Rod End
Rod End
Speed at the end of a rod when it reaches the bottom?
A rod of length 2.5m is hinged at one end. It starts from rest (30 degrees above the horizontal) and swings to the bottom of its arc. Find the speed of the END of the rod when it reaches the bottom. The moment of inertia of a rod that pivots about one end is I = 2/3mL.
Let:
I be the moment of inertia,
w be the angular velocity at the bottom,
m be the mass of the rod,
L be the length of the rod,
a be the initial angle above the horizontal,
v be the velocity of the end of the rod,
g be the acceleration due to gravity.
The moment of inertia of a rod about one end is:
I = mL^2 / 3 ...(1)
Equating potential energy at the top to kinetic energy at the bottom:
Iw^2 / 2 = mgL(sin(a) + 1) / 2
Substituting for I from (1):
mL^2 w^2 / 3 = mgL(sin(a) + 1)
Lw^2 / 3 = g (sin(a) + 1)
Substituting v / L for w:
v^2 / 3L = g(sin(a) + 1)
v = sqrt( 3gL(sin(a) + 1) )
= sqrt(3 * 9.81 * 2.5 * 1.5)
= 10.51 m/s.
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Bad Tie Rod End
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