Question

In the figure shown, the mass of the plank is m and that of the solid cylinder is $8\text{m}$. Springs are light. The plank is slightly displaced from equilibrium and then released. There is no slipping at any contact point. The ratio of mass of the plank and stiffness of the spring i.e. $\frac{m}{k}=\frac{2}{{\pi }^2}$ . The period of small oscillations of the plank (in seconds) is

Answer 1

If the plank is displaced slightly by x towards left, then

$2kx-f_1=m{a}_2$......(i)
$f_1-f_2=8m{a}_1$........(ii)

${\tau }_{cm}=I_{cm}\alpha$
or $(f_1+f_2)R=\frac{8m{R}^2\alpha }{2}$
$(f_1+f_2)=\frac{8m{a}_1}{2}$.....(iii)

From (ii) and (iii),
$2f_1=\left(8m\right)\times \frac{3a_1}{2}$
and $f_2=2m{a}_1$
Hence, $f_2$ will be in forward direction.
Also, for no slipping, at the point of contact between plank and cylinder.
$2a_1=a_2$..........(iv)
$f_1=\left(8m\right)\times \frac{3a_2}{8}=\left(3m\right)a_2$......(v)

From (i) and (v),
$2Kx-\left(3m\right)a_2=m{a}_2$
$a_2=\frac{2K}{4m}\times x$
or ${\omega }^2=\frac{{\pi }^2}{4}$
or $\omega =\frac{\pi }{2}$
or $T=4s$