Short Answer:

Angular momentum is the property of a rotating body that defines the quantity of its rotational motion. It is the moment of momentum of a body in rotational motion and is equal to the product of its moment of inertia and angular velocity.

It is a vector quantity, having both magnitude and direction (along the axis of rotation). The direction of angular momentum is given by the right-hand rule. The SI unit of angular momentum is kilogram meter² per second (kg·m²/s), and it plays an important role in analyzing rotating systems like flywheels, rotors, and turbines.

Detailed Explanation :

Angular Momentum

Angular momentum is a fundamental concept in rotational dynamics that describes the quantity of rotation of a body about a fixed axis. It is the rotational equivalent of linear momentum in linear motion. Just as linear momentum is the product of mass and velocity, angular momentum is the product of moment of inertia and angular velocity.

It represents the rotational state of a body and determines how difficult it is to change its rotational motion. The greater the angular momentum, the harder it is to stop or change the direction of rotation. Angular momentum is an important concept in mechanical engineering, physics, and aerospace applications where rotating or spinning bodies are involved.

Definition

Angular momentum is defined as:

“The moment of momentum of a rotating body about an axis is called angular momentum.”

It is given by the relation:

where,
= Angular momentum (kg·m²/s)
= Moment of inertia (kg·m²)
= Angular velocity (rad/s)

Thus, angular momentum depends on two factors:

1. Moment of inertia (I): It represents the body’s resistance to change in its angular velocity, depending on the distribution of mass about the axis.
2. Angular velocity (ω): It represents how fast the body is rotating around the axis.

Vector Form of Angular Momentum

Angular momentum is a vector quantity, meaning it has both magnitude and direction.
In vector form, it is expressed as:

where,
= position vector of the particle (distance from the axis),
= linear momentum of the particle .

The direction of angular momentum is perpendicular to the plane of rotation and is given by the right-hand rule — if the fingers of your right hand curl in the direction of rotation, the thumb points in the direction of angular momentum.

Units and Dimensions

• SI Unit: kilogram meter² per second (kg·m²/s)
• CGS Unit: gram centimeter² per second (g·cm²/s)
• Dimensional Formula:

Law of Conservation of Angular Momentum

The law of conservation of angular momentum states that:

“If no external torque acts on a system, the total angular momentum of the system remains constant.”

Mathematically,

This means that the product of the moment of inertia and angular velocity remains constant in the absence of external torque:

This principle is very useful in understanding many physical phenomena. For example:

• A figure skater spins faster by pulling her arms close to her body because her moment of inertia decreases, and to conserve angular momentum, her angular velocity increases.
• A rotating flywheel maintains constant speed unless acted upon by external torque.

Relation Between Torque and Angular Momentum

Torque is the rate of change of angular momentum with respect to time.

where,
= torque (N·m),
= angular momentum (kg·m²/s).

This equation shows that just as force changes linear momentum, torque changes angular momentum. If no torque acts, angular momentum remains conserved.

Angular Momentum for Different Cases

1. For a Single Particle:
   The angular momentum of a particle rotating about an axis is:

where  is the angle between the linear velocity and the position vector. For circular motion, , so:

1. For a Rigid Body:
   The total angular momentum is the sum of the angular momenta of all particles:

This is commonly used in the analysis of rotating machines.

Examples of Angular Momentum

1. Spinning Flywheel:
   A flywheel stores rotational energy due to its angular momentum and resists changes in rotational speed.
2. Earth’s Rotation:
   The Earth has constant angular momentum as it spins about its own axis, and due to conservation, its rotation speed remains nearly constant.
3. Figure Skating:
   When a skater pulls their arms inward, their speed of rotation increases to conserve angular momentum.
4. Rotating Machinery:
   Turbines, rotors, and gears use angular momentum principles to maintain motion stability and balance.
5. Planetary Motion:
   Planets orbiting the Sun conserve angular momentum as they move along elliptical paths.

Importance in Mechanical Engineering

• Energy Storage: Flywheels and rotors store energy through angular momentum.
• Motion Control: Helps maintain steady rotation in machines and reduce vibration.
• Gyroscopic Effect: Used in stabilizing vehicles, ships, and aircraft.
• Design of Rotating Parts: Helps engineers analyze torque, power, and resistance in rotating systems.
• Robotics and Space Technology: Used in orientation and motion control of satellites and robotic arms.

Conclusion

Angular momentum is the measure of the quantity of rotation of a body about an axis and is given by the product of moment of inertia and angular velocity. It is a vector quantity that plays a vital role in the dynamics of rotating systems. The law of conservation of angular momentum states that if no external torque acts on a system, its angular momentum remains constant. This concept is fundamental in engineering mechanics, used to analyze rotating machinery, planetary motion, and energy transfer in dynamic systems.