Question

A uniform thin cylindrical disk of mass M and radius R is attaached to two identical massless springs of spring constatn 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 initially at its equilibrium position with its centre of mass (CM) at a distance L from the wall. The disk rolls without slipping with velocity `vecV_0 = vacV_0hati.` The coefficinet of friction is `mu.`

The net external force acting on the disk when its centre of mass is at displacement x with respect to its equilibrium position is.
A. `=-kx`
B. `-2kx`
C. `-2kx//s`
D. `-4kx//3`

Answer 1

Correct Answer - D
(d) When the disc is at a
disctance x from the mean position (equilibrium
position), the forces
acting on the disc are
shown in the figure

`:. -2kx + f =- Ma_c. …(1) Mean where `a_c = acceleration of center positon`
of mass Also the torque acting
on the disc about its center of
mass C is `pi = fxxR = I xx alpha_c`
`:. f= (Ialpha)/(R ) = ((1)/(2)MR^2)/(R ) xx(a_c)/(R )`
`[:. I = (1)/(2)MR^2 and a_c = Ralpha_c for rolling without slipping ]`
`:.f=(1)/(2)Ma_c ...(ii)`
From (i) &(ii)`
`-2kx+(1)/(2)Ma_c = -Ma_c`
`rArr (3)/(2)Ma_c = 2kx rArr Ma_c = (4kx)/(3)`
`rArr` Net external force acting on the disce when its centre of
mass is at displacement x with respect to the equilibrium
position `=(4kx)/(3)` directed towards the euqilibrium.