When Is Angular Momentum Not Conserved

When Is Angular Momentum Not Conserved?

Angular momentum, a fundamental concept in physics, describes the rotational motion of an object. It's a vector quantity, meaning it has both magnitude and direction. For many systems, angular momentum is conserved—meaning it remains constant over time. However, there are specific circumstances where this conservation law breaks down. Understanding these exceptions is crucial for a complete grasp of classical and quantum mechanics. This article delves into the instances where angular momentum is not conserved, exploring the underlying physics and providing illustrative examples.

The Conditions for Conservation of Angular Momentum

Before exploring the exceptions, let's briefly review the conditions under which angular momentum is conserved. According to the law of conservation of angular momentum, the total angular momentum of a closed system remains constant if no external torques act upon it. This principle is derived from Newton's second law for rotational motion:

τ = dL/dt

where:

• τ represents the net external torque acting on the system.
• L represents the angular momentum of the system.
• dt represents a change in time.

If the net external torque (τ) is zero, then the change in angular momentum (dL/dt) is also zero, implying that the angular momentum (L) remains constant. This is the crux of the conservation law. A "closed system" implies a system isolated from external influences.

Situations Where Angular Momentum Is Not Conserved

When an external torque acts on a system, the angular momentum is no longer conserved. Let's examine various scenarios where this occurs:

1. External Torques: The Primary Reason

The most straightforward reason for non-conservation of angular momentum is the presence of an external torque. Any interaction with the surroundings that exerts a net torque will cause a change in the system's angular momentum. Consider these examples:

• A spinning top slowing down: As a spinning top slows and eventually falls, friction between the tip and the surface exerts a torque, reducing its angular momentum until it eventually comes to rest. The frictional torque is the external influence disrupting the conservation of angular momentum.

• A skater pulling their arms in: While a figure skater spins, they can change their rotational speed by altering their moment of inertia. When they pull their arms inward, their moment of inertia decreases, leading to an increase in angular velocity to conserve angular momentum. However, if friction from the ice acts on their skates, an external torque is introduced, and the conservation law is violated. The total angular momentum is not exactly preserved due to energy loss via friction.

• A rocket launching: The expulsion of hot gas from a rocket's nozzles produces a torque that accelerates the rocket and changes its angular momentum. The exhaust gases exert an external force and produce a torque on the system.

2. Non-conservative Forces: Friction and Drag

Friction and drag are classic examples of non-conservative forces. They convert mechanical energy into thermal energy, leading to a decrease in angular momentum without a corresponding increase elsewhere in the system.

• A ball rolling on a rough surface: A ball rolling across a rough surface will eventually come to rest due to frictional forces that act as an external torque. The frictional torque steadily reduces the ball's angular momentum.

• A spinning satellite in the Earth's atmosphere: A satellite orbiting Earth encounters atmospheric drag. This drag force creates a torque that slows the satellite's rotation, gradually decreasing its angular momentum. The effect is more pronounced at lower altitudes.

3. Internal Torques and Non-rigid Bodies

While the presence of external torques is the primary cause of non-conservation, it's important to address internal torques within a system. Consider the case of non-rigid bodies:

• A collapsing star: In astrophysics, consider a rotating star undergoing gravitational collapse. Although it might seem like a closed system, internal torques arise due to the redistribution of mass within the star. These torques affect its angular momentum. While the total angular momentum of the system (star plus any ejected material) might be conserved, the angular momentum of the star itself is not conserved.

• A person jumping on a rotating platform: Imagine a person standing on a rotating platform, and suddenly jumps off the platform. The person creates an internal torque in the system. While the total angular momentum of the system (person + platform) may remain constant (ignoring friction), the angular momentum of the platform alone changes.

4. Relativistic Effects

In the realm of relativity, the conservation of angular momentum becomes more nuanced. While angular momentum is conserved in special relativity for closed systems, general relativity introduces complexities, especially in strong gravitational fields.

• Rotating black holes: The angular momentum of a rotating black hole (Kerr black hole) is a significant parameter in its description. However, its interaction with surrounding matter (accretion disks) results in complex exchanges of angular momentum, and the black hole's angular momentum will not necessarily remain constant in such environments.

• Frame-dragging: In general relativity, a rotating massive object drags spacetime around it. This effect, called frame-dragging (or Lense-Thirring effect), influences the angular momentum of other objects in its vicinity. The influence is small but measurable.

5. Quantum Mechanics: Spin and Orbital Angular Momentum

Quantum mechanics introduces further subtleties to the conservation of angular momentum. While the total angular momentum is conserved, the distinction between spin and orbital angular momentum complicates the picture:

• Spin-orbit coupling: In atoms, the interaction between the electron's spin angular momentum and its orbital angular momentum (spin-orbit coupling) can affect the individual angular momenta. While the total angular momentum remains conserved, the individual components may not.

• Measurement and uncertainty: The act of measuring the angular momentum in quantum mechanics can disrupt its conservation. The uncertainty principle dictates that precise measurements of angular momentum in one direction introduce uncertainty in other directions.

Practical Applications and Implications

Understanding when angular momentum is not conserved has crucial implications in various scientific and engineering fields:

• Spacecraft design: Understanding how external torques (from atmospheric drag, solar radiation pressure, gravity gradients) affect spacecraft angular momentum is critical for designing stable and maneuverable spacecraft.

• Robotics: The control of robotic manipulators and other robotic systems requires precise knowledge of angular momentum, and how external and internal torques affect their motion.

• Fluid dynamics: In fluid dynamics, the angular momentum of fluid elements plays a vital role in turbulence and other flow phenomena. Understanding the influences and interactions on angular momentum is essential in this field.

• Nuclear physics: The angular momentum of nuclei and elementary particles is a crucial parameter in nuclear reactions and decay processes.

Conclusion

The conservation of angular momentum is a cornerstone of physics, providing a powerful tool for analyzing a wide range of physical systems. However, this conservation law only applies under specific conditions. The presence of external torques, non-conservative forces, internal torques in non-rigid bodies, relativistic effects, and quantum mechanical subtleties all provide scenarios where angular momentum is not conserved. A comprehensive understanding of these exceptions is crucial for accurate predictions and control in various fields of science and engineering. This detailed understanding allows for both accurate modeling and technological innovation in areas that rely on precise control over rotational motion and energy transfer. The seemingly simple principle of angular momentum conservation has a rich and nuanced application in both the macroscopic and microscopic world.