The rate of change of angular momentum of a system of particles about the centre of mass is equal to the sum of external torque about the centre of mass when the centre of mass is `:`

Options

A. Fixed with respect to an inertial frame.
B. in linear acceleration.
C. in rotational motion.
D. is in a translational motion.

Explanation

To solve the question, we need to understand the relationship between angular momentum, torque, and the center of mass of a system of particles. The question states that the rate of change of angular momentum of a system of particles about the center of mass is equal to the sum of external torque about the center of mass when the center of mass is in a specific condition.

### Step-by-Step Solution:

1. **Understanding Angular Momentum and Torque**:
- The angular momentum \( L \) of a system of particles about a point (in this case, the center of mass) is defined as:
\[ L = r \times p \]
where \( r \) is the position vector from the center of mass to the particle, and \( p \) is the linear momentum of the particle.

2. **Rate of Change of Angular Momentum**:
- The rate of change of angular momentum with respect to time is given by:
\[ \frac{dL}{dt} = \tau_{\text{net}} \]
where \( \tau_{\text{net}} \) is the net external torque acting on the system about the center of mass.

3. **Condition for the Center of Mass**:
- The equation \( \frac{dL}{dt} = \tau_{\text{net}} \) holds true when the center of mass is in a specific frame of reference. This frame is typically an inertial frame, where Newton's laws of motion are valid.

4. **Inertial Frame**:
- An inertial frame is one that is either at rest or moving at a constant velocity. In such frames, the laws of physics, including the conservation of momentum and angular momentum, apply without the influence of fictitious forces.

5. **Conclusion**:
- Therefore, the rate of change of angular momentum of a system of particles about the center of mass is equal to the sum of external torque about the center of mass when the center of mass is fixed with respect to an inertial frame.

### Final Answer:
The rate of change of angular momentum of a system of particles about the center of mass is equal to the sum of external torque about the center of mass when the center of mass is fixed with respect to an inertial frame.

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