A stepped pulley having mass `m` radius of gyration `k` is connected with two ideal springs of stiffness `k_(1)` and `k_(2)` as shown in figure. If the pulley shown in the figure rolls without sliding, find the angular frequency of its oscillation.

Answer

`sqrt((4k_(1) r^(2) + k_(2) (R - r)^(2))/(m(k^(2) + r^(2))))`

Explanation

The tarque about P is
`tau = k_(1) x_(1) 2r + k_(2) x_(2) (R - r) = I_(p) alpha`

But putting `x_(1) = 2 r theta, x_(2) = (R - r) theta`
and` I_(p) mk^(2) + mr^(2)`, we have alpha `= (4k_(1) r^(2) + k_(2) (R - r)^(2))/(m(k^(2) + r^(2)) theta`
`omega = sqrt((4k_(1) r^(2) + k_(2) (R - r)^(2))/(m(k^(2) + r^(2))))`