Question

A cotton reel of mass $m$ and moment of inertia $I$ is kept at rest on a smooth horizontal surface. The reel has inner and outer radius $r$ and $R$ respectively. A horizontal force $F$ starts acting as shown in figure. Find the
(a) acceleration of the centre of mass of reel
(b) angular acceleration of the reel
(c) net acceleration of point of contact

Answer 1

(a) From force equation $F=ma\Rightarrow a=\frac{F}{m}....\left(i\right)$
(b) We can apply torque equation about centre of mass. From torque equation
$Fr=I\alpha \Rightarrow \alpha =\frac{Fr}{I}...\left(ii\right)$
(c) acceleration of ${}^{\prime }{P}^{\prime }:{\overrightarrow{a}}_P={\overrightarrow{a}}_{P.O}+{\overrightarrow{a}}_O$
$\Rightarrow a_P=(-\alpha R)+a=a-\alpha R=(a-\frac{Fr}{I})R$
Hence acceleration of point of contact will be $(a-\frac{Fr}{I})R$