Conservation of Angular Momentum

A physical quantity associated with the system of particles, momentum, is defined as time required to bring the object in motion to rest by application of constant external force. On the basis of types of motion, Momentum is majorly categorized as Linear Momentum and Angular Momentum.

Angular Momentum: Brief

The rotational analog of linear momentum, angular momentum (I) , of a particle comes into play when it moves along the curved path. Mathematically, Linear Momentum is the product of mass (m) and velocity (v) of the particle in linear motion, whereas angular momentum is the vector product of position vector (r) and corresponding linear momentum (P) of the particle in motion.

Linear Momentum P=mv

Angular Momentum I= r × P = I=rp Sinθ

Where r and p, are the magnitude of position vector and linear momentum respectively and is the angle between these two vectors in rotation.

Relationship between Torque applied and Angular Momentum

We know that, Angular Momentum I= r × P

The application of external Torque results in a change in the Angular Momentum. To calculate this change, let us differentiate the mathematical expression of angular momentum.

dIdt= d(rP)dt

By applying the differentiation property of cross product, we get

dIdt= drdtP+ r dPdt ————–(1)

Since, r is the position vector

Therefore, drdt signifies change in displacement and hence is equivalent to velocity v of the particle.

Also, P=mv

Substituting these values in (1) we get

dIdt= vmv+ r dmvdt 

In above equation, in vmv the angle between these two velocity vectors is 0, hence the term vmv becomes 0.

• dIdt= 0+ r dmvdt = r dmvdt

We know that, rate of change of momentum is Force applied, hence

 dmvdt = F

• dIdt = r dmvdt = r F

The cross product of position vector and force applied is Torque.

• dIdt= r F =

Where is Torque applied on the body

• Rate of change of angular momentum is torque applied on the body.

Theoretical and Mathematical explanation of Conservation of Angular Momentum

From the expression derived above, we know that the rate of change of angular momentum is the torque applied on the body.

Thus, angular momentum is conserved if the torque applied on the body is 0.

i.e. dIdt = = 0

The continuous rotation of Earth around the sun can be explained best by the law of conservation of angular momentum.

For a point mass system, the angular momentum is simply calculated as the product of position vector and linear momentum, but for an extended system like planets (say), the angular momentum is calculated by Moment of Inertia (MOI). MOI relates the moving mass of the body, the distance of the body from the centre of the moving system and the angular velocity.

Mathematically, Angular momentum for the extended object can be represented as I= I ×

The mathematical differences of the extended objects and point objects leaves no impact on the theoretical aspects of conservation of momentum. In both cases, as long as there is no net application of external force and torque, the momentum before and after a particular time period remains the same, i.e. remains conserved.

For example, consider a system, wherein a ball is tied to the string and rotated at a particular pace. If we reduce the length of the string, as per the law of conservation of angular momentum, there should be no visible change, as change in mass in this case is not possible. This means the decrease in length of r should compensate with increase in velocity v, in order to conserve the angular momentum.

Let us consider another example of an archer. The target is a cylinder of mass m1 with radius r and an arrow of mass m2. The velocity with which the archer releases the arrow is v1. Initially the target is at rest, hence there is no linear momentum, as velocity is 0. On releasing the arrow, the arrow develops linear momentum and some angular momentum in relation to the rotating axis of the cylinder. On hitting the target, the arrow sticks to the target and becomes one single system, which is arrow and the rolling cylinder, and hence, the net momentum, before and after collision, remains conserved.