c. The magnitude of the resulting angular momentum of the bar about its center of mass is the same for both Case 3 and Case 4.
d. The angular momentum of a system is conserved if there are no external torques acting on the system.
(c) Magnitude of Angular Momentum
In both Case 3 and Case 4, the magnitude of the resulting angular momentum of the bar about its center of mass is the same. This is because the initial angular momentum of the system is zero, and the angular momentum is conserved during the collision.
In Case 3, the object sticks to the bar, so the final angular momentum of the system is simply the angular momentum of the bar with the object attached. This can be calculated using the formula:
L = Iω
where:
L is the angular momentum (kg m²/s)
I is the moment of inertia (kg m²)
ω is the angular velocity (rad/s)
The moment of inertia of a uniform bar about its center of mass is given by the formula:
I = (1/12)ML²
where:
M is the mass of the bar (kg)
L is the length of the bar (m)
In Case 4, the object bounces off the bar, so the final angular momentum of the system is the sum of the angular momentum of the bar and the angular momentum of the object. The angular momentum of the object can be calculated using the formula:
L = mvL
where:
L is the angular momentum (kg m²/s)
m is the mass of the object (kg)
v is the velocity of the object (m/s)
L is the distance from the object to the center of mass of the bar (m)
The distance L is equal to half the length of the bar, since the object bounces off the bar at its center.
Since the initial angular momentum of the system is zero, and the angular momentum is conserved during the collision, the final angular momentum of the system in Case 4 is equal to the final angular momentum of the system in Case 3. Therefore, the magnitude of the resulting angular momentum of the bar about its center of mass is the same for both Case 3 and Case 4.
(d) Justification
The angular momentum of a system is conserved if there are no external torques acting on the system. In both Case 3 and Case 4, the only forces acting on the system are the internal forces between the object and the bar. Since internal forces do not produce any net torque on the system, the angular momentum of the system is conserved. Therefore, the magnitude of the resulting angular momentum of the bar about its center of mass is the same for both Case 3 and Case 4.