac{2}{3}M{R}^2}⟹\alpha =\frac{3F}{2MR}$

Since $\alpha$ is constant, we have

angular displacement , $\theta =\frac{1}{2}\alpha t^2=2\pi$

or $t^2=\frac{4\pi }{\alpha }$

Also we have,

$s=ut+\frac{1}{2}a{t}^2⟹s=0+\frac{1}{2}a{t}^2⟹s=\frac{1}{2}\times \frac{F}{M}\times \frac{4\pi }{\alpha }⟹s=\frac{F}{2M}\times \frac{4\pi }{\frac{3F}{2MR}}⟹s=\frac{F}{2M}\times \frac{8\pi MR}{3F}⟹s=\frac{4\pi R}{3}$

is the distance traveled by the centre of mass of the solid sphere during the time it makes one full revolution