Question

A wooden plank of length L and mass M is hanging vertically attached to the ceiling by a frictionless hinge. The plank is in equilibrium, when struck by a bullet of mass m that has velocity given by the equation below. The bullet hits the plank at a distance D (D< L) from the ceiling and remains stuck in the plank.

~v = vxˆi + vyˆj -> velocity of the bullet.

a) What is the angular acceleration of the plank when it makes an
angle θ (θ<90ο) with the ceiling? (Express your answer using
known quantities m, M, D, L, θ, vx, vy and g. Not all may be
necessary.)

b) Assuming that the momentum of the bullet is sufficient to kick the plank all the way to the ceiling, what is the angular
velocity of the plank just before it hits the ceiling? (Express your answer using known quantities m, M, D, L, θ, vx, vy
and g. Not all may be necessary.)

Answer 1

(A) The angular acceleration of plank when it makes an angle with the ceiling which is given as :

For an equilibrium, we have

= I

D F sin = [(1/3) M L²]

where, = making an angle with the ceiling = 90⁰

M = mass of the wooden plank

L = length of wooden plank

D = distance b/w the bullet hits & ceiling

then, we get

D (m g) sin 90⁰ = [(1/3) M L²]

= 3 m g D / M L²

(B) The angular velocity of the plank just before it hits the ceiling which is given as :

we know that, = v / r

where, v = velocity of the bullet = v_x + v_y

then, we get

= (v_x + v_y) / D