Angular Momentum

Last updated on May 16th, 2022 at 08:24 pm

When an object moves with an angular speed ω along the circumference of a circular path of radius r, then we say that it has angular momentum. Angular momentum is the rotational equivalent of linear momentum and is defined as the moment of Linear momentum.

It depends on 2 quantities, one is rotational inertia or moment of inertia and the other one is the angular speed of the rotating object.

In an earlier post, we have discussed Linear momentum. Just to brush up, when an object of mass m moves with a speed v in a straight line, we say that it has a linear momentum p=mv. 

Angular momentum is the rotational equivalent of linear momentum. It’s designated with the symbol L. We will use the symbols in this article interchangeably with the name of the quantity under discussion.

Like linear momentum, L is a vector quantity and has direction as well as magnitude.

Angular Momentum formula or equation

The magnitude of L is given by replacing m and v in the expression for linear momentum(p) with their angular counterparts I and ω respectively.

[ I is the moment of inertia or rotational inertia and ω is the angular velocity]

Angular momentum L is defined as the cross product of rotational inertia, I, and angular velocity, ω.

Angular Momentum (L) = Rotational Inertia (I) x Angular Velocity (ω)
(this is the formula or equation of Angular Momentum)

Inertia versus Moment of Inertia | Compare Inertia and Moment of Inertia

Linear momentum or simply momentum is a product of mass and linear velocity.

And you know, this mass is the determining factor of inertia (property of an object to retain its state of motion – when either it’s moving linearly or in static condition), that’s why mass is also called inertia for linear motion.

If we try to find out the equivalent of linear inertia for the rotational motion then we get the notion of Rotational Inertia.

Here comes Newton’s first law which is also called the law of inertia. This law applies to rotating objects as well.

When an object is rotating about an internal axis, the object tends to keep rotating about that axis. Rotating objects tend to continue their rotating condition and non-rotating objects tend to remain non-rotating.

An object resists changes in its rotational motion status. This resistance is called rotational inertia (or the moment of inertia).

The greater an object’s rotational inertia, the more difficult it is to change the rotational speed of the object.

A force is required to change the linear state of motion of an object.

Similarly, torque is required to change the rotational state of motion of an object.

In the absence of a net torque, a rotating object keeps rotating, while a non-rotating one stays non-rotating.
This moment of inertia is designated with the sign I.

Derivation of Angular Momentum formula | Derive the relation between angular momentum and moment of inertia

Angular momentum is the rotational equivalent of linear momentum (p) or in other words, it is the moment of linear momentum.

Angular momentum = L =moment of linear momentum

L= r X p ……………. (1)

Expanding equation (1) we get:

L = r X p = r p sinθ = r mv sinθ = r m ωr sinθ = mr² ω sinθ
= I ω sinθ = I X ω = cross product of moment of inertia and angular velocity.

L = I X ω [ Thus we derived the equation or formula of Angular Momentum]

[ Here, linear momentum = p = mv

linear velocity (v) = ωr where ω is the angular velocity and

mr² represents the moment of inertia ]

Thus we express L as follows:
L = I ω sinθ ………………… (2.1)

And when θ = 90 degrees, L = I ω  ……………. (2.2)

This expression is applicable to any object undergoing angular motion, whether it is a point mass moving in a circle or a rotating disk.

Angular Momentum SI unit

 SI unit of angular momentum: kg.m^2/s

Angular Momentum of a point mass in a circular motion

Now let’s say a point mass m moving in a circle of radius r.
 The moment of inertia varies for different shapes and in this case for a point mass, it is I=mr^2.

In addition to this, the linear speed of the mass is v=ωr. [Ref: our post on linear motion and circular motion – 3 relations]
That means, ω = v/r

Combining these results for a point mass, we find L = I ω =(mr^2) (v/r) = rmv…………….. (3)

Noting that mv is the linear momentum p, we find that the angular momentum of a point mass can be written in the following form:
L =rmv =r p ………………….. (4)

It is important to recall that these 2 expressions (equations 3 and 4) apply specifically to a point particle moving along the circumference of a circle.

Note: Equation 4 represents a specific case of the generic expression of L (equation 1).

Rate of change of Angular Momentum with Time is called Torque

The rate of change in angular momentum with time is also known as torque. Get the detailed derivation of formula etc. here: Torque & Conservation of angular momentum

Related study

Linear Momentum

Numerical problems on torque, angular momentum

Law of conservation of angular momentum