Question

A uniform plank of mass $m=2kg$ free to move in the horizontal direction only, is placed at the top of a solid cylinder of mass $M=2kg$ and radius $R$. The plank is attached to a fixed wall by means of a light spring of spring constant $K=11N/m$. There is no slipping between the cylinder and the plank, and between the cylinder and the ground. The angular frequency $\left(rad/s\right)$ of small oscillation of the system

• A. 2.0
• B. 2.00
• C. 2

Answer 1

Answer: (A), (B), (C)

Total energy of the system,
$E=\frac{1}{2}M{V}^2+\frac{1}{2}I{\omega }^2+\frac{1}{2}m(2V{)}^2+\frac{1}{2}K(2x{)}^2$
$\Rightarrow E=\frac{1}{2}M{V}^2+\frac{1}{2}\left({\frac{MR}{2}}^2\right){\left(\frac{V}{R}\right)}^2+\frac{1}{2}M(2V{)}^2+\frac{1}{2}K(2x{)}^2$
$\Rightarrow E=\frac{11}{4}M{V}^2+2K{x}^2$
$E=\text{constant}\Rightarrow \frac{dE}{dt}=0$
$\therefore \frac{dE}{dt}=\frac{11}{2}MV\frac{dV}{dt}+4Kx\frac{dx}{dt}=0$
$\Rightarrow a=-\frac{8}{11}\frac{K}{M}x$
$\therefore {\omega }^2=\frac{8}{11}\frac{K}{M}=\frac{8}{11}\times \frac{11}{2}=4$
$\Rightarrow \omega =2rad/s$