Conservation of Angular Momentum – Calculation

Angular momentum is an attribute of a rotational system carrying inertia whose axis may or may not go via the system that is under observation. Our planet possesses an orbital angular momentum due to its annual motion around the Sun and a spin angular momentum due to rotation about its axis every 24 hours.

Angular momentum has been categorised as a vector quantity and requires both magnitude and a particular direction to make functional sense. The value of angular momentum is its linear momentum times the perpendicular distance from its centre of rotation to the line about which it is currently rotating.

Angular momentum is a significant physical quantity and is conserved, i.e. the total angular momentum calculated for a closed system is a constant quantity. Bicycles, disks, and cosmos studies have benefited from angular momentum and its conservation principles. Without the knowledge of angular momentum, it would not be possible to understand why different disk shapes rotate differently or why the speed of planets changes at certain points in their orbit.

Conservation of angular momentum

Angular momentum is a substantial quantity; that is, the total angular momentum of a random complex system is the addition of the angular momentum of each particle in the system. In the case of either a constant rigid body or a liquid, the cumulative angular momentum can be found by integrating the angular momentum density over the three dimensions (i.e. angular momentum per unit volume as the volume tends to zero) on the overall body. 

A rotational analogue of Newton’s Third Law of Motion can be phrased as, “When an external torque is applied on a closed system, an equal and opposite torque is exerted on the initial force applicant, which is equal in magnitude but opposite in direction.” 

Therefore, angular momentum can be replaced between bodies within a closed system, that is, a system free of interference from external forces. Its cumulative angular momentum pre- and post-replacement will remain the same (conserved).

You know that angular momentum results from the torque applied to a body, similar to how linear momentum emanates from the applied force. Hence, you can also say that the rotational equivalent of Newton’s first law states, “Unless an external torque is applied, a rigid body will not change its state of current rotation.”

Therefore, with zero external impact, the real angular momentum of this system will continue to be static.

Conservation of angular momentum can also identify if the force applied to a body is a central force. A central force is any force applied to an object to direct it towards a specific point that is the centre of the system.

When a central force is applied to a body, no external torque can be applied to the body, and hence, the angular momentum is conserved. Therefore, if the angular momentum is conserved, there is no torque on the body, and a force is acting on it to direct it towards some point, then that force can be a central force.

There is also the case with cosmic objects where the gravitational forces are put in a way toward the primary object, and orbiting objects conserve angular momentum by replacing the distance and velocity with each other as the objects traverse around the primary. The motion of central force also has its uses in analysing the Bohr Model of an atom.

Calculations of conservation of angular momentum

There are two conditions when angular momentum is applied to an object:

• Point object: The movement of an object about a fixed point.

L = r x p

Where,

• r is the radius 
•  p is the linear momentum.

• Extended object: The object rotates about a fixed point.

L = I x

Where,

• L is the angular momentum.
• I is rotational inertia.
• is the angular velocity.

kg.m2.s-1 is the SI unit of angular momentum.

In classical mechanics, the laws of Newtonian motion can replace the language of angular momentum. It is useful for the motion of celestial bodies like those in the solar system that have motion in central potential; that is, their distance governs their motion to the Sun. It can be said that the orbit a planet will follow is decided by the energy that the planet possesses, its angular momentum and the angle created by the major axis of the orbit with the relative coordinate frame it is in.

Right-hand rule of conservation of angular momentum

The right-hand rule of conservation of angular momentum can be used to find the direction of angular momentum. According to this rule, if a body is rotating in a given direction, then by holding your right hand parallel to the body and curling your fingers in the direction of the body’s rotation, your right thumb will give the direction of the angular momentum of the body. 

Conclusion

Angular momentum refers to the amount of torque required to stop a rotating body in its motion within a unit of angular displacement. It forms the basis of many studies in rotational kinematics and planetary motion.

Angular momentum of a body can be found using the right-hand rule, which can be applied by curling the fingers in the direction of the rotational motion, which would then imply that the thumb now points in the direction of the angular momentum.