Conservation of Angular Momentum

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What is Conservation of Angular Momentum?

It is a very important quantity in Physics that is because it is a conserved quantity that is the total momentum or the total angular momentum of a closed system that remains constant.


In three dimensions the angular momentum that is for a point which is a particle is a pseudovector r × p the product which is a cross product of the particle's position vector r that is relative to some origin vector along with its momentum vector. The latter is denoted as that is  p = mv in mechanics which is Newtonian.

This is the definition that can be applied to each point in continua like fluids and solids or physical fields.


Conservation of Angular Momentum

Unlike the discussed momentum the angular momentum does depend on where the origin is chosen since we know that the particle's position is measured from it. Just like for velocity which is angular velocity there are two special types of angular momentum they are the spin angular momentum and orbital angular momentum. The angular momentum which is the spin angular momentum of an object is defined as the angular momentum about its centre of mass coordinate.


The total angular momentum which is of an object is the sum of the spin that is orbital angular momenta. The angular momentum which is orbital to the vector which is of a point particle is always parallel and directly proportional to the orbital angular velocity vector that is denoted by ω of the particle. where the constant which is of proportionality that really depends on both the mass of the particle and its distance from origin.


Angular Momentum Vector

The spin which is of angular momentum vector of a body which is rigid  is proportional but not always parallel to the spin angular velocity vector that is denoted by Ω that are making the constant of proportionality a second-rank that tensor rather than a scalar.


The Angular momentum is a quantity extensive that is the total angular momentum of any composite system is the sum of the angular momenta which is of its constituent parts. We will discuss that for a continuous rigid body the total momentum or the angular momentum is the volume integral of angular momentum density that is the angular momentum which is per unit volume in the limit as volume shrinks to zero over the entire body.


State Law of Conservation of Angular Momentum

It is the analogue which is rotational of linear momentum that it is denoted by l and angular momentum of a particle which is in rotational motion is defined as:

l = r × p  

This whole thing or the whole process is a cross product of r that is  the radius of the circle which is really formed by the body in rotational motion and p that is the linear momentum of the body. The magnitude which is of a cross product that is of two vectors which is is always the product of their magnitude that multiplied with the sine of the angle which is between them and therefore in the case of angular momentum the magnitude is given by:

l = r p sin θ


The Law of conservation which is of angular momentum that has many applications, including the following:

  • The Electric generators.

  • The Aircraft engines, etc.

To learn more about that to the conservation of momentum or the angular momentum and the other which is related topics too with the help of interactive video lessons.


The Angular momentum which we have earlier talked of a system that is conserved as long as there is no net external torque which is acting on the system. The earth has been rotating on its axis from the time the solar system was formed due to the law of conservation of momentum or we can say the angular momentum.


There are two ways in which we can calculate the angular momentum which is of any object if it is a point object in a rotation then our angular momentum is equal to the times radius that the linear momentum of the object.


If we talk about this whole thing the we  have an extended object which is like our earth for example we can see that the angular momentum is given by moment of inertia that is how much mass is in motions in the object and how far it is from the centre times the angular velocity.


But in both cases which we are discussing as long as there is no net force acting on it. the angular momentum which is before and is equal to angular momentum after some given time and imagine rotating a ball which is tied to a long string the angular momentum.


Now when we somehow are able to decrease the radius of the ball by shortening the string while it is in rotation that the r which is the radius will reduce now according to the law of conservation which is of angular momentum denoted by L should remain the same there is no way for mass to change so therefore:

We  should increase to keep the angular momentum constant so that this is the proof for the conservation which is of angular momentum.

The Angular momentum basically depends on the rotational velocity of an object but also its rotational inertia. When an object changes its shape (rotational inertia), its angular velocity will also change if there is no torque which is external.

An example which is when an ice skater changes and spins her rotation velocity by that are holding her arms outwards or pulling them inwards.


(image will be uploaded soon)


When she pulls her arms in her inertia which is rotational is reduced. Since there is no external net torque which is on the ice skater her angular momentum really remains constant which is because her angular velocity increases the magnitude.

FAQ (Frequently Asked Questions)

Q1. Explain Why Angular Momentum is not Conserved?

Ans: the Angular momentum is also called the  vector which is pointing in the direction that is of the angular velocity. In the same way that the momentum or the linear momentum is always conserved when there is no net force acting that is angular momentum which is conserved when there is no net torque.

Q2. Explain What Kinetic Energy is Conserved in Angular Momentum?

Ans: The energy which is of rotational kinetic energy that is the kinetic energy due to the rotation of an object and is part of its total energy which is kinetic. The law of conservation of angular momentum clearly states that when no external torque that acts on an object that has no change of angular momentum will occur.

Q3. Is the Angular Momentum Said to be Conserved in Orbit?

Ans: The Angular momentum is said to be conserved as long as no net torque is applied to it.  At all points in the orbit of an angular momentum we can say that it  is conserved that too for an elliptical orbit as the letter which denotes r increases the speed must be reduced to compensate for that and vice versa.