Two identical uniform bars are held on a horizontal surface by sliding a vertical peg through their centers as shown above. In both cases, objects of equal mass slide with negligible friction toward the bars with equal speeds. In Case 3, shown above left, the object collides and sticks to the bar, and in Case 4, shown above right, the object bounces off the bar and reverses direction.

Question

Two identical uniform bars are held on a horizontal surface by sliding a vertical peg through their centers as shown above. In both cases, objects of equal mass slide with negligible friction toward the bars with equal speeds. In Case 3, shown above left, the object collides and sticks to the bar, and in Case 4, shown above right, the object bounces off the bar and reverses direction.

- Immediately after the collision, is the magnitude of the resulting angular momentum of the bar about its center of mass greater in Case 3, greater in Case 4, or the same for both? Briefly justify your answer.

Answer 1

To determine whether the magnitude of the resulting angular momentum of the bar about its center of mass is greater in Case 3 or Case 4, we need to consider the conservation of angular momentum.

Angular momentum is the product of moment of inertia and angular velocity. In the given scenario, the bars are identical and stationary, and the object is initially moving towards them with equal speeds. Since the bars are initially not rotating, their angular momenta are zero.

Now let's consider what happens in each case:

Case 3: In this case, the object collides and sticks to the bar. As a result, both the bar and the object start rotating together. The conservation of angular momentum states that the total angular momentum before the collision is equal to the total angular momentum after the collision. Since the object has a non-zero angular momentum, the bar must also acquire an angular momentum in the opposite direction to balance out the total angular momentum. Therefore, the magnitude of the resulting angular momentum of the bar about its center of mass will be greater in Case 3 compared to its initial zero angular momentum.

Case 4: In this case, the object bounces off the bar and reverses its direction. The conservation of angular momentum still applies. After the collision, the object will have an equal but opposite angular momentum compared to before the collision. However, since the bar does not acquire any rotation (as it remains stationary), its angular momentum remains zero.

Therefore, the magnitude of the resulting angular momentum of the bar about its center of mass is greater in Case 3 compared to Case 4.