1. What is Angular Momentum?
Angular momentum describes the quantity of rotational motion a body has. Just as linear momentum measures how difficult it is to stop a moving object, angular momentum tells us how difficult it is to stop a rotating object. It depends on both how fast the object is rotating and how its mass is distributed around the axis of rotation.
Objects like spinning tops, rotating wheels, planets, and even orbiting satellites possess angular momentum. This quantity helps explain why a spinning top stays upright or why a figure skater spins faster when pulling in their arms.
1.1. Intuitive Understanding
Angular momentum increases when either the rotational speed increases or when more mass is located farther from the axis of rotation. That is why a heavier wheel spinning fast has a large angular momentum compared to a small toy wheel spinning at the same speed.
1.1.1. Daily Life Examples
- A spinning bicycle wheel resists changes in its orientation due to high angular momentum.
- A diver or skater changes their spin rate by adjusting their body posture.
- The Earth keeps rotating on its axis because it has enormous angular momentum.
2. Definition and Formula for a Particle
Angular momentum of a particle about a point is defined as the cross product of its position vector and its linear momentum:
\( \vec{L} = \vec{r} \times \vec{p} \)
Where:
- \( \vec{r} \) is the position vector from the axis
- \( \vec{p} = m \vec{v} \) is the linear momentum
2.1. Magnitude of Angular Momentum
If the angle between \( \vec{r} \) and \( \vec{p} \) is \( \theta \), then:
\( L = r p \sin \theta \)
This shows that angular momentum depends on both the perpendicular distance to the motion and the linear momentum.
3. Angular Momentum of a Rigid Body
For a rigid body rotating about a fixed axis, the angular momentum depends on the moment of inertia and the angular velocity:
\( L = I \omega \)
This form is particularly useful when dealing with wheels, discs, and other rotating bodies.
3.1. Dependence on Mass Distribution
A body with a larger moment of inertia (mass farther away from the axis) has greater angular momentum at the same angular velocity. This explains why it is harder to stop or change the direction of a spinning heavy wheel.
4. Direction of Angular Momentum
Angular momentum is a vector quantity. Its direction is determined by the right-hand rule, the same rule used for torque and angular velocity. Curl your fingers in the direction of rotation; your thumb points in the direction of angular momentum.
4.1. Right-Hand Rule
If a wheel rotates counterclockwise, the angular momentum points upward; if it rotates clockwise, the angular momentum points downward. This direction helps analyse rotational stability and gyroscopic effects.
5. Relation Between Torque and Angular Momentum
Torque is the rate of change of angular momentum. This is the rotational analogue of Newton's second law for linear momentum.
\( \vec{\tau} = \dfrac{d\vec{L}}{dt} \)
This equation means that a net torque is required to change the angular momentum of a body.
5.1. When Torque is Zero
If no external torque acts on a system, its angular momentum remains constant. This leads to the important principle of conservation of angular momentum, which is explored in the next topic.
6. Examples Demonstrating Angular Momentum
Understanding angular momentum helps explain many physical events and daily observations.
6.1. Spinning Figure Skater
When the skater pulls their arms in, their moment of inertia decreases, causing them to spin faster to conserve angular momentum.
6.2. Rotating Bicycle Wheel
A spinning wheel resists changes in its orientation. This property, used in gyroscopes, arises from high angular momentum.
6.3. Astronomical Systems
Planets orbit the Sun with stable angular momentum unless acted upon by external forces.
7. Why Angular Momentum Matters
Angular momentum is essential for understanding rotation, stability, gyroscopic effects, and the behaviour of rotating systems in physics, engineering, and astronomy. It forms the foundation for the next topic — conservation of angular momentum.