Question

The pulley shown in the figure has a moment of inertia I about its axis and mass m. Find the time period of vertical oscillation of its center of mass. The spring has spring constant K and the string does not slip on the pulley?

Answer 1

See below.

Explanation:

Assuming the pulley of uniform material.

`I = 1/2 m r^2`

Pulley dynamics

`m alpha = mg+T_1+T_2`
`I dot omega = r(T_1-T_2)`

then solving

`{(T_1+T_2 = -mg + m alpha),(T_1-T_2 = I/r dot omega):}`

we obtain

`T_1 = 1/2(-mg+m alpha+I/r dot omega)`

but

`T_1 = -k x, alpha = ddot x/2` and `dot omega = ddot x/r` so

`(m/2+m/2) ddot x + 2 k x - mg = 0` or

`m ddot x + 2k x - m g = 0`

now considering the homogeneous differential equation

`m/(2k) ddot x_h + x_h = 0`

obtaining

`Omega = sqrt((2k)/m)` and

period `bbT = (2pi)/Omega = 2pisqrt(m/(2k))`