Question

The figure shows a rigid assembly of a thin hoop (of mass m = 0.27 kg and radius R = 0.17 m) and a thin radial rod (of length L = 2R and also of mass m = 0.27 kg). The assembly is upright, but we nudge it so that it rotates around a horizontal axis in the plane of the rod and hoop, through the lower end of the rod. Assuming that the energy given to the assembly in the nudge is negligible, what is the assembly's angular speed about the rotation axis when it passes through the upside-down (inverted) orientation?

Hoop Rod Rotation axis

Answer 1

Using the parallel axis of theoram

I = mL^2/12 + m*(L/2)^2 + mR^2/2 + m(R+L)^2

given L = 2R

I = 10.83*mR^2

Center of mass of this rigid assembly

where base of rod is (0, 0)

Xcm = 0

Ycm = [m*(L/2) + m*(R + L)]/(m + m)

Ycm = 2R

initial potential energy =MgH = 2m*g*(2R)

final potential energy = MgH = 2m*g*(-2R)

Loss in PE = 2m*g*4R = 8mgR

Using energy conservation

dKE = dPE

0.5*I*w^2 = 8mgR

w = sqrt(16*mgR/(10.83mR^2))

w = sqrt(16*9.81/(10.83*0.17)) = 9.233 rad/sec