7. A stick of length L and mass M is hanging at rest from its top edge from a ceiling hinged at that point so that it is free to rotate. Find the distance from the top of the stick where an impulse, FΔt, is applied such that there is no horizontal component to the force of the hinge on the stick. This point is called the center of percussion.

Let the impulse 'FΔt' acts as a distance 'x' from the hinge 'H'. Assume no impulsive reaction is generated at 'H'. Let the angular velocity of the rod about 'H' just after the applied impulse be 'W'. Also consider that the center of percussion is the point on a bean attached to a pivot where a perpendicular impact will produce no reactive shock at the pivot.

Applying impulse momentum theorem for linear momentum.

FΔt = m(Wl/2), since velocity of center of mass of rod = Wl/2

Similarly applying impulse momentum theorem per angular momentum about H

FΔt * x = I * W

Where FΔt * x represents the impulsive torque and I is the moment of inertia

F Δt.x = (ml² . W)/3

Substituting FΔt

M(Wl/2) * x = (ml². W)/3

1/x = 3/2l

x = 2l/3