two independent particles are originally moving withthe angular momenta l1 and l2 in a region of space with no external torques.A constant external torque then acts on partcle

When we consider two independent particles moving with angular momenta \( l_1 \) and \( l_2 \), we need to understand how angular momentum behaves, especially when external torques come into play. In a system where no external torques are acting, the total angular momentum of the system remains conserved. However, when a constant external torque is applied to one of the particles, it introduces some interesting dynamics.

Angular Momentum Conservation

Angular momentum (\( L \)) is a vector quantity defined as the product of a particle's moment of inertia and its angular velocity. For a system of particles, the total angular momentum is the vector sum of the angular momenta of each particle:

• For particle 1: \( L_1 = l_1 \)
• For particle 2: \( L_2 = l_2 \)

The total angular momentum of the system is given by:

Total Angular Momentum: \( L_{\text{total}} = L_1 + L_2 = l_1 + l_2 \)

Effect of External Torque

Now, when a constant external torque (\( \tau \)) acts on one of the particles, say particle 1, it will change the angular momentum of that particle over time. The relationship between torque and angular momentum is given by:

Torque Equation: \( \tau = \frac{dL}{dt} \)

This means that the torque applied to a particle will result in a change in its angular momentum. Specifically, for particle 1, we can express this as:

Change in Angular Momentum: \( \frac{dL_1}{dt} = \tau \)

Time Evolution of Angular Momentum

Integrating this equation over time gives us the change in angular momentum for particle 1:

Angular Momentum Over Time: \( L_1(t) = l_1 + \tau t \)

Here, \( l_1 \) is the initial angular momentum, and \( \tau t \) represents the change due to the external torque over time \( t \).

Implications for the System

Since the total angular momentum of the system must still be conserved, any change in the angular momentum of particle 1 must be balanced by a corresponding change in the angular momentum of particle 2. This can be expressed as:

Conservation of Total Angular Momentum: \( L_{\text{total}} = L_1(t) + L_2(t) = \text{constant} \)

Thus, if particle 1's angular momentum increases due to the external torque, particle 2's angular momentum must decrease to keep the total angular momentum constant:

Relationship: \( l_1 + \tau t + L_2(t) = l_1 + l_2 \)

From this, we can derive:

Change in Particle 2's Angular Momentum: \( L_2(t) = l_2 - \tau t \)

Visualizing the Concept

To visualize this, think of a spinning top (particle 1) that you start to push with your hand (the external torque). As you push, the top spins faster, but if you have another top (particle 2) nearby, it will start to slow down slightly due to the conservation of angular momentum in the system.

Final Thoughts

In summary, when a constant external torque acts on one of two independent particles, it alters the angular momentum of that particle while necessitating a compensatory change in the angular momentum of the other particle to conserve the total angular momentum of the system. Understanding these dynamics is crucial in fields ranging from classical mechanics to advanced physics applications.