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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Answer 1 · 2018-03-20T10:44:09

See below.

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))#