Oscillation of cylinder attached to two springs

[Abhishekdas]

Oscillation of cylinder attached to two springs...

Homework Statement

A solid uniform cylinder of mass m performs small oscillations due to the action of two springs attached at its topmost point. Spring have spring constant k. Find time period of these oscillations in the absence of sliding...

Homework Equations

R*alpha= accleration of CM.
Time period = 2pi/(w)

The Attempt at a Solution

Let the cylinder be displaced by a small angle θ.
Assuming extension of spring and the arc R*θ to be in the same line (because θ is very small) we can equate them.
So extension of spring in one case and contaraction of spring in the other case = Rθ

Force due to each spring = kRθ so total force is 2kRθ.
Let friction be f be in the direction of spring force.

Now,

2kRθ+f=mR*alpha (pure rolloing so acc of CM = angular acc)...1

Torque equation is

R*(2kRθ - f)= I*alpha

Subtituting I=mR²/2 and simplifying,

we get alpha = 8kθ/3m

So angular frequency w = 8k/3m and hence T = pi*(3m/2k)^1/2

but answer in book is pi/2*(3m/k)^1/2...

So can anyone please point out the mistake? if any...Thank you...

 

[anathema]

Your solution looks correct to me. However, it is always a good idea to double check your calculations and make sure all the units are consistent. Also, the answer given in the book may be in a simplified form, so it would be helpful to show your steps of simplification to see where the difference may lie.

Another thing to consider is the initial conditions of the system. Is the cylinder released from rest? If so, the initial angular velocity would be zero and this may affect the final answer.

Overall, I believe your approach and solution are correct. Keep up the good work!