Angular Momentum: Formula, Conservation Law & Examples

Introduction to Angular Momentum

Angular momentum is one of the key concepts in physics. As like we have linear momentum for linear motion, angular momentum arises when we are talking about the circular motion or a rotational path. It explains how objects behave when they rotate or move in a circular path. It also tells us the mechanism of something that is spinning or rotating. We ourselves are experiencing angular momentum as we are on earth that rotates around the sun. However, many of us are unaware of universal facts.

Angular momentum reveals the physics of microscopic particles like electrons, the world of daily life, and the massive space of stars and galaxies. It is used in interdisciplinary realms like mechanics, engineering, astronomy, particle physics, and quantum physics.

Here I have tried to explain angular momentum in a simpler explanatory way so that readers can understand it easily.

Rotational Motion and Moment of Inertia

Angular momentum is the momentum of a mass in rotational motion. Thus, rotational motion and moment of inertia are the two basic concepts to be understood before studying angular momentum. 

Rotational Motion

Rotational motion comes into account for rigid bodies. When those bodies rotate, they maintain a constant motion, at a constant distance from an imaginary axis. Thus the rotation will be about the common axis and at constant velocity. This point of the axis may be different for different shapes of the bodies.
Examples:

• A wheel turning around its axle.
• A planet rotating on its axis in its orbit.
• The fan blades spin around a fixed rod.

Some terms used to describe rotational motion are:

• Angular displacement (θ) (radian): As like linear displacement, it is the angle swept by a rotating body from initial position at a certain time period. It is measured in radian (rad) and for one complete rotation the value of theta will be 2𝜋.
• Angular velocity (ω) (rad/sec): It is also analogous with the linear velocity and is defined as the rate of change of angular displacement. It can also be said as the rate at which the body performs its rotation. 

ω = Δθ/Δt

• Angular acceleration (α) (rad/sec^2): Similar to linear acceleration, it is the rate of change of angular velocity with time. It is the rate at which the spinning object accelerates.

α = Δv/Δt

Moment of Inertia

Moment of inertia is the inertia of a body in rotational motion. It can also be called as the force required for the opposition of the inertia of the rigid body at rest, to turn it about its rotational axis. This turning force is assigned as torque in rotational motion. We can say that it is a moment of mass in  rotational form. It is the measurement of rotational equivalent of mass and is carried out for rigid bodies. The formula is given by,

I=∑mr^2 [Equation 1]

Where, 

m = mass of the rigid body

r = distance of the mass from its axis

Equation  (1) shows that, m ∝ 1/r^2. Thus greater the distance of the mass from its imaginary rotational axis, greater will be the moment of inertia and hence greater torque is required to turn it. Thus, objects of different shapes rotate uniquely.

What Is Angular Momentum?

Angular momentum (L) can be called the moment of momentum. Thus, it is a basic physical quantity that quantifies the rotating mass and tells how much motion a rotating body contains. It depends on:

• How fast something is rotating (ω)
• How massive it is
• How its mass is distributed around the axis (I)
  

For rotating objects, L=Iω [Equation 2]

This is the simple form used for rigid bodies like wheels, fans, and rotating rods.

For a moving particle,

L→ = r→ × p→ [Equation 3]

Where:

• r = position vector
  
• p = m v = linear momentum
  

Meaning in Simple Words

Angular momentum increases when:

• The object spins faster
  
• The object is heavier
  
• The object’s mass is farther from the axis
  

A spinning wheel with heavy edges has more angular momentum, which is why it stays stable.

Linear vs Rotational Dynamics — The Connection

Angular momentum is closely related to linear momentum. Some common terms and their difference in measurements are given below:

Because of this parallel relationship, many rotational problems can be solved using linear concepts.

Conservation of Angular Momentum

It states that, in the absence of external torque in a system, the total angular momentum remains constant.

Mathematically:

Ii​ωi​ = If​ωf​ [Equation 4]

This principle explains many natural and man-made phenomena.

Some simple examples demonstrating this law are as follows:

• Ice Skater Spin

When the skater stretches arms, moment of inertia increases  which slows the rotation. Similarly, on pulling the arms in, I decrease and hence ω increases . This makes the skater spin faster.

• Planets Around the Sun

When a planet is nearer to the Sun, it moves faster. Also, when it is far from the sun, it moves slower. Hence, the constant orbital angular momentum is maintained.
This keeps orbital angular momentum constant.

Torque and Angular Momentum

Torque is the force that keeps the rotation fixed about an axis. It also opposes the property of the objects trying to remain at rest and brings them in rotational motion, 

τ = rFsin⁡θ [Equation 5]

It tells how strongly a force causes rotation.

Relation Between Torque and Angular Momentum

τ = dL/dt [Equation 6]

Thus,

If torque is zero → angular momentum remains unchanged

If torque is applied → angular momentum changes

Some common examples of torque acting are:

• Opening a Door
  A push near the hinge is weak; far from the hinge is stronger.
• Using a Wrench
  A longer wrench gives more torque and hence turning the bolts and screw becomes easier.
• Pedaling a Bicycle
  Torque applied to pedals turns the wheel.

Angular Momentum in Rigid Bodies

Rigid bodies have irregular shapes however, their shape remains same under rotation. Some common rigid bodies are, wheels, disks,rod, fan blades, gyroscope etc. For rigid bodies, angular momentum is calculated as: 

L=Iω 

Here, I is the moment of inertia and ω is the angular velocity. The angular momentum varies with the shape of the rigid bodies, as the moment of inertia will vary with the shape.

For example:

• Solid sphere can rotate easily
• Ring is harder to rotate
• Long rod rotates according to its axis

This is why vehicle parts, turbines, and machinery are designed with shapes that give suitable angular momentum. The theorems used for rigid bodies are:

• Parallel Axis Theorem
  Used when rotating around an axis away from center.
• Perpendicular Axis Theorem
  Used for flat plates.

Engineers can thus calculate the rotational energy of several machines.

Angular Momentum in Orbital Motion

Angular momentum controls the motion of planets, satellites and also the galaxy clusters and superclusters. 

Orbital Angular Momentum

For a planet:

L=mvr [Equation 7]

Kepler’s second law simply explains that a planet sweeps equal areas in equal times because its angular momentum is conserved.

Nearer to the Sun

• r decreases
• v increases
  → angular momentum stays constant.

Away from the Sun

• r increases
• v decreases
  → again constant.

For example, planets near the sun move faster than those which are farther. Satellites also travel faster in low orbits than in high orbits. We can also study the spatial orientation of angular momentum of galaxies in several clusters and superclusters, by using various filters and magnitudes in data collection. The universe is full of rotating systems. Thus, angular momentum takes care of all these systems.

Angular Momentum in Quantum Mechanics

At atomic and sub-atomic level also, we can see the effects of angular momentum. The types are given below:

• Orbital Angular Momentum
  Electrons also revolve around the nucleus. This has proved circular motion at the microscopic level too. They have quantized angular momentum.
  
• Spin Angular Momentum
  Particles have an internal rotation-like property called spin.

Important features

• Angular momentum is allowed only to take certain values.
• It is fundamental for understanding atoms.
• It determines the magnetic behavior of materials.
• It is essential in technologies like MRI and quantum computing.

Even though quantum angular momentum is hypothetical, the effects are being experienced as real.

Applications in Real Life

Some real-life applications of angular momentum are as follows:

Sports

Gymnasts use the concepts of angular momentum to perform flips and turns. Divers control their spins by changing body posture. Athletes throw javelins and footballs with a spin to make them stable.

Everyday Machines

Fans, mixers, washing machines, grinders rotate by implying the concepts of angular momentum. Flywheels store rotational energy for smooth turns of vehicles and machines.

Vehicles

Motorcycles and bicycles stay stable in the path because the rotating wheels give the required angular momentum. Rotating shafts, axles, and turbines are used in cars to create good turns.

Space and Astronomy

Satellites adjust orientation using spinning reaction wheels. Day and night are seen on earth due to the rotation of earth in its orbit. While merging and the formation of gravitational waves, neutron stars or black holes spin faster.

Toys

Toy spinners rotate using rotational energy. Yo-yo balls go up and automatically return using rotational energy.

Engineering

Turbines rotate showing angular momentum. Motor engines, generators, dynamos and electric fans are based on rotational energy.

Nature

Earth’s angular momentum makes hurricanes spin. When water swirls down a drain, it moves in a circular motion.

Chemistry

In chemistry too, the concept of angular momentum is used for atomic, and molecular interaction and diatomic bonds.

Conclusion

Physics regards angular momentum as the core of various natural to complex phenomena. It explains about the rotation of the objects and the stability they gain in their paths. To understand the simple daily activities of complex engineering, cosmology, astrophysics, quantum mechanics and particle physics it is crucial. By understanding angular momentum, we can know the spatial orientation of the galaxies and planets, physics behind the motors and electric fans and also the motion of vehicles and athletes in circular paths. Thus, angular momentum is a beautiful natural concept that connects simple circular motions to the vague laws of the universe.

References

Zare, R. N., & Harter, W. G. (1988). Angular momentum: understanding spatial aspects in chemistry and physics. New York, 120.

Borrelli, A. (2011). Angular momentum between physics and mathematics.

Edmonds, A. R. (1996). Angular momentum in quantum mechanics (Vol. 4). Princeton university press.

https://en.wikipedia.org/wiki/Angular_momentum

https://byjus.com/physics/angular-momentum

https://www.geeksforgeeks.org/physics/angular-momentum