Question

A solid cylinder of mass m and radius R is
kept in equilibrium on horizontal rough surface.
Three unstretched springs of spring constant k, 2k,
3k are attached to cylinder as shown in the figure.
Find the period of small oscillations. Given that
surface is rough enough to prevent slipping of
cylinder.

Attached the picture in the replies. Also, the answer is supposed to be 2*pi*sqrt(3m/23k)

Answer 1

What have you written yet?
If you use a dynamics approach, there will be 3 kinds of relations to write:
- kinematic ones to find the relative displacements of moving points of the springs and angular velocity
- Newton's 2nd law to express acceleration
- torque equation to express angular acceleration

If you use an energy approach, you will have to express the KE of the body and the PE of the three springs, instead.

Answer 2

The given answer is correct.

Answer 3

i would do this by taking the torque around the contact point between the cylinder and ground. you have springs pulling it either way. from Hookes Law and for a SMALL angular displacement, you get the following displacements:



at each point you can then calculate a force on the cylinder in terms of K, R and this SMALL rotation x. so for point P, the force from Hooke's Law is k * 2Rx.

this allows you also to calculate a torque about the point of contact between cylinder and ground. so for point P, the torque is k * 2Rx times 2R = 4kR^2 x. acting clockwise.

add up the torques, noting that the one at Q opposes the ones at P and O.

applying Newton's 2nd for rotation, T = I alpha, where alpha is x'' in the drawing above should give an equation in shm relating SMALL roration x to angular acceleration x'', ie the alpha in the law just stated. T = ∑ torques, I = mom of inerita of cylinder ABOUT contact point.

I for the cylinder is 1/2 mR^2 + mR^2 (central axis theorem -- but pls check that) = 3/2 mR^2.

the final equation should be in the classic shm form: x + P x'' = 0. where P will be something reflecting k and m. just plug in the standard shm solution: x = cost wt and w should come out as some funtion of k/m as it does for a simple spring arrangement.

i say SMALL angular displacement because that allows the assumption x = sin x, the same as is made in the pendulum case, and simplifies the trig. that is why we can get the displacements for the springs in the drawing above simply by applying radius * x. in truth, for larger displacements, the springs will depart from the horizontal and start pulling upwards/ pushing downwards.......