Angular momentum

Angular momentum 

Angular momentum is the rotational equivalent of linear momentum. In general, the broader an object’s moment of inertia is, the harder it will be for something to change its rotation speed. Conservation of angular momentum is one of several conservation laws in physics.

For example, the angular momentum comes into play while ice skating as an ice skater opens her legs and arms for less speed. But when she needs higher velocity, she closes her arms and legs. The angular momentum formula is:

L =mvr

L =r*p

L = Angular Momentum

m= Mass of the object

v = linear velocity of the object

r = radius of the object 

p = linear momentum

The angular momentum of a rigid object results from both the speed with which it turns and its mass. An object’s angular momentum can be considered a measurement of how hard it is to stop the object from rotating if applied force tries to stop the rotation. 

Derivation 

Angular momentum is a physical quantity representing the product of a body’s rotational inertia and its angular velocity about a particular axis. It has been derived from some terms that come into action according to its definition. Linear momentum is given as 

P =mv                               (1)

Now, in the next equation, the angular momentum is equal to the moment of inertia and angular speed.

L =I                                (2)

After these two equations, we know that I = mr2 and = v/r for the circular motion. So, we will replace the terms given above in the equations. From this, we will get the final equation,

L = mr2vr

L =mvr

Hence angular momentum is the product of three terms, and these are mass, velocity, and radius of the body.

Dimensional Analysis :

The dimensional analysis of Angular Momentum is [ML2T-1] as it is derived from Mass(M), Velocity(V), and Radius (R). So, the dimensions of the following are :

Mass = [M]

Velocity = [LT-1]

Radius = [L]

So, the dimensions are [ML2T-1]

Angular Momentum Formula :

The formula of angular momentum is taken into account with an object classified into two groups. These groups are taken as :

• Point Object: For a point object, the angular momentum formula will be :

L =r*p

L = Angular Momentum

r = radius between the object and the point around which it revolves

p = linear momentum

• Extended Object:  For an extended object, the angular momentum formula will be :

L =I

L = Angular Momentum

I = Rotational Inertia

= Angular velocity

Right-Hand Rule 

The right-hand rule is most commonly applied to the angular momentum of point objects and rigid bodies, where it simplifies expressions for their momentum and kinetic energies.

The right-hand rule describes the orientation and movement of an object in an inertial reference frame, specifically when it is moving or rotating. Both angular momentum and angular velocity have a right-hand direction of rotation associated with them.

Conditions of Right-Hand Rule 

The right-hand rule is for angular momentum, and the fingers and their curling direction shows us how angular momentum works. 

• When the right hand is taken into a direction, the fingers will give the direction of “r” given in the Angular Momentum Formula.
• Now, when the fingers are curled in a particular direction, then it will show the direction of the linear momentum, which is denoted by “p.”
• The thumb will denote the direction of angular momentum, denoted by “L.”

Angular Momentum Quantum Number 

Angular momentum theory is a significant area in theoretical physics. The angular momentum quantum number describes the size and shape of the orbital.

• The angular momentum quantum number or the azimuthal quantum number is denoted as ℓ. It has a value ranging from 0 to n-1, where n is the principal quantum number. 
• The angular momentum quantum number ℓ determines the shape and orientation of the orbital.
•  For example, for a px orbital, ℓ would have a value of ‘1’, and the shape would be consisting of two lobes with a node positioned at the centre of the lobes. The potential energy in such interaction depends on the distance between particles. However, since this is usually not known, it is customary to quote energies relative to a “defining distance.”

It is an essential quantity in physics because it is a conserved quantity – the angular momentum of a system remains constant unless acted on by an external torque.

Conclusion 

In this article, we learned about angular momentum and the topics associated with it in the syllabus. Besides this, we got an idea about the angular momentum formula and its derivation. We also understood how to use the right-hand formula for calculating the angular momentum.