A rod of mass M and length L can rotate about a hinge at its left end and is initially at rest. A putty ball of mass m, moving with speed V, strikes the rod at angle θ from the normal and sticks to the rod after the collision. What is the angular speed ωf of the system immediately after the collision, in terms of system parameters and I?

w_f = m*V*cos (Q_n) / L * (m+M)

Explanation:

Given:

- mass of the putty ball m

- mass of the rod M

- Velocity of the ball V

- Length of the rod L

- Angle the ball makes before colliding with rod Q_n

Find:

What is the angular speed ωf of the system immediately after the collision,

Solution:

- We can either use conservation of angular momentum or conservation of Energy. We will use Conservation of angular momentum of a system:

L_before = L_after

- Initially the rod is at rest, and ball is moving with the velocity V at angle Q from normal to the rod. We know that the component normal to the rod causes angular momentum. Hence,

L_before = L_ball = m*L*V*cos (Q_n)

- After colliding the ball sicks to the rod and both move together with angular speed w_f

L_after = (m+M) * L*v_f

Where, v_f = L*w_f

L_after = (m+M) * L^2 * w_f

- Now equate the two expression as per conservation of angular momentum:

m*L*V*cos (Q_n) = (m+M) * L^2 * w_f

w_f = m*V*cos (Q_n) / L * (m+M)