## What is Angular Momentum?

**Angular momentum is the moment of linear momentum of a body with respect to an axis of rotation.**

Angular momentum of a particle about O is defined as

Where **P** is the linear momentum and **r** is the position vector of the particle from the given point O. The angular momentum of a system of particles is the vector sum of the angular momenta of the particles of the system. So,

** **

Let a particle P of mass m moves at velocity **v**. Its angular momentum about a point O can be written as

Where is the perpendicular distance of the line of motion from O.

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## Proof of L=I*ω

Suppose a particle is going in a circle of radius r, and at some instant, the particle’s speed is v.

** **

The origin may be chosen anywhere on the axis. We choose it at the center of the circle.

Now, angular momentum of particle about centre,

Also, is perpendicular to **r** and **P** and hence is along the axis. So, component of along the axis is mvr itself.

Now, consider a rigid body rotating about an axis AB.

Let the angular velocity of the rigid body be .

Consider the i^{th} particle going in a circle of radius with its plane perpendicular to AB. The linear velocity of this particle at this constant is .

Now, the angular momentum about AB (for considered particle)

Where I is the moment of inertia of the rigid body about axis AB.

## Problems based on angular momentum

### Example 1: Angular momentum of a Rotating disk

**The diameter of a disc is 1 m. It has a mass of 20 kg. It is rotating about its axis with a speed of 120 rotations in one minute. Its angular momentum is is __________.**

**Ans.- **Here, the body rotates about an axis, so we will use .

Here, ; mass = m = 20 kg ;

### Example 2: Particle moving in a circle

**A particle of mass m is moving along a circle of radius r with a time period T. Its angular momentum is _____________.**

**Ans.- **Since the particle is moving along a circle

### Example 3: Angular momentum of a projectile

**A particle is projected at time t = 0 from a point O with a speed u at an angle to horizontal. Find the angular momentum of the particle at time .**

**Ans.- **Velocity of particle at time t is .

Position vector of particle at time t is .

Hence, angular momentum of the particle about the origin,

…(i)

Here,

&

Now putting values of and into equation (i),

.

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