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Angular Displacement


Displacement of a particle is the change in the position of the particle in a particular direction. Just as Position co-ordinate is measured in units of length, the displacement is also measured in units of length. i.e, It is the shortest distance from the final and initial positions of a point.

In the study of linear motion, the displacement of a particle of the body along a line is called the linear displacement. This concept has its analog in rotational motion: angular displacement. In this section we will learn more about angular displacement.

What is Angular Displacement?

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The angle through which the radius vector representing the position of a particle rotates is called angular displacement or the change in position of the particle in a circular path with respect to its centre is called angular displacement. The angular displacement of a body with respect to a reference line is denoted as $\theta$.

Angular Displacement Formula

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Angular Displacement
Let a particle rotate around the circumference of the circle whose center is at 'O'. Let us assume that the initial position of the particle at time $t = 0$ is A. The final position of the particle at time $t = t$ is B. Then the angular displacement of the particle is given by the difference in its final and the initial angles

$\Delta Q = Q_{f}-Q_{i}$
Angular Displacement Formula
Consider the figure, where the radius line $r$ is moved around an arc length of $s$, and hence is moved through an angular displacement $\theta$.

$\theta$ = $\frac{s}{r}$

$s$ = Arc length
$r$ = Radius of the path (m)
$\theta$ = Angular displacement

Angular Displacement Units

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The angular displacement can be measured in degree. But the most convenient unit to measure it is radians. Radian is the SI unit of angular displacement.

One radian is defined as the angle subtended at the centre of a circle by an arc which is equal to length of the arc divided by the radius of the circle.

Consider the situation in which the arc length $s$ is the circumference of the circle, i.e., $s = 2\pi r$

We know that

$\theta$ = $\frac{s}{r}$ = $\frac{2\pi r}{r}$ = $2\pi$ radians.

To convert angle in radians to degree:

We know that the angular displacement in the entire circle is $360^{0}$, so we now have

$2\pi$ rad = $360^{0}$

$1$ rad = $\frac{360^{0}}{2\pi}$

$1$ rad = $\frac{180^{0}}{\pi}$

To convert an angle in degree to radian:

we have

$1^{0}$ = $\frac{\pi}{180}$ radians

Angular Velocity and Acceleration

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The rate of change of angular displacement of a particle moving on a circle is called the angular velocity represented by $\omega$. Angular velocity = Angular displacement per unit time or rate of change of angular displacement. The SI unit of angular velocity is radian/sec.

$\omega$ = $\frac{d\theta}{dt}$

The angular acceleration ($\alpha$) of the rigid body is the time rate of the angular velocity,

$\alpha$ = $\frac{d \omega}{dt}$

The basic unit of the angular acceleration is radian per second square.

How to Calculate Angular Displacement?

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The following are the problems of angular displacement.

Solved Examples

Question 1: A boy rides a merry-go-round. His seat is at a distance of $2$m from the center. If the boy moves along an arc length of $2.5$m, what is his angular displacement?
We know that the linear and angular displacement is related as 

$s = r \theta$,

 $r = 2$ m
 $s = 2.5$ m, 

So, the Angular displacement is given by,

$\theta$ = $\frac{s}{r}$

$\theta$ = $\frac{2.5}{2}$

$\theta$ = $1.25$ radians 

             = $1.25 \times 57.296^{0}$

             = $71.62^{0}$.   


Question 2: The angular speed of a motor bike moving in a circular track is 35 rad/sec. Find the angular displacement of the motor car at time t = 5 seconds?
$\omega$ = $35$rad/sec
$t$ = $5$ sec 

we know that,

$\omega$ = $\frac{\theta}{t}$

$\theta$ = $\omega \times t$

$\theta$ = $35 \times 5$

$\theta$ = $175$radian

The Angular displacement of the motor car is $175$ radians.