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? Justify your answer.
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. This is because angular momentum is conserved in both cases. In Case 3, the object collides and sticks to the bar, so the angular momentum of the object is transferred to the bar. In Case 4, the object bounces off the bar and reverses direction, so the angular momentum of the object is reversed and the bar does not gain any angular momentum. Therefore, the magnitude of the resulting angular momentum of the bar about its center of mass is the same for both cases.
In order to determine the magnitude of the resulting angular momentum of the bar after the collision, we need to consider the conservation of angular momentum.
Angular momentum is given by the equation:
L = Iω
Where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
In both cases, the bars are held on a horizontal surface by sliding a vertical peg through their centers. This means that the bars are able to rotate freely about their central axis.
Initially, before the collision, the bars are at rest and have zero initial angular momentum.
In Case 3, after the object collides and sticks to the bar, the system becomes a single object consisting of two bars and the attached object. The moment of inertia of this system is greater than that of the single bar in Case 4. Therefore, the angular velocity of the system in Case 3 will be smaller compared to the angular velocity of the bar in Case 4, given that they both have the same angular momentum.
Since angular momentum is conserved, and the moment of inertia is greater in Case 3, the resulting angular velocity will be smaller, resulting in a smaller magnitude of angular momentum compared to Case 4.
Therefore, the magnitude of the resulting angular momentum of the bar about its center of mass is greater in Case 4 than in Case 3.
To determine the magnitude of the resulting angular momentum of the bar about its center of mass immediately after the collision, we can use the principle of conservation of angular momentum.
The principle of conservation of angular momentum states that the total angular momentum remains constant unless an external torque is applied. In this case, there is no external torque acting on the bars, so the initial angular momentum of the system must be equal to the final angular momentum.
Let's analyze each case separately:
Case 3: The object collides and sticks to the bar.
In this case, the system (object + bar) initially has zero angular momentum because the object is sliding directly towards the center of mass of the bar. When the object collides and sticks to the bar, it causes the bar to start rotating about its center of mass. Since the total angular momentum is conserved, the angular momentum gained by the bar must be equal in magnitude but opposite in direction to the angular momentum initially possessed by the object. Therefore, the magnitude of the resulting angular momentum of the bar about its center of mass is non-zero and greater than zero in Case 3.
Case 4: The object bounces off the bar and reverses direction.
Similarly, in this case, the system initially has zero angular momentum because the object is sliding directly towards the center of mass of the bar. When the object bounces off the bar and reverses direction, it imparts an angular impulse to the bar, causing it to start rotating about its center of mass. Again, the total angular momentum is conserved, so the magnitude of the resulting angular momentum of the bar about its center of mass will be non-zero and greater than zero in Case 4, just like 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.