The figure shows a rigid assembly of a thin hoop (of mass m = 0.17 kg and radius

Question

The figure shows a rigid assembly of a thin hoop (of mass m = 0.17 kg and radius R = 0.13 m) and a thin radial rod (of length L = 2R and also of mass m = 0.17 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? (4 significant figures )

Explanation / Answer

the moment of inertia of the loop about the axis shown in figure I1 = 0.5*m*R^2 + m(3R)^2 = 19*m*R^2/2

The moment of inertia of rod is I2 = (1/3)*m*L^2 = (1/3)*m*(2R)^2 = (4/3)*m*R^2

tehn moment of inertia isI = I1+I2 = 9.5*m*R^2 + (4/3)*m*R^2

I = 10.83*m*R^2

height of Center of mass = 2R

When the system is upside down,height of center of mass = -2R

deltah = 4R

From law of conservation of energy

2*m*g*deltah = 2*m*g*(4R) = 8mgR

0.5*I*w^2 = 8mgR

w^2= 96*g/(65R) =

w = 10.55 rad/s

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