A uniform rod of mass m and length l is struck at one end by a force F perpendicular to the rod for a short time interval t . Assuming t is so brief that the rod does not appreciably change its orientation while the force is applied, compute the kinetic energy of the rod after the force has stopped acting.

Step-by-step solution

The linear impulse applied to the rod is J = F t . This impulse provides a translational velocity to the center of mass, given by v = J m = F t m . The angular impulse about the center of mass is the product of the linear impulse and the perpendicular distance from the center of mass, which is L = J l 2 = F t l 2 . The moment of inertia of the uniform rod about its center of mass is I = m l^2 12 . The translational kinetic energy is K_ trans = p^2 2m = (F t)^2 2m = F^2 t^2 2m . The rotational kinetic energy is K_ r