Angular Momentum

Angular momentum is all about rotation. It's a vector quantity that describes how much an object is spinning. Just like linear momentum describes how fast an object is moving in a straight line, angular momentum describes how fast an object is rotating.

To calculate angular momentum, you need to multiply an object's moment of inertia (how resistant it is to rotation) with its angular velocity (how fast it's spinning).

Angular momentum isn't just for circular motion either. You can use it to study non-circular motion too, as long as you're looking at the direction of motion perpendicular to the radius vector.

So, in a nutshell, angular momentum is all about how much and how fast an object is rotating. The more you know about it, the better you can understand how things move in the world around us.

The direction of angular momentum

The direction of angular momentum can be determined using the right-hand rule, where four fingers represent the direction of motion while the thumb represents the direction of angular velocity momentum.

If the direction of motion is anticlockwise, the angular momentum is positive. If the direction of motion is clockwise, the angular momentum is negative.

Angular momentum formula

Imagine a point particle moving in a circle, like the one in figure 2. Now picture a plane that's created by the particle's movement and the direction it's moving in.

To find the angular momentum of the particle, you need to multiply its mass (m) by the position and velocity vectors, and the angle (θ) between them.

The angle θ can range from 0 to 180 degrees, depending on the particle's movement. And if you want to write the formula for angular momentum in terms of momentum, you can use the particle's mass (m) and its linear momentum (p).

So, to sum it up, angular momentum is all about the movement of a point particle in a circular motion. By understanding how the particle's position and velocity vectors interact, you can calculate its angular momentum and gain a deeper understanding of how things move in the world around us.

If you're dealing with circular motion, you can modify the formula for angular momentum to make it more accurate. Instead of using linear velocity, you can use angular velocity.

As you can see in figure 3, the angular momentum (L) of a point particle moving in a circular path is perpendicular to the plane formed by the radius vector (r) and the velocity vector (v).

To get a more suitable equation, you can substitute angular velocity (ω) for linear velocity in the formula. This modified equation can also be written in terms of the moment of inertia.

Angular momentum is measured in kgm2/s, and the equation looks similar to the formula for linear momentum. But instead of mass (m), you use moment of inertia (I), which is the reciprocal of mass. And instead of acceleration (a), you use angular velocity (ω), which is the reciprocal of linear acceleration in linear motion.

Understanding the relationship between angular momentum and angular velocity is crucial for studying circular motion and gaining a deeper understanding of how things move in the world around us.

Angular motion and Newton’s laws

You're absolutely right! Newton's second law of linear motion relates the acceleration of object to acting on it, and the inverse of its mass. This can also be applied to angular momentum, where an object is rotating.

In the case of rotational motion, Newton's second law for angular motion states that the angular accelerationα) of a rotating object is directly proportional to the sum of (T) acting on the object's axis of rotation.

On the other hand, angular acceleration is inversely proportional to the moment of inertia (I) with respect to the axis of rotation. The moment of inertia is a measure of how much resistance an object has to rotational motion, and it is similar to mass in linear motion.

The equation that relates these variables is shown below:

T = Iα

where T is the external torque acting on the object, I is the moment of inertia of the object with respect to the axis of rotation, and α is the angular acceleration of the object.

By using this equation, we can calculate the amount of torque required to produce a certain amount of angular acceleration in an object Understanding the relationship between torque, moment of inertia, and angular acceleration is important for analyzing rotational motion in a wide range of applications, from machines and motors to celestial bodies in space.

Newton’s law generalised for angular motion

Yes, you are absolutely right! This expression is known as the angular momentum principle or the law of conservation of angular momentum. It states that the rate of change of angular momentum of a body with respect to some point in space is equal to the sum of external torque impacting the body with respect to that point.

The equation for this principle is as follows:

dL/dt = Στ

where L is the angular momentum of the body with respect to the point, τ is the external torque acting on the body, and t is time.

This principle is analogous to Newton's second law for linear motion, where the rate of change of linear momentum of a body is equal to the net force acting on the body.

The law of conservation of angular momentum is extremely important in many areas of physics, including astronomy, mechanics, and fluid dynamics. It can be used to analyze the motion of rotating objects, such as planets, stars, and galaxies, as well as the behavior of fluids in motion.

Conservation of angular momentum

Yes, you are correct! If the net external torque acting on a body or a system of bodies is zero, then the angular momentum of the system is conserved. This is known as the law of conservation of angular momentum. Mathematically, it can be expressed as follows:

Στ = 0  (net external torque is zero)
dL/dt = 0  (rate of change of angular momentum is zero)

Thus, the total angular momentum of the system remains constant. This principle is applicable to a wide range of physical systems, such as rotating objects, spinning tops, and even subatomic particles like electrons.

The law of conservation of angular momentum is a fundamental principle in physics and has many practical applications. For instance, it is used in designing satellites and spacecraft, where conservation of momentum is critical for maintaining stability and controlling the orientation of the spacecraft. It is also used in the study of celestial mechanics to analyze the motion of planets, asteroids, and other celestial bodies.

Angular Momentum

What is angular momentum?

Angular momentum is the tendency of a rotating object to keep on its rotational movement.

How to calculate orbital angular momentum?

We can calculate orbital angular momentum using the equation L [kgm2 / s] = [ I (I+1)]1/2(h/2π)

Is angular momentum the same as linear momentum?

No, it isn’t.

What is an angular momentum example?

An ice skater rotating about its centre of mass.

How to find angular momentum?

We can determine angular momentum by using the equation L[kgm2/s] = Iω.