Angular Momentum

Angular momentum is referred to the property of any rotating object which is given by moment of inertia times angular velocity.

It is described as the property of a rotating body which is further given by the product of the moment of inertia and the angular velocity of the rotating object.

It is a vector quantity, which indicates angular momentum has both the magnitude, as well as the direction.

The angular momentum is a vector quantity and is represented by the symbol L^→

It is represented in the SI units: Kg.m².s^-1

The dimensional formula for angular momentum is given by: [M][L]2[T]-1

Angular Momentum Formula

Angular momentum can be experienced by material in only two situations. They are given below:

• Point object: The object which accelerates around a fixed point. For instance, Earth revolves around the sun where the sun is considered to be the fixed point. Thus, the angular momentum is given by:

L^→ = r×p^→

Where,

L^→ represents the angular velocity

r represents the radius [ that is the distance between the object (example; earth) and the fixed point(example; sun) ]

p^→ represents the linear momentum.

• Extended object: The object, which is rotating about a fixed point or on its own axis. For example, Earth rotates on its axis. Here the angular momentum is given by:

L^→ = I×ω^→

Where,

L^→ represents the angular momentum.

I represents the rotational inertia.

ω^→ represents the angular velocity.

Angular Momentum Quantum Number

Angular momentum quantum number is similar to the Azimuthal quantum number as an angular quantum number also describes the shape and size of an atomic orbital. Its value generally ranges from 0 to 1.

Angular Momentum and Torque

Consider the point mass which is attached to a string, the string is further tied to a point, and now when we exert a torque on the point mass, it will thus start rotating around the center.

The particle having mass m will travel with a perpendicular velocity V which is the velocity that is perpendicular to the radius of the circle and r represents the distance of the particle for the center of its rotation.

The magnitude of L^→

L = r.m.v sin ϕ

= r p⊥

= r.m.v⊥

= r⊥p

= r⊥mv

Φ represents the angle between r^→ and p^→

p⊥ and v⊥ are referred to as the components of p^→ and we know that v^→ is perpendicular to r^→.

Note: In The equation L = r⊥mv the angular momentum of the body only varies when a net torque is applied to it. Therefore, when no torque is applied, then the perpendicular velocity of the body will thus depend upon the radius of the circle that is the distance from the center of mass of the given body to the center of the circle.

Thus, it can be said that;

1. For a shorter radius, the velocity will be high.

2. But, for a higher radius, the velocity will be low.

Right-Hand Rule

The direction of angular momentum is often given by the right-hand rule, which further states that;

 Primarily, position your right hand such that the fingers lie in the direction of r.

 Then curl the fingers around the palm in such a way that they start to point towards the direction of Linear momentum(p).

 The extended thumb will provide us with the direction of angular momentum(L).

Examples of Angular Momentum

Ice-skater: An ice skater generally goes for a spin by keeping her hands and legs far apart from the center of the body. But when she wants more angular velocity to spin, then she keeps her hands and leg closer to her body. Therefore, her angular momentum is conserved, and thus she spins faster.

Gyroscope: A gyroscope commonly uses the principle of angular momentum to sustain its orientation. It uses a spinning wheel that has 3 degrees of freedom. When the wheel is rotated at a very high speed then it locks on to its orientation, and hence won’t deviate from its alignment. This is convenient in space applications where the attitude of a spacecraft is a vital factor that needs to be controlled.

Conservation of Angular Momentum

Angular momentum is referred to the rotational analog of linear momentum, it is represented by the symbol l, and the angular momentum of a particle in rotational motion is defined as follows:

l = r × p

This is a cross product of r which is the radius of the circle and is formed by the object in rotational motion, and p represents the linear momentum of the body. The magnitude of angular momentum is given by,

l = r p sinθ

Conservation of Angular Momentum Applications

The Law of conservation of angular momentum has several applications which further include; Aircraft engines, Electric generators, etc.

Difference Between Angular Momentum and Momentum

Momentum	Angular Momentum
Momentum or linear momentum is referred to as the mass in motion and is useful in measuring the quantity of motion of an object.	Angular momentum is defined as the momentum of rotation and is considered to be the rotational analog of linear momentum.
The SI unit for momentum is represented in kg m/s.	The SI unit for momentum is represented in kg m^2/s.
Momentum is defined as the product of the mass of an object and its velocity.	Angular momentum is defined as the product of the Moment of inertia for mass and its angular velocity.
Momentum is generally conserved when there are no external forces act.	Angular momentum is generally conserved when no net torques are involved.