Two particle are initially moving with angular momentum `vec(L)_(1)` and `vec(L)_(2)` in a region of space with no external torque. A constant external torque `vec( tau)` then acts on one particle, but not on the other particle, for a time interval `Delta t`. What is the change in the total angular momentum of the two particles ?

Options

A. `vec( DeltaL) = vec( L)_(1) - vec(L) _(2)`
B. `vec( DeltaL) = (1)/(2)(vec( L)_(1) - vec(L) _(2))`
C. `vec(Delta L) = vec( tau) Delta t `
D. `vec( Delta L)` is not applicable for this system

Answer

To solve the problem, we need to analyze the situation using the principles of angular momentum and torque.

### Step-by-Step Solution:

1. **Understand the Initial Conditions**:
- We have two particles with initial angular momentum represented as \(\vec{L}_1\) and \(\vec{L}_2\).
- The total initial angular momentum of the system (both particles) can be expressed as:
\[ \vec{L}_{\text{initial}} = \vec{L}_1 + \vec{L}_2 \]

2. **Identify the External Torque**:
- A constant external torque \(\vec{\tau}\) is applied to one of the particles for a time interval \(\Delta t\).
- The other particle is not affected by any external torque.

3. **Apply the Torque-Angular Momentum Relation**:
- According to the relationship between torque and angular momentum, we have:
\[ \vec{\tau} = \frac{d\vec{L}}{dt} \]
- This implies that the change in angular momentum \(\Delta \vec{L}\) due to the applied torque over the time interval \(\Delta t\) can be expressed as:
\[ \Delta \vec{L} = \vec{\tau} \cdot \Delta t \]

4. **Calculate the Change in Total Angular Momentum**:
- Since the torque is only acting on one particle, the total change in angular momentum of the system (both particles) will be equal to the change in angular momentum of the particle experiencing the torque:
\[ \Delta \vec{L}_{\text{total}} = \Delta \vec{L} = \vec{\tau} \cdot \Delta t \]

5. **Final Result**:
- Therefore, the change in the total angular momentum of the two particles is:
\[ \Delta \vec{L}_{\text{total}} = \vec{\tau} \Delta t \]

### Conclusion:
The change in the total angular momentum of the two particles when a constant external torque acts on one of them for a time interval \(\Delta t\) is given by \(\vec{\tau} \Delta t\).