Angular Momentum Formula

Angular Momentum Formula

Angular momentum is a fundamental concept in physics that describes the rotational motion of objects. Just as linear momentum measures the quantity of motion in a straight line, angular momentum measures the quantity of rotational motion. Understanding angular momentum provides insight into the dynamics of many systems in nature and engineering, making it an essential topic in the study of physics. Learn more about angular momentum, its definition, formula, and examples in this article prepared by experts at Extramarks.

What is Angular Momentum?

Angular momentum is a measure of the amount of rotation an object has, taking into account its mass, shape, and rotational speed.

Angular momentum is defined as follows:

Any rotating object’s moment of inertia multiplied by its angular velocity.

It is the attribute of a spinning body determined by the product of its moment of inertia and angular velocity. It is a vector quantity, therefore the direction, as well as the magnitude, are taken into account.

Angular Momentum Formula

The angular momentum formula is given for single particle and extended objects. The formula of angular momentum is given as below:

Angular Momentum for Single Particle

Angular momentum can be experienced by a single particle when the object is accelerating around a fixed position. For example, in the case of the earth and the sun, the earth is revolving around the sun in its orbit, where the sun is fixed at its position.

Angular momentum, in that case, is given by the formula:

\( \vec{L} = r \times \vec{p} \)

where,

• • \( \vec{L} \) is Angular Momentum
  • \( r \) is Radius of Rotational Path
  • \( \vec{p} \) is Linear Momentum of Object

Angular Momentum for Extended Object

Angular momentum can be experienced by a point object when the object is rotating about a fixed position. For example, in case of the earth rotating at its axis,.

Angular Momentum, in that case, is given by the formula:

\[\vec{L} = I \times \vec{\omega}\]

where,

• \( \vec{L} \) is Angular Momentum
• \( I \) is Rotational Inertia
• \( \vec{\omega} \) is the angular velocity of Object

Angular momentum of a system of particles

The angular momentum of a system of particles is the vector sum of the individual angular momentum of each particle. The angular momentum of a particle is calculated as l = r×p, where r represents the particle’s distance from the origin. where p is the particle’s linear momentum. The angular momentum of the system with n particles is,

L = l₁ + l₂ + l₃ +…+ l_n

Angular Momentum Quantum Number

The angular momentum quantum number, often denoted अस l is a crucial concept in quantum mechanics that describes the angular momentum of an electron in an atom. This quantum number is part of the set of quantum numbers used to characterize the quantum state of an electron.

The angular momentum quantum number l is associated with the shape of the electron’s orbital, which is the region in space where there is a high probability of finding the electron. It determines the magnitude of the orbital angular momentum of an electron.

Values of l $l$

The angular momentum quantum number $l$ can take on any integer value from 0 up to$n-1$ n−1, where $n$ is the principal quantum number. Mathematically, it is represented as: l = 0, 1, 2,…,n−1

For a given principal quantum number n, there are $n$ possible values of $l$.

Angular Momentum Magnitude

The magnitude of the orbital angular momentum for an electron in a given orbital is given by:

L=√l(l+1)​ℏ

where

$\hbar$ℏ (h-bar) is the reduced Planck constant, ℏ=h/2π

Right Hand Rule

The Right Hand Rule is a mnemonic used to determine the direction of the angular momentum vector in rotational motion. It is a fundamental tool in physics, especially when dealing with angular momentum, torque, and magnetic fields. Here’s how it applies specifically to angular momentum:

Application of the Right Hand Rule

Determining the Direction of Angular Momentum
1. Identify the Rotation Axis: Determine the axis about which the object is rotating.
2. Curl Your Fingers: Point the fingers of your right hand in the direction of the rotation (the direction in which the object is spinning).
3. Thumb Direction: Extend your thumb. The direction in which your thumb points is the direction of the angular momentum vector (\(\vec{L}\)).

Detailed Steps

1. Pointing Fingers in Rotation Direction: Imagine the object rotating around an axis. For example, if a wheel is spinning clockwise when viewed from above, point the fingers of your right hand in the direction the wheel is moving.

2. Thumb Up or Down: Extend your thumb perpendicular to the curl of your fingers. This thumb now points along the axis of rotation. If your fingers curl in a clockwise direction, your thumb points downward. If they curl counterclockwise, your thumb points upward.

Solved Examples on Angular Momentum Formula

Example 1: A 2 kg mass is moving in a circular path with a radius of 3 meters at a constant speed of 4 m/s. Calculate its angular momentum.

Solution:

Given Data:
Mass (m) = 2 kg
Radius (r) = 3 meters
Speed (v) = 4 m/s

Formula for Angular Momentum (L) for a point mass moving in a circle:
\[L = m \cdot v \cdot r\]

Substitute the values:
\[L = 2 \, \text{kg} \times 4 \, \text{m/s} \times 3 \, \text{m}\]

Calculate:
\[L = 24 \, \text{kg} \cdot \text{m}^2/\text{s}\]

Answer: The angular momentum of the mass is \( 24 \, \text{kg} \cdot \text{m}^2/\text{s} \).

Example 2: A solid disk with a mass of 10 kg and a radius of 0.5 meters is rotating about its central axis at an angular velocity of 10 rad/s. Calculate its angular momentum.

Solution:

Given Data:
Mass (m) = 10 kg
Radius (r) = 0.5 meters
Angular velocity (\(\omega\)) = 10 rad/s

Moment of Inertia (I) for a solid disk rotating about its central axis:
\[I = \frac{1}{2} m r^2\]

Substitute the values:
\[I = \frac{1}{2} \times 10 \, \text{kg} \times (0.5 \, \text{m})^2\]

Calculate the moment of inertia:
\[I = \frac{1}{2} \times 10 \times 0.25 = 1.25 \, \text{kg} \cdot \text{m}^2\]

Formula for Angular Momentum (L) for a rotating object:
\[L = I \cdot \omega\]

Substitute the values:
\[L = 1.25 \, \text{kg} \cdot \text{m}^2 \times 10 \, \text{rad/s}\]

Calculate:
\[L = 12.5 \, \text{kg} \cdot \text{m}^2/\text{s}\]

Answer: The angular momentum of the rotating disk is \( 12.5 \, \text{kg} \cdot\text{m}^2/\text{s} \).

Example 3: Calculate the angular momentum of an electron in the second energy level (n=2) of a hydrogen atom.

Solution:

Given Data:
Principal quantum number (n) = 2
The angular momentum quantum number (l) can take integer values from 0 to n-1. For the second energy level, l can be 0 or 1.

Formula for Angular Momentum (L) in quantum mechanics:
\[L = \sqrt{l(l+1)} \hbar\]
where \(\hbar\) is the reduced Planck constant (\(\hbar \approx 1.054 \times 10^{-34} \, \text{Js}\)).

Calculate for \(l = 1\):
\[L = \sqrt{1(1+1)} \hbar = \sqrt{2} \hbar\]

Substitute \(\hbar\):
\[L = \sqrt{2} \times 1.054 \times 10^{-34} \, \text{Js}\]

Calculate:
\[L \approx 1.49 \times 10^{-34} \, \text{Js}\]

Answer: The angular momentum of the electron in the second energy level with \(l = 1\) is approximately \( 1.49 \times 10^{-34} \, \text{Js} \).