A thin uniform rod of mass M and length L is positioned vertically above an anchored frictionless pivot point, as shown in the figure, and then allowed to fall to the ground. With what speed does the free end of the rod strike the ground?
A.
B.
C.
D.
E.
(GR0177 #26)
Solution:
Conservation of energy of the system before and after the rod falls onto the ground:
Mgy = ½Iω2
y = L/2 (center of the mass of the rod is at the center of the rod)
Moment inertia of uniform rod: I = ¹/₃ML2
v = rω
vf = ωL → ω = vf /L
Mg(L/2) = ½(¹/₃ML2)(vf /L)2
gL = ¹/₃ vf 2
vf = (3gL)1/2
Answer: C
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