Conservation of Angular Momentum: Principle, Formula & Examples
Angular momentum is a fundamental concept in Physics that describes the rotational analog of linear momentum. It helps explain how and why objects rotate, what determines their spinning motion, and under what conditions this rotational motion is conserved. Angular momentum is crucial in many natural and technological phenomena, from the spinning of planets and galaxies to the functioning of gyroscopes and everyday rotating objects.
Definition of Angular Momentum
For a single particle, angular momentum (L) is defined as the cross product of its position vector (r) and its linear momentum (p):
For a rigid body, the total angular momentum is the sum of the angular momenta of all its constituent particles. If the body rotates about a fixed axis, all points have the same angular speed (ω). In this case, angular momentum can also be written as:
Key Formulas and Units
| Physical Quantity | Symbol | Formula | SI Unit |
|---|---|---|---|
| Angular Momentum (Particle) | L | r × p | kg·m2/s |
| Angular Momentum (Rigid Body) | L | I ω | kg·m2/s |
| Moment of Inertia | I | Σ mi ri2 | kg·m2 |
Properties and Direction
Angular momentum is a vector quantity: it has both magnitude and direction. The direction of the angular momentum vector is along the axis of rotation, determined by the right-hand rule. Curl the fingers of your right hand in the direction of rotation; your thumb will point in the direction of L.
Types: Spin and Orbital Angular Momentum
A physical system can possess two kinds of angular momentum:
- Spin Angular Momentum: Rotation about its own center of mass (e.g., Earth's daily rotation).
- Orbital Angular Momentum: Revolution about an external point or axis (e.g., Earth's orbit around the Sun).
Law of Conservation of Angular Momentum
If no external torque acts on a system, its total angular momentum remains constant. This principle is fundamental in Physics and applies to isolated systems, from molecules and particles to rotating celestial bodies. Changes in the distribution of mass, like an ice skater pulling in their arms, result in a change in rotational speed to keep the angular momentum conserved.
Stepwise Approach to Solving Angular Momentum Problems
| Step | What To Do |
|---|---|
| 1 | Identify the type of system—particle, rigid body, or system of particles. |
| 2 | Determine the axis or point from which angular momentum is considered. |
| 3 | Calculate moment of inertia (I) if it’s a rigid body or sum individual angular momenta if a multi-particle system. |
| 4 | Apply the correct formula (L = r × p for particles or L = I ω for rigid bodies). |
| 5 | Check for conservation: is there a net external torque? |
| 6 | Solve step-by-step, minding both the units and vector directions. |
Example Problem
Example: A solid sphere (mass 0.50 kg, radius 0.60 m) spins around its center at a period of 1.2 s. What’s its angular momentum?
1. Angular speed ω = 2π/Period = 2π / 1.2 = 5.24 s-1
2. For a solid sphere: I = (2/5) M R2
I = (2/5) × 0.50 × (0.60)2 = 0.072 kg·m2
3. L = I ω = 0.072 × 5.24 = 0.377 kg·m2/s
Comparison: Angular vs. Linear Momentum
| Feature | Angular Momentum | Linear Momentum |
|---|---|---|
| Definition | L = r × p | p = m v |
| SI Unit | kg·m2/s | kg·m/s |
| Motion Type | Rotational | Translational |
| Conservation | If net external torque is zero | If net external force is zero |
Applications and Examples
Angular momentum is observed in everyday experiences and advanced science. For instance, the Earth's stable axis points towards the North Star due to conserved angular momentum. In astronomy, conservation explains galaxy formation and why planets speed up as they approach the Sun in their orbits. On a playground, if a merry-go-round receives forces that produce torque, its angular momentum changes.
Further Study and Vedantu Resources
- Angular Momentum About Fixed Axis
- Rotational Kinetic Energy
- System of Particles & Rotational Motion
- Learn About Torque
- Conservation of Angular Momentum
Mastering angular momentum is essential for understanding rotational motion in Physics. Practice with more examples, focus on vector directions, and always check for external torques when applying conservation laws.
FAQs on Angular Momentum: Concepts, Formulas & Applications
1. What is angular momentum in simple words?
Angular momentum is the rotational equivalent of linear momentum. It describes how much rotation an object has around a point or axis. The faster an object spins and the farther its mass is from the axis, the higher its angular momentum.
2. Write the formula for angular momentum.
The formula for angular momentum (L) is:
- For a particle: L = r × p (where r is the position vector, and p is linear momentum)
- For a rigid body: L = Iω (where I is moment of inertia, and ω is angular velocity)
3. What is the SI unit of angular momentum?
The SI unit of angular momentum is kilogram metre squared per second (kg·m²/s).
4. What are the conditions for conservation of angular momentum?
Angular momentum is conserved when the net external torque acting on a system is zero. This means:
- No external forces or torques change the system’s rotation
- The objects do not lose or gain mass
5. What is the difference between linear and angular momentum?
Linear momentum deals with motion in a straight line, while angular momentum concerns rotational motion. Key differences:
- Linear momentum: p = m·v (kg·m/s)
- Angular momentum: L = r × p or Iω (kg·m²/s)
- Linear momentum is due to movement along a path; angular momentum is due to rotation around an axis
6. Give an example of angular momentum conservation from daily life.
A classic example is a figure skater spinning faster when pulling in their arms. As the skater reduces their moment of inertia, the angular velocity increases to conserve angular momentum.
7. What does ‘moment of inertia’ mean in angular momentum?
Moment of inertia (I) is a measure of how mass is distributed with respect to the axis of rotation. A larger moment of inertia means the object resists changes in rotational speed more, directly affecting its angular momentum (L = Iω).
8. How is angular momentum related to torque?
The rate of change of angular momentum equals the net external torque applied on a system: τ = dL/dt. If external torque is zero, angular momentum remains constant.
9. What is the dimensional formula of angular momentum?
The dimensional formula of angular momentum is:[M1L2T-1], meaning mass × (length)2 per time.
10. What is an angular momentum operator?
In quantum mechanics, the angular momentum operator is a mathematical operator used to calculate angular momentum of particles. Its symbol is ĤL or L̂, and it follows specific commutation relations important in atomic structure and quantum states.
11. What is the conservation law statement for angular momentum?
The law of conservation of angular momentum states: If no external torque acts on a system, its total angular momentum remains constant. Mathematically, I1ω1 = I2ω2 for a closed system.
12. Can angular momentum change if mass changes?
Yes, if a system’s mass changes (like a rocket burning fuel), the angular momentum can change even without external torque, as both moment of inertia and velocity might be affected.
