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?

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1 Answer
Mar 20, 2018

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))