Angular and Linear Velocity

June 13, 2024

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In the real physical world, any physical entity whether at macroscopic level or micro level keeps on moving following some path, either it moves in a straight or circular path etc. In this article we are going to understand the velocity associated with straight and circular paths.

As the title of the topic depicts, we are going to understand in depth about angular and linear velocity, its definition, basic quantities involved and the relationship between them. We will also discuss some real world problems for creating better realization of the topic.

Table of Contents

Introduction

To start with the topic, we must have an idea of the velocity of a body which measures the degree of fastness of a body in a particular direction. When a body moves in a straight line, then the velocity is said to be linear velocity but when the body is moving in circular, eliptical fashion or moving while spinning/rotating then the velocity associated with it are angular as well as linear velocity or a combination of both.For Example, A car moving on straight road with velocity 40 meter/sec is the linear velocity of car but If we want to study the motion of wheels of car which are making certain number revolutions about the axis of wheel while moving, then angular displacement and angular velocity came into the picture. In the next section, we are going to study in brief about the definition formula involved and the relationship between them.

Linear velocity

The rate of change of displacement with respect to time when an object moves along a straight line is measured as linear velocity. It is a vector quantity. The linear velocity dimensional formula is [M]0[L]1[T]-1.

When an object moves in linear or straight manner then the velocity is termed as linear and equals to the rate of change of displacement over the time. Mathematically, linear velocity can be expressed as-

where,

v, linear velocity

x, denote displacement of a body

t, time taken by body to cover x displacement

Angular Velocity

Angular velocity comes into play when the body is rotating. A body which exhibits rotational motion undergoes rotation around a fixed axis passing through a rotor.For example-When a windmill’s blades revolve around an axis that passes through the rotor, rotational motion is displayed. The term “angular velocity” refers to the speed at which rigid bodies rotate about a fixed axis.

In simple words, angular velocity is defined as the time rate at which an object rotates about a fixed axis.

The Greek letter omega,⍵ (also, ) stands for angular velocity. The SI unit for angular velocity is radians per second because it is expressed as an angle per unit of time. [M0 L0 T-1] is the dimensional formula for angular velocity. Every point on an object that is revolving about an axis has the same angular velocity. The tangential velocity of points distant from the axis of rotation is, nevertheless, different from that of points closer to the axis of rotation. Rotational velocity and also an angular frequency vector are other names for angular velocity.

Formula for calculating angular velocity is expressed as follows-

where,

is angular velocity

is angular displacement of rotating body

t, is the amount of change in time

The instantaneous angular velocity, of a rotating body can be calculated by the formula stated below-

Direction of Angular velocity

Since, Angular velocity is a vector quantity. Therefore it has got direction as well. Determination of the direction of angular velocity is a difficult task for a rotating body, whose direction of point object keeps on changing.The rotating object’s axis is the only position where it has a fixed direction. The Right Hand Rule is used to identify the direction of angular velocity using the axis of rotation.

When you curl your fingers in the rotational direction of the disc, the thumb will point in the direction of angular velocity. The plane of rotation is always perpendicular to the direction of angular velocity.

Relationship between Linear Velocity and Angular Velocity

To determine the relationship between linear and angular velocity, consider a rigid rotating body made up of point objects. If the point object moves an arc length in time its linear velocity is given by following equation-

,——————(1)

The angular displacement expressed as ratio of arc length and the radius of curvature, r,

,

on rearranging,

, ——————(2)

Substituting (2), in(1), we get

On further simplification, we get,

, —————-(3)

Equation (3) shows the relationship between linear and angular velocity.

Let’s take real life example to understand the concept of relationship between angular and linear velocity,

Angular velocity of wheels of car

Angular velocity of a wheel of car measures how fast the wheel of the car is rotating.A wheel of a car traveling with velocity v is rotating at angular velocity, . As a result, the car advances at a linear speed of , where r is the wheel radius. The greater the wheel angular velocity, the faster the car will move.

The following problem will help us better understand the application of the relationship between angular and linear velocity.

Q- On a highway, an automobile wheel with a 20-inch radius rotates at eight rotations per second . What is the tire’s angular velocity and linear velocity?

Solutions- The angular velocity is given by the following equation:

Since a whole rotation (360°) equals 2, we multiply by 2 to account for the car’s eight revolutions per second.

When we substitute the values in the equation, we obtain

Linear velocity

The tire’s angular velocity is 16 rad/sec and linear velocity is 25.5meter/sec as a result.