A uniform 4.00kg square solid wooden gate 2.00m on each side hangs vertically from a frictionless pivot at its upper edge. A 1.30kg raven flying horizontally at 4.50m/s flies into this door at its center and bounces back at 2.50m/s in the opposite direction.
What is the angular speed of the gate just after it is struck by the unfortunate raven?
L = Iα = r x mv
p = mv
Moment of inertia of a door = (1/3)MR2
Conservation of angular momentumγ
The Attempt at a Solution
So I know how to get the answer to this using conservation of momentum, but I was wondering if you can combine linear momentum and angular momentum.
For example, initially only the raven is moving which has linear momentum and finally, the raven and the door are moving:
mv = Iα + mv'
Or using only conservation of angular momentum:
(rxmv) = Iα + (rxmv')
In this situation the answers are the same because the raven hits the center of the door which is 1 meters from the axis of rotation (r = 1), but if the door had a different length, would I have to use the second conservation equation? Such as if the length of the door were 4 meters instead, r = 2.