Question

If the acceleration of the cylinder of mass $m$ and radius $R$ and that of plank of mass $M$ placed on smooth surface if pulled with a force $F$ as shown in figure is $a=\frac{F}{M+xm}$, find the value of $x$. Given that sufficient friction is present between cylinder and the plank surface to prevent sliding of cylinder.

Answer 1

For Cylinder
$T-f=ma$ .....(i)
$a_P=a_Q$
$\therefore R\alpha -a=a.....\left(ii\right)$
$\alpha =\frac{\tau }{I}=\frac{f\cdot R}{\frac{1}{1}m{R}^2}=\frac{2f}{mR}....\left(iii\right)$
For plank
$F-T-f=Ma......\left(iv\right)$
We have four knowns $f,T,a$ and $\alpha$
Solving the equations we get,
$a=\frac{F}{M+3m}$