1. What is Conservation of Angular Momentum?
The principle of conservation of angular momentum states that the angular momentum of a system remains constant if no external torque acts on it. This means that even if objects within the system change their shape, position, or distribution of mass, the total angular momentum stays the same as long as external influences are absent.
This idea is similar to the conservation of linear momentum but applies to rotational motion. It helps explain everyday phenomena such as spinning skaters, rotating planets, and even how cats land on their feet.
1.1. Why Does Angular Momentum Stay Constant?
Angular momentum is conserved because internal forces within a system cannot change its total rotational motion. Only an external torque can alter angular momentum. If no external torque acts, the system continues rotating with the same angular momentum.
1.1.1. Simple Examples
- A spinning skater pulls in their arms and spins faster.
- A collapsing neutron star spins rapidly as its radius shrinks.
- A rotating stool experiment where a person holding weights pulls them inward to increase spin speed.
2. Mathematical Statement
If the net external torque on a system is zero, then the angular momentum \( \vec{L} \) remains constant:
\( \dfrac{d\vec{L}}{dt} = 0 \Rightarrow \vec{L} = \text{constant} \)
For a rotating rigid body:
\( I \omega = \text{constant} \)
This equation shows that if the moment of inertia changes, the angular velocity adjusts to keep the product constant.
2.1. Understanding Through Moment of Inertia
When a spinning object changes its mass distribution, its moment of inertia \( I \) changes. To conserve angular momentum, the angular velocity \( \omega \) must change inversely.
2.2. Real-Life Example
A dancer or skater spins faster when they bring their arms close to the body (reducing \( I \)) and slows down when they extend their arms (increasing \( I \)).
3. Examples Demonstrating Conservation of Angular Momentum
Conservation of angular momentum appears in many physical systems ranging from everyday life to cosmic events.
3.1. Spinning Skater
A figure skater begins a spin with arms extended. When they pull their arms inward, their moment of inertia decreases, so their angular velocity increases to maintain constant angular momentum.
3.2. Planets and Stars
Planets orbit the Sun with almost constant angular momentum. When stars collapse under gravity to form neutron stars, their radius becomes extremely small, causing them to spin rapidly.
3.3. Rotating Stool Experiment
If a person sits on a rotating stool holding weights and then pulls the weights inward, their spinning speed increases. This is a classic physics demonstration.
4. Zero External Torque Condition
For angular momentum to be conserved, the net external torque must be zero. This occurs when:
- External forces act through the axis of rotation.
- External forces cancel out symmetrically.
- No external force is applied at all.
4.1. Why Internal Forces Don’t Matter
Internal forces come in equal and opposite pairs (Newton’s Third Law), so they cannot change the total angular momentum of the system.
5. Applications of Angular Momentum Conservation
This principle is vital in many fields of physics, engineering, and astronomy.
5.1. Examples in Technology
- Gyroscopes used in navigation rely on stable angular momentum.
- Rotating machinery maintains steady motion due to angular momentum conservation.
- Reaction wheels help satellites maintain orientation in space.
5.2. Examples in Astronomy
- The Earth's rotation remains steady because no large external torque acts on it.
- Formation of galaxies and star systems follows this principle.
6. Why Conservation of Angular Momentum Matters
This principle explains many rotational phenomena and ensures predictable behaviour of rotating bodies. It forms the basis for understanding rotational dynamics, gyroscopic stability, astronomical processes, and advanced mechanics.
Understanding this topic prepares you for exploring rolling motion and rotational kinetic energy in the upcoming sections.