Question

A $2kg$ uniform cylinder is placed on a plank of mass $4kg$ which in turn rests on a smooth horizontal plane. A constant horizontal force of $20N$ is applied on the plank. If no slipping occurs between cylinder & plank, find the ratio of the magnitude of acceleration of the plank and the cylinder.

Answer 1

Suppose acceleration of cylinder w.r.t. plank is $b$ and acceleration of plank is $a$.
As given, there's no slipping,
$b$ = $R\alpha ........\left(1\right)$
Equation of Rotation motion, $f.R=I\alpha$
where, $I_{cylinder}=\frac{M_c{R}^2}{2}$
angular acceleration $\alpha =\frac{b}{R}$
$\Rightarrow f=\frac{M_c{R}^2}{2.R}.\frac{b}{R}=\frac{M_{c}b}{2}..........\left(2\right)$

Linear motion of cylinder
$f=M_c(a-b).............\left(3\right)$
for plank
$20-f=4a.............\left(4\right)$

Putting $M_c=2kg$ in equation $\left(2\right)$, we get
$f=bf=2(a-b)f=2(a-f)$
$\Rightarrow f=\frac{2a}{3}$
using in (4) $20-\frac{2a}{3}=4a\Rightarrow a=\frac{30}{7}{\text{m/s}}^2$
and $\text{b}=\frac{20}{7}{\text{m/s}}^2$

Acceleration of cylinder :
$=a-b=\frac{10}{7}{\text{m/s}}^2$
Therefore : $a_1=\frac{10}{7}{\text{m/s}}^2,a_2=\frac{30}{7}{\text{m/s}}^2$
and $a_2/a_1=3$