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Angular Displacement

Two quantities distance and displacement are seemed to be similar in meaning but both are different from each other. The first quantity is scalar while the second one is a vector quantity. Distance shows covered surface in motion while displacement also shows the covered distance but it shows the complete change in the position of moving objects. This is described by both quantities that are a magnitude and direction of motion. It’s a change from initial to final state of motion.

When an object is rotated around axis then it is very difficult to analyse its motion because at every point of the path, the quantities like velocity, acceleration is changed. For a rigid body, particles are in constant motion, so the rotation of a rigid body on a circular path is called the rotational motion. But when an object is moved on a curved or circular path then this change in its position from initial to final state is shown by the angular displacement. This rotational quantity is angled at which a body rotates around the axis. Let’s discuss the angular displacement, its formula, and problems based on this.

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What is Angular Displacement?

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We know that the Displacement is the shortest distance from the initial position to final position irrespective of the path taken by it to reach the final position. It is the virtual straight line connecting initial position and the final position.

Here we can see the distance traveled (Actual Path) is AB + BC, but displacement is AC.
where, AC = AB + BC.
Now, What is Angular displacement?, In simple words we can say displacement covered in terms of angle. Thus, the displacement of the body moving in the curved path is represented by Angular displacement.
It is defined as the angle in radians through which a point has been rotated about a specified axis. It is the distance an object moves in a curved path. It is represented by the length of the arc of curved path.
The arc is measured in the angle and hence angular displacement is also measured as an angle.

Angular Displacement Units

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The Angular displacement is measured as angles and the angles are measured in radians. So, the unit of angular displacement is radians.

The angular displacement is represented in polar co-ordinates and not in x-y co-ordinates.

Angular Displacement Formula

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Let's say, a body is moving in circular direction as shown in fig,
Angular Displacement Equation
For $\theta$ angle and the radius of curved path is r. The linear displacement is related to the angular displacement as: S = r $\theta$ ...............(1)where r = radius of the curvature,
$\theta$ = angular displacement.
The equation (1) is the angular displacement equation.

Angular Velocity and Acceleration

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Like the Linear motion, Circular motion too has all the equivalent quantities as we have in linear motion. Now, we will briefly discuss the angular velocity and angular acceleration.

Angular Velocity Definition

The Angular velocity is the measure of how fast a body is changing its angle. The Angular velocity is measured in radians per second. The angular velocity is represented by $\omega$ and hence
$\omega$ = $\frac{\theta}{t}$.


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

Angular Acceleration Definition

The Angular acceleration is the measure of how fast a body is changing its angular velocity. The Angular acceleration is measured in rad per sec2.
The angular acceleration is represented $\alpha$ and hence
$\alpha$ = $\frac{\omega}{t}$ = $\frac{\theta}{t^2}$,

$\alpha$ = $\frac{d \omega}{dt}$ = $\frac{d^2(\theta)}{dt^2}$.

Relation Between Linear Speed and Angular Speed

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The Linear Speed and the angular Speed of any body moving in curved path is given by:
V = $\omega$ r
where V = Linear speed,
$\omega$ = angular speed,
r = radius of the curved path.

How to Find Angular Displacement?

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With the help of the formulae given above, we will study how to find angular displacement, angular velocity and angular acceleration.

Solved Examples

Question 1: A boy is running along a Circular track of 15 m diameter. If in 20 seconds he covers half the track, then find his Angular speed and Linear speed?

The angular speed of the body is calculated as $\omega$ = $\frac{\theta}{t}$,

Here $\theta$ = 180 degrees = $\Pi$ radians. (since the boy has completed half the circular track)
    or 1 radian = 57.296 degrees.
t = 20 seconds.

So $\omega$ = $\frac{\theta}{t}$
      $\omega$ = $\frac{180}{20}$,
      $\omega$ = 9 degree/s
      $\omega$ = 0.157 radians/s.

Now to find his Linear speed, we need to convert the radians in meters and then using the relation of
Linear and angular speed we can find the Linear speed of the boy.
V = r $\omega$
   = $\frac{15}{2}$ meters $\times$ 0.157 radians/s.
V = 1.1775 m/s.
So, the Linear speed of the boy is 1.1775 m/s.


Question 2: A insect is on the top of a motor bike wheel with a diameter of 600 mm. The wheel turns anticlockwise until the insect is crushed.
1. What is the insect’s angular displacement?
2. What arc length does it travel?
1. The insect is on top of the wheel and for it to be crushed it had to come directly under the wheel and so it has to travel the semi circle before it is crushed.

So, θ = 180 deg = $\Pi$ radians
           1 degree = $\frac{\Pi}{180}$
                        =  0.01745 radians.
$\theta$ = 180 degree = 180 $\times$ 0.01745 radians.
Hence the Angular displacement of insect is 3.14 radians.

2. Now for finding the Linear displacement we need to find the radius of wheel and it is given by,
r = 300mm,
$\frac{d}{2}$ = 3 mm = 0.3 m

S = r $\theta$,
S = 0.3 m $\times$ 3.14 radians,
S = 0.942 m
So, the Linear displacement is 0.942 m.


Question 3: A Girl rides a merry-go-round. Her seat is at a distance of 1 m from the center. If the girl moves along an arc length of 1.5 m, what is her angular displacement?
We know that the linear and angular displacement is related as
S = r $\theta$,
where r = 1 m
         S = 1.5 m,

So, the Angular displacement is given by,
$\theta$  = $\frac{S}{r}$,
$\theta$ = $\frac{1.5}{1}$,
$\theta$ = 1.5 radians
             = 1.5 $\times$ $57.296^{\circ}$
             = $85.94^{\circ}$.


Question 4: A man is circling on his motor bike with the angular speed of 14 rad/s. Find the angular displacement of the bike and also its linear displacement if the radius of the track is 3 meters and the time for which displacement is to be calculated is 10 seconds?
Angular Speed = 14 rad/s,
Radius of track = 3 m,
Time = 10 seconds

1. The linear speed is calculated as; V = $\omega$ r,
So, the Linear speed is given by 
V = 14 $\times$ 3,
V = 42 m/s.

2. Angular displacement is given by
$\omega$ = $\frac{θ}{t}$,
So $\theta$ = $\omega$ t,
         = 14 $\times$ 10,
         = 140 rad/s.

Linear Displacement is given by
S = r$\theta$,
    = 3 $\times$ 140
    = 420m.

Question 5: A boy is playing in merry-go-round. His seat is at the 1.5 meters from the center of the merry-go-round. Find the angular displacement of the boy, if arc covered by merry-go-round is 2.5 meters?

r = 1.5 meters,
s = 2.5 meters,
θ = ?

Using the relation between the Angular displacement and the Linear displacement;
So, the angular displacement is given by,
θ = $\frac{S}{r}$
θ = $\frac{2.5}{1.5}$,
θ = $\frac{5}{3}$
   = 1.66 radians;
The angular displacement of the boy sitting in the merry-go-round is 1.66 radians.


Question 6: The angular speed of a motor car moving in a circular track is 45 rad/sec. Find the angular displacement of the motor car at time t = 8 seconds?
$\omega$ = $\frac{θ}{t}$,
$\omega$ = 45 rad/s,
θ = ?
t  = 8 seconds

so, $\theta$ = $\omega$ t,
     $\theta$ = 45 $\times$ 8,
     $\theta$ = 360 radians,
The Angular displacement of the motor car is 360 radians.


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