Problem:
A uniform rod AB of length 2l and mass m is rotating in a horizontal plane about a vertical axis through A with angular velocity w. When the midpoint of the rod strikes a fixed nail and is brought immediately to rest, find the impulse exerted by the nail.

Solution:

Introduction:
In this problem, we have a uniform rod AB of length 2l and mass m rotating about a vertical axis through point A with an angular velocity w. The midpoint of the rod, which is point O, strikes a fixed nail and comes to rest immediately. We need to find the impulse exerted by the nail on the rod.

Impulse:
Impulse is defined as the change in momentum of an object. When an object comes to rest, its final momentum becomes zero, and the impulse exerted on it is equal to its initial momentum.

Initial Momentum:
The initial momentum of the rod can be calculated by considering it as a point mass located at its center of mass, which is point O. The velocity of point O is given by the product of angular velocity w and the distance of point O from the axis of rotation, which is l. Therefore, the initial momentum P_initial can be calculated as:
P_initial = m * l * w

Final Momentum:
When the midpoint of the rod strikes the nail and comes to rest, the final momentum of the rod becomes zero.

Impulse Exerted by the Nail:
The impulse exerted by the nail on the rod is equal to the change in momentum, which is the final momentum minus the initial momentum. Therefore, the impulse J can be calculated as:
J = P_final - P_initial
Since the final momentum is zero, we have:
J = -P_initial

Calculating the Impulse:
Substituting the value of initial momentum P_initial, we get:
J = -m * l * w

Conclusion:
The impulse exerted by the nail on the rod is given by -m * l * w. The negative sign indicates that the impulse is in the opposite direction of the initial momentum, which is necessary to bring the rod to rest.