Angular Momentum in Case of Rotation About a Fixed Axis

Imagine riding a bicycle. As you pedal, the wheels start spinning, and their speed depends on how fast you pedal. If you suddenly stop pedaling, the wheels keep rotating for a while before gradually slowing down. This phenomenon occurs due to rotational motion, where the spinning wheels possess angular momentum, which gradually decreases due to friction.

Angular Momentum

The motion of a rigid body is composed of translatory motion and rotational motion. It is known that force is needed to bring the body into translatory motion. Torque is analogous to force in rotational motion.

Torque brings a change in the angular momentum of the system. Angular momentum is analogous to linear momentum in the case of translatory motion. 

In the figure, a particle is given whose position vector with respect to the origin O is “r”. “p” denotes the linear momentum of the particle moving around that point. In that case, the angular momentum is given by, 

l = r × p 

⁛ This is the cross-product of the position vector and the linear momentum vector. 

The magnitude of the angular momentum is given by:

∣L∣ = ∣r∣⋅ ∣p∣ ⋅ sin(θ)

where θ is the angle between the position vector and the momentum vector. Differentiating the equation above gives us the rate of change of angular momentum

Angular Momentum of a System of Particles

Often, bodies are not point masses. They contain more than one particle. Let us assume that the body consists of n particles each of which has a position vector r_i and momentum denoted by p_i. In that case, the angular momentum of the system is given by the vector sum of the individual momentum of the particles. 

L = l₁ + l₂ + l₃ + … l_n

Each particle has an angular momentum which is given by, 

l_i = r_i x p_i

L = r₁ x p₁ + r₂ x p₂ + r₃ x p₃ …. r_n x p_n

Angular Momentum for Symmetrical and Asymmetrical Bodies

Angular momentum is like the spinning force of an object. How smoothly or unpredictably something spins depends on its shape and mass distribution.

1. Symmetrical Bodies (like a spinning wheel or basketball) have mass evenly spread out, so they rotate smoothly and predictably.

Example: A spinning top stays upright as long as it keeps rotating due to its balanced angular momentum.

2. Asymmetrical Bodies (like an uneven rock or a gymnast flipping) have mass spread unevenly, making their rotation wobbly or unpredictable.

Example: A falling tree branch tumbles in different directions because its weight is not balanced.

Differentiation of Angular Momentum

The rate of change of angular momentum describes how an object’s spinning motion varies over time. Simply put, if an object’s angular momentum changes, it indicates that an external force or torque is influencing its rotation.

For example, When a skater spinning with arms outstretched pulls their arms in, their speed increases. This change happens because the rate of change of angular momentum is influenced by how mass is distributed and any external forces acting on it.

Conservation of Angular Momentum

The principle of conservation of angular momentum states that if no external torque (rotational force) acts on a system, its overall angular momentum remains constant. This indicates that if an object changes its shape or distribution of mass while spinning, its rotation speed will adjust to maintain balance.

For Example ,

Imagine a ballerina spinning on one foot with her arms extended.

1. When her arms are stretched out, she spins slowly.
2. When she pulls her arms close to her body, she spins much faster.

This happens because her mass moves closer to the center, reducing her moment of inertia. To conserve angular momentum, her rotation speed increases.

Where,

• L = Angular Momentum (remains constant if no external force is applied)
• I = Moment of Inertia (depends on how mass is distributed)
• ω = Angular Velocity (spin speed)

Since angular momentum is conserved, we can write

I₁ω₁ = I₂ω₂

Where,

• I₁ and ω₁ ​ are the initial moment of inertia and angular velocity.
• I₂ and ω₂ are the moment of inertia and angular velocity after the shape changes.

If I₂ decreases (bringing arms in), ω₂​ increases (spinning faster).

Key Terms Related to Angular Momentum

• Moment of Inertia (I): A measure of an object’s resistance to changes in its rotation, depending on how its mass is distributed relative to the axis of rotation.
• Angular Velocity (ω): The rate at which an object rotates around an axis.
• Angular Momentum (L): A measure of the rotational motion of an object, which depends on its mass, distribution, and speed of rotation.

Also Check,

• Dynamics of Rotational Motion
• Center of Mass
• Motion of center of mass
• Moment of Inertia
• Rolling Motion
• Kinematics of Rotational Motion
• Angular Acceleration
• Torque and Angular Momentum

What is the angular momentum in case of rotation about a fixed axis?

Angular momentum in rotation about a fixed axis is the measure of an object’s rotational motion around that axis. It depends on the object’s mass, how far it is from the axis, and how fast it is spinning.

What is the angular position of a rotation about a fixed axis?

The angular position is the angle that an object has rotated from a reference point, usually measured in degrees or radians. It tells you how far the object has turned around a fixed axis.

What is the rotation of a body about a fixed axis called?

The rotation of a body about a fixed axis is called rotational motion. It happens when an object spins around a stationary line or axis, like the Earth rotating around its axis.

Is rotation necessary for angular momentum?

Yes, rotation is required for angular momentum because it only exists when an object is spinning. Without rotation, there is no angular momentum.

What are the conditions for angular momentum?

For angular momentum to exist, the object must be rotating or moving in a circular path. Additionally, there must be a fixed point or axis around which the object is spinning.

Find angular momentum of particle revolving around the origin at a distance of 5 m with linear momentum of 50Kg/s. 

Angular Momentum is given by, 

l = r x p sin(θ)

Given: r = 5 m, p = 50Kgm/s and θ = 90°. 

l = r x p sin(θ) 

l = (5)(50)sin(90)

l = 250

Find the angular momentum of a particle revolving around the origin at a distance of 10m with linear momentum of 10Kg/s. The angle between the position vector and momentum is 30°. 

Angular Momentum is given by, 

l = r x p sin(θ)

Given: r = 10m, p = 10Kgm/

l = r x p sin(θ) 

l = (10)(10)sin(30)

l = 50

Find the angular momentum of a particle revolving around the origin at a distance of 1.5m with a linear momentum of 100Kgm/s. The angle between the position vector and momentum is 60°. 

Angular Momentum is given by, 

l = r x p sin(θ)

Given: r = 5 m, p = 50Kgm/s

l = r x p sin(θ) 

l = (1.5)(100)sin(60)

l = 75√3

Find the angular momentum of the system of two particles revolving around the origin at a distance of 5 m with a linear momentum of 50Kg/s and 10 Kg/s. 

Angular Momentum is given by, 

l = r x p sin(θ)

Given: r = 5 m, p₁ = 50Kgm/s, p₂ = 10Kgm/s and the angle is a right angle. 

Since the system consists of two particles. The total angular momentum will be the sum of the angular momentum of these particles.  

l = rp₁ + rp₂

l = (5)(50) + (5)(10)

l = 250 + 50 

l = 300