A uniform 4.00 kg square solid wooden gate 1.50 m on each side hangs vertically

Question

A uniform 4.00 kg square solid wooden gate 1.50 m on each side hangs vertically from a frictionless pivot at its upper edge. A 1.20 kg raven flying horizontally at 5.00 m/s flies into this door at its center and bounces back at 2.50 m/s in the opposite direction.

Part A

What is the angular speed of the gate just after it is struck by the unfortunate raven?

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A uniform 4.00 kg square solid wooden gate 1.50 m on each side hangs vertically from a frictionless pivot at its upper edge. A 1.20 kg raven flying horizontally at 5.00 m/s flies into this door at its center and bounces back at 2.50 m/s in the opposite direction.

Part A

What is the angular speed of the gate just after it is struck by the unfortunate raven?

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Explanation / Answer

mass of gate = Mg = 4 kg

side of the gate = L = 1.50 m

mass of raven = Mr = 1.20 kg

initial velocity of raven = Vi = 5 m/s

final velocity of raven = Vf = - 2.50 m/s

moment of inertia of the gate is given as

I = Mg L2/3 = 4 (1.5)2 /3 = 3 kgm2

Let angular velocity be "w"

Initial momentum of gate = 0

final momentum of gate = I w = 3 w

Initial momentum of raven = Mr Vi (L/2) = 1.2 x 5 x 1.5/2 = 4.5 kgm/s       (since raven hits the center at distance L/2)

final momentum of raven = Mr Vf (L/2) = 1.2 x (-2.50) x 1.5/2 = - 2.25 kgm/s

Using conservation of momentum ::

initial Total momentum = final Total momentum

initial momentum of ( gate   + raven )     = final momentum of (raven + gate )

0 + 4.5 = - 2.25 + 3w

w = 2.25 rad/s

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