Question

A tengetial force F acts at the top of a thin spherical shell of mass m and radii R. Find acceleration of the shell if it rolls without slipping.

Answer 1

Let f be the frictional force acting on the shell, opposite to the direction of F.

The net linear force is given by,

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

where a is net linear acceleration.

The net torque is given by,

$\sf{\longrightarrow (F+f)R=I\alpha}$

For a spherical shell, moment of inertia, $\sf{I=\dfrac{2}{3}\,mR^2.}$ Then,

$\sf{\longrightarrow (F+f)R=\dfrac{2}{3}\,mR^2\alpha}$

$\sf{\longrightarrow F+f=\dfrac{2}{3}\,mR\alpha}$

Since $\sf{a=R\alpha,}$

$\sf{\longrightarrow F+f=\dfrac{2}{3}\,ma\quad\quad\dots(2)}$

Adding (1) and (2), we get,

$\sf{\longrightarrow F=\dfrac{5}{6}\,ma}$

$\sf{\longrightarrow\underline{\underline{a=\dfrac{6F}{5m}}}}$

Answer 2

Explanation:

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