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Physics Fundamentals BLOG for K12

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Angular Momentum & Moment of Inertia for Rotational Motion

Angular Momentum

In our last post we have discussed on 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. When the same object moves with an angular speed ω along the circumference of a circle of radius r, then we say that it has an angular momentum, L. This L depends on 2 quantities, one is rotational inertia or moment of inertia and the other one is angular velocity. Let’s discuss in details till we get the concept of torque.

Inertia vs 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 its rotating and non-rotating objects tend to remain non-rotating.

An object resists to changes in its rotational motion. 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 a 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.

Equation of Angular Momentum – derivation

The magnitude of angular momentum L is given by replacing m and v in the expression for linear momentum p with their angular analogues I and ω. [ I is the moment of inertia or rotational inertia and ω is the angular velocity]

Derivation of angular momentum formula:

Angular momentum is the rotational analogue of linear momentum(p) or in other words it is the moment of linear momentum.
Angular momentum = l =moment of linear momentum = r X p ……………. (1)

Expanding equation (1) we get:
Angular momentum = l = r X p = r p sinθ = r mv sinθ = r m ωr sinθ = mr^2 ω sinθ = I ω sinθ = I X ω = cross product of rotational inertia and angular velocity.
[ Here, linear momentum = p = mv
linear velocity (v) = ωr where ω is the angular velocity]

Thus we express the angular momentum as follows: L = I ω sinθ ………………… (2.1)
And when θ = 90 degree, L = I ω  ……………. (2.2)
 SI unit of angular momentum:kg.m^2/s

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

Definition of the Angular Momentum

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

angular momentum(L) = rotational inertia (I) X rotational velocity (ω)

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

Angular Momentum of a point mass in 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 (equation 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 angular momentum (equation 1).

Rate of change of Angular Momentum with Time – 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

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