Question

A solid homogeneous cylinder of radius ‘r’ rolls without slipping on the inside of a stationary large cylinder of radius R. Find the equation of motion.

Answer 1

Let the angular velocity of the system about the point os suspension at any time be ‘’
So, vc = (R – r)
Again vc = r1 [where, 1 = rotational velocity of the sphere]
1 =
r
vc = 



  
r
R r  …(1)
By Energy method, Total energy in SHM is constant.
So, mg(R – r)(1 – cos) + (1/2) mvc
2+(1/2) I1
2 = constant
 mg(R – r) (1 – cos) +(1/2) m(R – r)2 2 +(1/2) mr2
2
r
R r


 
2 = constant
 g(R – r) 1 – cos) + (R – r)2 2



 
5
1
2
1
= constant
Taking derivative, g(R – r) sin 
dt
d

10
7 R – r)22
dt
d

 g sin  = 2 ×
10
7
(R – r)
 g sin  =
5
7
(R – r)
  =
7(R r)
5gsin


=
7(R r)
5g





= 2 =
7(R r)
5g


= constant
So the motion is S.H.M. Again  = 
7(R r)
5g

T = 2
5g
7(R  r)