The figure shows a rigid assembly of a thin hoop (of mass m = 0.16 kg and radius

Question

The figure shows a rigid assembly of a thin hoop (of mass m = 0.16 kg and radius R = 0.17 m) and a thin radial rod (of length L = 2R and also of mass m = 0.16 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?

Explanation / Answer

Here,

mass of hoop = Mh = 0.16 kg
radius of hoop = R = 0.17 m

Length of rod = 2R
Mass of Rod = Mr = 0.16 kg

Moment of inertia of rod :
Ir = M*L^2/3
Ir = (4/3)*Mr^2
Ir = 1.33 * 0.16 * (2*0.17)^2
Ir = 0.02459 kg.m^2

Moment of inertia of Hoop :
Ih = Mh*r^2
Ih = 0.16*0.17^2
Ih = 0.004624 Kg.m^2

Net moment of inertia of system
I = Ir + Ih
I = 0.02459 + 0.004624
I = 0.029214 Kg.m^2

From Conservation of energy :
Rotational Kinetic Energy of system = POtential Energy

0.5 * I * w^2 = mgh

solving for angular momentum w :

w = sqrt(2mgr/I)
w = sqrt(2 *0.16 *0.17 / 0.029214 )
w = 1.364 rad/s

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