Short Answer:

The conservation of angular momentum is a fundamental principle in mechanics which states that if no external torque acts on a rotating body or system, the total angular momentum remains constant. This means that the angular momentum before and after any event (like rotation, collision, or motion) will be the same, provided no external torque is applied.

In simple words, the conservation of angular momentum means that a rotating object keeps spinning with the same total angular momentum unless an external twisting force (torque) acts on it. This principle explains many real-life phenomena, such as a spinning skater pulling in their arms to rotate faster.

Detailed Explanation :

Conservation of Angular Momentum

The law of conservation of angular momentum is one of the most important laws in rotational mechanics. It describes how the rotational motion of a body behaves when no external torque acts on it. The concept is similar to the law of conservation of linear momentum, but instead of linear motion, it applies to rotational motion.

Mathematically, angular momentum is the rotational equivalent of linear momentum. Just as linear momentum is conserved in the absence of external forces, angular momentum is conserved in the absence of external torques.

Definition of Angular Momentum

The angular momentum (L) of a rotating body is the product of its moment of inertia (I) and angular velocity (ω).

Where,
= angular momentum (kg·m²/s)
= moment of inertia of the body about the axis of rotation (kg·m²)
= angular velocity (rad/s)

Angular momentum depends on how mass is distributed around the axis of rotation and how fast the object is rotating.

Statement of the Law

The law of conservation of angular momentum states that:

“If the resultant external torque acting on a system is zero, the total angular momentum of the system remains constant.”

This means that in the absence of an external torque, a rotating object will continue to spin at a constant rate or will adjust its angular velocity if its moment of inertia changes, so that the total angular momentum remains unchanged.

Mathematically,

This equation shows that if the moment of inertia of a body changes, its angular velocity will adjust automatically to conserve angular momentum.

Derivation of Conservation of Angular Momentum

According to Newton’s second law for rotation, the torque acting on a rotating body is equal to the rate of change of angular momentum:

If no external torque acts on the body,

Hence,

This means that the angular momentum does not change with time, or

Thus, the angular momentum of a system remains constant when no external torque acts on it.

Explanation of the Principle

The conservation of angular momentum implies that:

When the moment of inertia of a rotating system increases, the angular velocity decreases.
When the moment of inertia decreases, the angular velocity increases.

This adjustment occurs so that the product (angular momentum) remains constant.

This behavior can be seen in many physical and mechanical systems. For example, when a diver pulls in his arms and legs while spinning in air, his rotation speed increases because his moment of inertia decreases.

Examples of Conservation of Angular Momentum

Spinning Skater:
A figure skater spins faster when they pull their arms close to their body. This happens because pulling the arms in reduces the moment of inertia , and to conserve , the angular velocity increases.
Diver or Gymnast:
A diver tucks in during a somersault to spin faster and stretches out before entering the water to slow down. This is due to the conservation of angular momentum.
Rotating Planet or Star:
When a star collapses under gravity to form a neutron star, its radius decreases drastically. As a result, its rotation speed increases tremendously to conserve angular momentum.
Rolling Wheel on a Smooth Surface:
A freely rolling wheel continues to spin with the same angular momentum unless acted upon by an external torque such as friction or braking force.
Motion of Earth and Moon:
The Earth–Moon system maintains constant total angular momentum in its orbit unless acted upon by external gravitational forces.

Conditions for Conservation of Angular Momentum

For the angular momentum of a system to be conserved:

The external torque acting on the system must be zero.
The system must be isolated, meaning there are no external influences.
The internal forces (like tensions or contact forces between components) must act in equal and opposite pairs so that their torques cancel out.

If these conditions are not met, the angular momentum will change over time according to the equation:

Applications in Engineering and Physics

Gyroscopes and Stabilizers:
The conservation of angular momentum principle is used in gyroscopes and flywheels for stability in ships, airplanes, and spacecraft.
Rotating Machinery:
Engineers design turbines, rotors, and engines by applying angular momentum conservation to balance forces and ensure smooth rotation.
Spacecraft Motion:
In space, rockets use the expulsion of gases (which carry angular momentum) to rotate and control orientation without external torque.
Astronomy:
The principle explains the rotation of planets, stars, and galaxies in the universe and their stable orbits.
Energy Storage Systems:
Flywheels store rotational energy by conserving angular momentum, which can later be converted into useful mechanical work.

Importance of Conservation of Angular Momentum

It is a fundamental law of nature, holding true for all isolated systems.
It helps in predicting rotational motion in both natural and artificial systems.
It ensures energy efficiency and stability in rotating machinery.
It provides insight into cosmic phenomena such as the rotation of planets, stars, and galaxies.

Without this principle, analyzing rotational motion in mechanical and celestial systems would be very complex.

Conclusion

The conservation of angular momentum states that when no external torque acts on a system, its total angular momentum remains constant. It is mathematically expressed as . This principle is fundamental to rotational mechanics and explains how bodies behave during rotation, collisions, and motion in isolated systems. From spinning skaters to rotating planets and machinery, the conservation of angular momentum governs all rotational phenomena in nature and engineering.