?The figure shows a rigid assembly of a thin hoop (of mass m = 0.21 kg and radiu

Question

?The figure shows a rigid assembly of a thin hoop (of mass m = 0.21 kg and radius R = 0.15 m) and a thin radial rod (of length L = 2R and also of mass m = 0.21 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

the position of the center of mass

yc = my + my / (m + m)

yc = m * R + m * 3R / (m + m)

yc = 2 * R

then the rotational inertia of the rigid assembly about a horizontal axis in the plane of the rod and hoop through the low end of the rod

I = Irood + Ihoop

I = (1/3) * m * (2R)^2 + ( 0.5 * m * R^2 + m * (3R)^2)

put the value of R = 0.15 m

I = 0.244 m

from the conservation law of mechanical energy

M * g * yc = 0.5 * I * w^2

w = sqrt( 2 * ( m + m) * g * 4 *R / I)

w = sqrt( ( 2 * 2 * 9.8 * 4 * 0.15 / (0.224)) = 9.82 rad/s

angular speed is 9.82 rad/s

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