Angular Momentum

What is Momentum?

Momentum is defined as the product of a particle's mass and velocity. Momentum is a vector quantity, which means it has a magnitude as well as a direction. According to Isaac Newton's second law of motion, the time rate of change of momentum equals the force acting on the particle.

What is Angular Momentum?

Angular momentum is the property that characterizes the rotatory inertia of an object in motion about an axis that may or may not pass through that object. The rotation and revolution of the Earth are excellent examples of angular momentum. For example, the Earth's annual revolution around the Sun reflects orbital angular momentum, whereas its daily rotation about its axis reflects spin angular momentum.

Angular momentum is broadly classified as follows: 

• The spin angular momentum.
• The orbital angular momentum.

A body's total angular momentum is the sum of its spin and orbital angular momentum.

SI unit: kg⋅m/s

Dimension: MLT−1

Other units: slug⋅ft/s

Symbol: Denoted by L

Angular Momentum Formula

An object can encounter angular momentum in two ways. They are:

A point object is an object that accelerates around a fixed point. The Earth, for example, revolves around the sun. Here the angular momentum is given by:

L= Angular Momentum

v = linear velocity of the object

m = mass of the object

P = linear momentum

Furthermore, angular momentum can be expressed as the product of a rotating body's moment of inertia (I) and angular velocity (). In this case, the angular momentum can be calculated using the following expression:

Where, 

• L→is the
• angular momentum.
• I is the rotational inertia.
• ω is the angular velocity.

The right-hand thumb rule indicates that the direction of the angular momentum vector is the same as the axis of rotation of the given object.

The Law of the Right Hand

The right-hand rule states that if someone positions his or her hand so that the fingers come in the direction of r, then the fingers on that hand curl towards the direction of rotation, and the thumb points towards the direction of angular momentum (L), angular velocity, and torque.

Angular velocity

Angular velocity, also known as rotational velocity, is a pseudovector representation of how quickly an object's angular position or orientation changes over time. 

Torque:

Torque is defined as the rotational equivalent of linear force in physics. It is also known as the moment of force. It denotes a force's ability to cause a change in the rotational motion of the body.

T = torque

F = linear force

r = the distance measured from the axis of rotation to the point at which linear force is applied.

Theta = the angle between F and r

Solved Examples

Q1. When two identical nonrotating cylinders fall on top of the first, a cylinder of mass 250 kg and radius 2.60 m rotates at 4.00 rad/s on a frictionless surface. Because of friction, the cylinders will eventually all rotate at the same speed. What is the final angular velocity? 

Ans:We have a collision that causes a rotational change, so we conserve angular momentum.

 Lf = Li .     (1) 

The objects are rotating so their angular momentum is given by L = IΩ. As a result, in this case, equation (1) becomes 

3IΩf = IΩi . 

Solving, we find ωf = Ωi / 3 = 1.33 rad/s . 

Q2. A 22g bug crawls from the center to the outside edge of a 150g disk with a 15.0 cm radius. The disk was rotating at a rate of 11.0 rad/s. What is the final angular velocity?

Ans:  Consider the bug to be a point mass.

Because the rotation of the bug will change as it moves, we apply the Law of Conservation of Angular Momentum.

 Lf = Li . (1) 

Assuming that the bug doesn't slip then it rotates at the same velocity as the disk. As a result, the bug rotates in a circle. An object moving in a circle has an angular momentum of r 2m, where r is the radius of rotation. Initially, r = 0 in the disk's center. At the final position, the disk's edge, r = R.

 The disk is rotating and thus has an angular momentum by IΩ. Consulting a table of moments of inertia, we find I = ½MR 2 .

Thus equation (1) applied to this problem is 

R 2mΩf + ½MR 2Ωf = (0) 2mΩi + ½MR 2Ωi . 

Dividing through by ½R 2 and solving for Ωf , we find

 Ωf = MΩi / (2m + M) = (0.15)(11.0 rad/s) / [2(0.022) + 0.150] = 8.51 rad/s .