see pic Problem -20P38 A thin wooden rod of length l and mass Mis suspended vert

Question

see pic

Problem -20P38 A thin wooden rod of length l and mass Mis suspended vertically from one end and rotates freely about O, the point of suspension. A bullet of mass m and unknown speed is fired horizontally into the lower end of the rod, where it becomes embedded. Assume that the rod does not move appreciably during the collision. The pendulum with embedded bullet then Swings up to a maximum angle 6, as shown at right. Consider the sys- tem bullet plus rod. (a) Is angular momentum of the sys tem about the point of suspension necessarily conserved in the collision between bullet and rod? Explain. (b) Is linear momentum of the system necessarily conserved in the collision Explain. (c) Is kinetic energy of the system necessarily conserved in the collision Explain. (d) What is the initial speed v of the bullet in terms of the given pa rameters and g, the gravitational field?

Explanation / Answer

a) yes it is since the rod doesnt move there is no external torque
which means angular momentum conservation

b) now since there is an external force which comes from thepin the rod is suspended from

c) It is not since it is an inelastic collision

d) initial angular momentum

M L v = ( I + mr^2) w

M L v = ( 1/3 M L^2 + mL^2) w
w = M V/(1/3 M L + m L)

now conservation of energy

1/2 (I + mr^2) w^2 = M g L/2 ( 1- cos theta) + m g L ( 1- cos theta)

1/2 ( 1/3 M L^2 + m L^2) M^2 v^2/( 1/3 M L + m L)^2 = (M/2 + m) g L ( 1- cos theta)

solve for v
1/2 L M^2 v^2/(1/3 M L + m L) = (M/2 + m) g L ( 1- cos theta)

v = sqrt( 2*(M/2 + m) g ( 1- cos theta)*(1/3 M L + m L)/M^2)

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