Conservation of Angular Momentum: Conditions, Applications, and Controversies

Angular momentum, a fundamental concept in physics, plays a crucial role in understanding the motion of rotating objects. This article delves into the conditions under which angular momentum is conserved, the role of external torques, and some controversies surrounding the concept.

Conditions for Conservation of Angular Momentum

The principle of conservation of angular momentum states that the angular momentum of a system remains constant if no external torque acts on it. This principle is central to our understanding of planetary motion, atomic behavior, and the overall dynamics of rotating bodies.

No External Torque, No Change in Angular Momentum

Consider a body rotating about a fixed axis. The body's angular momentum is conserved if there are no external torques acting on it. This means that the angular momentum remains constant unless an external force with a component perpendicular to the radial vector applies torque to the body.

Mathematically, angular momentum (L) is given by:

[L Iomega]

Where (I) is the moment of inertia and (omega) is the angular velocity. If there is no external torque ((tau 0)), the angular momentum remains constant:

[frac{dL}{dt} 0]

Role of External Torque

The presence of a non-zero external torque can lead to a change in the angular momentum. As soon as an external torque, even if it is very small, is applied to the body, the angular momentum will change. This is mathematically expressed as:

[frac{dL}{dt} tau]

This equation shows that the rate of change of angular momentum is directly proportional to the applied torque.

Application to Planetary and Atomic Spheres

The conservation of angular momentum is particularly evident in the motion of planets in the solar system. Planets rotate due to initial angular momentum imparted by the formation of the solar system. Since there is no significant external torque acting on them, they continue to rotate about their axes.

However, at the atomic scale, things are different. Although atoms have a spherical geometry, they do not rotate like planets. Instead, atoms are made up of fields and interact with each other through electric fields. These interactions tend to cancel out any tendency for rotation. As a result, angular momentum is not conserved in the same way as it is for planetary systems.

Controversies and Insights

The principle of conservation of angular momentum can sometimes lead to counterintuitive results. For example, consider a scenario where a spinning ring is dropped onto a stationary ring. According to the conservation law, if the radii of both rings are equal, angular momentum is conserved. This is because the term (r) can be canceled out from the equation.

However, this is not always the case. In a laboratory setting, if the radius of a rotating system changes, angular momentum is no longer conserved simply because of the changing radius. This is a point raised by physicist Friedrich W. Hehl, who argues:

"Why should string lengths matter if you give velocity? What a foolish question."

Hehl's argument highlights a fundamental principle, suggesting that the velocity of an object should not change just because the radius changes. Interestingly, this principle holds true in orbital motion, where changes in velocity due to gravitational forces can be considered.

External Forces and Torques in a Closed System

Angular momentum is not conserved if the sum of the external forces acting on a body is zero but the sum of the external torques is not. This situation often occurs in isolated systems, where the total torque may differ even if the net force is zero.

Example: If a small object is placed on a rotating disk and the object is moved closer to the center, the radius (r) decreases. According to Hehl's principle, angular momentum is not conserved in this scenario.

Mathematically, the conservation of angular momentum only holds if:

[sum tau 0]

When the sum of the external torques is not zero, the angular momentum of the system will change, despite the net force being zero.