Conservation of Angular Momentum with Examples

Angular momentum is an attribute characterised by the rotary inertia of a body or a system of several objects that are moving on an axis that may or may not go via the object or system. Our planet possesses an orbital angular momentum due to its annual motion around the sun and spin angular momentum due to its rotation each day around its axis.

Angular momentum has been categorised as a vector quantity that requires a specified magnitude and a particular description to describe it properly. The quantity of angular momentum of a body is the same as its linear momentum times the perpendicular to the distance from its centre of rotation to the line that is drawn in the same path as its instantaneous motion and passes via the object’s centre of gravity. 

Angular momentum is a significant physical quantity due to it being a conserved quantity, i.e. the total angular momentum calculated for a closed system is always a constant quantity. Angular momentum possesses both direction and magnitude simultaneously, and both values are preserved. Bicycles, disks, and the study of the cosmos have benefited from angular momentum for their working models.

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 momenta that each of the particles that are part of the system has. 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 duplicate of the third law of Newton’s 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 some other matter which is equal in magnitude but opposite in direction.” Therefore, angular momentum can be replaced between bodies within a closed system, which is a system that is free of interference from external forces, then cumulative angular momentum pre- and post-replacement will remain static (conserve).

We know that angular momentum is a result of torque that has been applied to a body, similar to how linear momentum is the result of force being applied to a body. Hence, we can also say that the rotational equivalent of Newton’s first law states can be given as, “Unless an external torque is applied, a rigid body will not undergo a change in its state of uniform rotation.” Therefore, by zero impact externally to take action up to it, the real angular momentum of this system will continue to be static. 

Conservation of angular momentum can also be used as a criterion to identify if the force applied to a body is a central force. A central force is any force that is applied to an object to direct it towards a specific point that is the centre of the system. When central force is applied to a body, no external torque can be applied to the body, and hence, the angular momentum is conserved. Hence, if the angular momentum of a body 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 at all times put in a way towards 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.

Examples of Conservation of Angular Momentum

In quantum mechanics, in the case of a planet, angular momentum is divided into two halves towards the spin of the heavenly body and its revolution about its orbit, and they are replaced by each other on most occasions by different mechanisms.

The conservation of angular momentum within the earth-moon system gives an end product of shifting of angular momentum from the earth towards the moon, because of which there is a tidal torque exerted by the moon towards the earth. This has its actions leading to the decrease of the rate of rotation of the earth, at the rate of 65.7 nanoseconds per day. This also results in the increase of the orbit size of the moon around the earth by around 3.82 cm annually.

The conservation of angular momentum elaborates the angular acceleration of an ice skater once a person begins to get her limbs closer to the vertical axis of rotation. By bringing a segment of the weight of their physique even close to the axis, there is a decrement in the moment of inertia of their body. This is due to the angular momentum being the product of the moment of inertia and the angular velocity. Had the angular momentum been static, then the skater would have increased their angular velocity.

There can be a phenomenon that can be experienced in the same manner, which results in the tremendously fast spin possessed by small stars (white dwarfs, neutron stars, and black holes), while they were created from considerably bigger and slower rotating stars. The shrinking of the size of an entity by n size leads to an increase in its angular velocity by the term of n2.

Right-Hand Rule of Conservation of Angular Momentum

A simple rule can be used to find the direction of angular momentum. This rule is called the right-hand rule of conservation 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 rotation of the body, the thumb of your right hand will give the direction of the angular momentum of the body. 

Conclusion 

Conservation is not, at all times, the complete elaboration of the working of a system, while it is a prominent constraint. In the case of a planet, angular momentum is divided into two halves towards the spin of the heavenly body and its revolution about its orbit, and they are replaced by each other on most occasions by different mechanisms.

The conservation of angular momentum has been in use for the analysis of the motion of central forces. When the cumulative force on some object is put at all times to direct at a point, the centre, further there can be no torque on the body that is right angle to the radius.

Conservation of angular momentum also supports the cause of hurricanes having spirals, and neutron stars have extremely high spin values. Henceforth, conservation limits the possible motion of a system but is highly unlikely to uniquely distinguish it.