Question

A horizontal force is applied on spherical shell placed on rough surface. For what height h of force
applied on shell will always perform pure rolling motion irrespective of magnitude of force applied?
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Answer 1

The spherical shell undergoes pure rotational motion, so the net linear forces acting on the shell is zero.

From FBD we get,

$\sf{\longrightarrow F=f\quad\quad\dots(1)}$

The moment of inertia of the shell,

• $\sf{I=\dfrac{2}{3}\,MR^2\quad\quad\dots(2)}$

where $\sf{M}$ is mass and $\sf{R}$ is radius.

The net torque acting on the shell is given by,

$\sf{\longrightarrow fR+F(h-R)=I\alpha}$

where $\alpha$ is angular acceleration.

From (1) and (2),

$\sf{\longrightarrow f(R+h-R)=\dfrac{2}{3}\,MR^2\alpha}$

Taking $\sf{f=MR\alpha,}$

$\sf{\longrightarrow MRh\alpha=\dfrac{2}{3}\,MR^2\alpha}$

$\sf{\longrightarrow\underline{\underline{h=\dfrac{2}{3}\,R}}}$

So the force $\vec{\sf{F}}$ should act at a height $\sf{\dfrac{2}{3}}$ times the radius of the shell, from the ground.