Question

The figure shows a rigid assembly of a thin hoop (of mass m = 0.23 kg and radius R = 0.16 m) and a thin radial rod (of length L = 2R and also of mass m = 0.23 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?

Answer #1

from the paralle axis theorem

I = ( mL^2/`12 + m( L/2)^2) + 0.5 mR^2 + m( L+R)^2

substitue 2R for L

I = ( m( 2R)^2/12 + m( R)^2 ) + 1/2 mR^2 + m(2R+R)^2

= 4mR^2/12 + mR^2 + 0.5 mR^2 + m ( 3R)^2

= 10.834 mR^2

Apply conservation of energy to the thin hoop - rod

Ki+PEi= Kf +PEf

0+ mg ( L/2) + mg (L+R) = 1/2 I w^2 + ( - mgL/2) + ( - mg(L+R)

1/2 I w^2 = mg (L/2) + mg ( L+R) +mgL/2 + mg (L+R)

1/2 I w^2 = mgL + 2mg (L+R)

substutey 2R for L

1/2 I w^2 = mg ( 2R) + 2 mg ( 2R+R)

1/2 I w^2 = 2mgR+ 6 mgR

w= sqrt 16 mgR/I

= sqrt 16 mgR/ 10.834 mR^2

= sqrt 16g/ 10.834 R

= sqrt 16 * 9.8/ 10.834 (0.16)

=9.51 rad/s

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