Question

A solid homogeneous cylinder of radius 'a' rolls without slipping inside a stationary hollow cylinder of large radius R. Write the Lagrangian and show that the motion of solid cylinder is simple harmonic.

Answer 1

It is simple harmonic only at small angles.

Answer 2

Oh.......... that i know!!!!!!!!

Answer 3

Have you tried to write the kinetic energy and the potential energy of the solid?

Use θ and dθ/dt to express them where θ is angle between vertical and line joining centre of big cylinder with centre of small cylinder.

Answer 4

\[K.E.= (1/2) m R^2\theta^.2 + (1/2) (I R^2 \theta^.2) /a^2\]
Right??????

Answer 5

Be careful that \(\omega\) of the cylinder is not \(\dot \theta\)
and that radius of path of C is R-a.

Answer 6

i got the answer \[\omega^2 = 2g/3(R-a)\]
thanks. ur picture was really very much helpful. i was confused. but it's clear now

Answer 7

What does this formula represent?
\(\omega\) must depend on \(\dot \theta\) since the small cylinder is rolling inside the big one.

Answer 8

The relation is:

\(a\;\omega=-(R-a)\;\dot\theta\)

Answer 9

yes i did like same and after that i got the final answer 'w'.

Answer 10

I did not realise you meant \(\omega_o=2\pi/T_o\), pulsation of the oscillations.