Solution: First consider the bar. Let vd
denote the speed of the center of mass of the bar
and ω
the angular speed of the bar.
The kinetic energy of the bar is then.
Tbar =
(1/2) m vd2
+ (1/2) Izzd
ω2
Izzd is the mass moment of inertia of the bar
about its center of mass and equals
(1/12) mR2
From
kinematics (for rolling) vc
= R ω Also
vd
= vc +
ω k x ( ˗R/2 ) j
So vd = R
ω i +
(R/2) ω i = (3R/2) ω i and vd =
(3R/2) ω
So
the kinetic energy for the bar is Tbar =
(1/2) m [ (3R/2)ω ]2
+ (1/24) mR2
ω2 = (7/6) mR2 ω2
Next
consider the ring. Tring
= (1/2) m vc2 +
(1/2) Izzc ω2 where
Izzc =
mR2
Tring =
(1/2) m(Rω)2 +
(1/2) mR2 ω2 =
mR2 ω2
For
the assembly of the bar and ring:
T = (7/6) mR2 ω2 +
mR2 ω2
= (13/6) mR2
ω2 (result)
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