Proving Angular Momentum Conservation: When External Torque is Zero

In physics, the concept of angular momentum is a fundamental principle that plays a vital role in the analysis of rotational motion. One of the key insights in this field is that the angular momentum of a system remains constant when the external torque applied to the system is zero. This article delves into the proof of this principle, explaining the definitions, theorems, and implications.

Definitions and Assumptions

To begin, let's define some key terms:

Angular Momentum

The total angular momentum (mathbf{L}) of a system of particles can be defined as:

[mathbf{L} sum_{i} mathbf{r}_i times mathbf{p}_i]

where (mathbf{r}_i) is the position vector of the (i)-th particle relative to a chosen origin and (mathbf{p}_i m_i mathbf{v}_i) is its linear momentum.

Torque

The torque (mathbf{tau}) acting on the system is defined as:

[mathbf{tau} frac{dmathbf{L}}{dt}]

where (mathbf{tau}) is the net external torque acting on the system.

Proof: Angular Momentum Conservation

Assumption of Zero External Torque

To prove that the angular momentum remains constant when the external torque is zero, we start with the assumption:

[mathbf{tau}_{external} 0]

Angular Momentum Rate of Change

Based on the definition of torque, if the net external torque is zero, then the rate of change of angular momentum is also zero:

[frac{dmathbf{L}}{dt} mathbf{tau}_{external} 0]

Integrating Both Sides

Since the derivative of angular momentum is zero, we can integrate both sides with respect to time:

[mathbf{L}t mathbf{L}t_0]

for any time (t) after time (t_0). This means that the angular momentum at any time (t) remains constant and is equal to its value at time (t_0).

Conclusion

Therefore, we conclude that when the external torque acting on a system is zero, the angular momentum of the system remains constant over time:

[mathbf{L}t text{constant}]

Implications

This principle is fundamental in physics, particularly for systems where external forces do not act, like in isolated systems. It is crucial for understanding phenomena in mechanics, astronomy, and many areas of physics where rotational motion is involved. Angular momentum conservation provides a clear and potent framework for analyzing and predicting the behavior of rotating objects under various conditions.

Summary

In summary, when the external torque acting on a system is zero, the angular momentum of that system remains constant due to the fundamental relationship between torque and the rate of change of angular momentum.