Question

A uniform thin cylindrical disk of mass $M$ and radius $R$ is attached to two identical massless springs of spring constant $k$ which are fixed to the wall as shown in the figure. The springs are attached to the axle of the disk symmetrically on either side at a distance $d$ from its centre. The axle is massless and both the springs and the axle are in horizontal plane. The unstretched length of each spring is $L$. The disk is initally at its equilibrium position with its centre of mass (CM) at a distance $L$ from the wall. The disk rolls without slipping with velocity $\overrightarrow{V_0}=V_0\hat{i}$. The coefficient of friction is $\mu$.
The net external force acting on the disk when its centre of mass is at development $x$ with respect to its equilibrium position is

• A. $-kx$
• B. $-2kx$
• C. $-\frac{2kx}{3}$
• D. $-\frac{4kx}{3}$

Answer 1

The correct option is D $-\frac{4kx}{3}$

$\frac{1}{2}$kx${}^2$ + $\frac{1}{2}$kx${}^2$ + $\frac{1}{2}$MV${}^2$(1+$\frac{1}{2}$) = constant

Differentiating with time,

2kx$\frac{dx}{dt}$ + $\frac{3}{2}$MV$\frac{dV}{dt}$ = 0

$\frac{dV}{dt}$ = - $\frac{4kx}{3M}$

$\frac{MdV}{dt}$ = -$\frac{4kx}{3}$