State the principle of conservation of angular momentum.

Short Answer:

The principle of conservation of angular momentum states that if no external torque acts on a system, the total angular momentum of that system remains constant. This means that the product of the moment of inertia and the angular velocity of a rotating body stays the same unless an external turning force (torque) is applied.

In simple words, a body continues to rotate with the same angular momentum as long as no external torque acts on it. This principle explains many phenomena in physics and engineering, such as the spinning of a flywheel, the motion of planets, and a skater spinning faster when pulling in their arms.

Detailed Explanation :

Principle of Conservation of Angular Momentum

The principle of conservation of angular momentum is a fundamental law in rotational dynamics. It states that the total angular momentum of a system remains unchanged unless an external torque acts on it. This law is the rotational analog of the law of conservation of linear momentum.

In other words, if a rotating system is free from any external influence, its angular momentum before and after any change in shape or position will be the same. The principle is derived from Newton’s laws of motion and is widely used in analyzing the motion of rotating bodies such as wheels, turbines, satellites, and planets.

Definition

The principle can be defined as:

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

Mathematically,

or,

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

This equation shows that if the moment of inertia changes, the angular velocity must change in such a way that their product remains constant as long as no external torque acts.

Derivation

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

If no external torque acts on the system, then

Hence,

which implies,

This proves that the angular momentum remains conserved when the external torque is zero.

Explanation

During rotational motion, if the shape or position of the rotating body changes (such as when a person or object brings its mass closer or farther from the axis of rotation), its moment of inertia (I) changes. To maintain constant angular momentum (), the angular velocity (ω) must change inversely.

  • If the moment of inertia decreases, angular velocity increases.
  • If the moment of inertia increases, angular velocity decreases.

This exchange keeps the total angular momentum unchanged.

Examples of Conservation of Angular Momentum

  1. Spinning Skater:
    A figure skater spins faster when she pulls her arms close to her body. By reducing her moment of inertia, her angular velocity increases to keep angular momentum constant.
  2. Diver Performing a Somersault:
    When a diver tucks his body, his moment of inertia decreases, and his spinning speed increases. On extending his body again, he slows down.
  3. Planetary Motion:
    Planets move faster when they are nearer to the Sun and slower when they are farther away. This is because of the conservation of angular momentum in orbital motion.
  4. Rotating Chair Experiment:
    A person sitting on a swivel chair can spin faster by pulling weights inward and slow down by stretching them outward, without any external torque applied.
  5. Flywheel or Rotor:
    A flywheel maintains constant angular momentum, helping engines run smoothly by resisting sudden changes in speed.
  6. Neutron Stars:
    When a large star collapses into a neutron star, its radius reduces drastically. Due to conservation of angular momentum, its angular velocity increases enormously, making it spin very fast.

Applications in Engineering

  1. Gyroscopic Devices:
    The principle is used in gyroscopes for stabilizing ships, aircraft, and spacecraft. They maintain direction due to conservation of angular momentum.
  2. Flywheels:
    Used in engines and power plants to store rotational energy and maintain uniform speed by conserving angular momentum.
  3. Rotating Machinery:
    Helps analyze and control the motion of rotors, turbines, and wheels.
  4. Satellite Motion:
    Used to control the orientation and stability of artificial satellites in space.
  5. Robotics and Aerospace Engineering:
    Angular momentum control is vital for precision in robotic arms and spacecraft maneuvering systems.

Mathematical Example

Consider a skater with a moment of inertia  rotating with angular velocity .
If she pulls her arms inward and her moment of inertia reduces to , then her new angular velocity is:

 

This shows that as the moment of inertia decreases, the angular velocity increases proportionally to conserve angular momentum.

Physical Significance

The conservation of angular momentum is one of the most important laws in physics and engineering. It helps explain the stability and behavior of rotating systems, both microscopic (like electrons in atoms) and macroscopic (like planets and turbines). The law ensures that the rotational energy of an isolated system is not lost but redistributed within the system if no external torque acts.

Conclusion

The principle of conservation of angular momentum states that in the absence of external torque, the total angular momentum of a system remains constant. It is expressed as . This principle explains various real-life and engineering phenomena such as the spinning of skaters, motion of planets, and working of gyroscopes and flywheels. It plays a vital role in rotational dynamics and forms a fundamental concept in mechanical and aerospace engineering for analyzing motion, stability, and energy conservation in rotating systems.