In a physics laboratory you do the following ballistic pendulum experiment: You shoot a ball of mass m horizontally from a spring gun with a speed v. The hall is immediately caught a distance below a frictionless pivot by a pivoted catcher assembly of mass M. The moment of inertia of this assembly about its rotation axis through the pivot is I. The distance r is much greater than the radius of the ball.
(a) Use conservation of angular momentum to show that the angular speed of the ball and catcher just after the ball is caught is = m/(mr2 + I).
(b) After the ball is caught, the center of mass of the ball-catcher assembly system swings up with a maximum height increase h. Use conservation of energy to show that", =V2(M + m)gh/(mr2 + I).
(c) Your lab partner says that linear momentum is conserved in the collision and derives the expression mv = (m + M)V, where V is the speed of the ball immediately after the collision. She then uses conservation of energy to derive that V = V2ih, so that mv = (m + M) v'2ih. Use the results of parts (a) and (b) to show that this equation is satisfied only for the special case when r
is given by 1 = Mr2.