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Angular Motion

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Any body in circular path covers certain angle from center taken as fixed axis. The distance is expressed in angles as there will be twist in the position.
The angular motion is the rotatory motion that a body takes when all the points on a body or object moves in circles about the central line or axis. Lets see more about this.
Angular Motion
Vehicles in angular motion

Angular Motion Equations

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Lets us take a body moving in a circular path having initial angle $\theta_o$ after some time it undergoes displacement $\theta$ covering the distance with angular velocity  $\omega$ from its initial angular velocity $\omega_o$.  The angular acceleration related to it is $\alpha$.
What is Angular Motion
They are many equations in angular motion. Here are some equations:
$\omega = \omega_o + \alpha\ t$
$\theta$ = $\frac{1}{2}$ ($\omega_o + \omega$) t
$\theta$ = $\omega_o$ t + $\frac{1}{2}$ $\alpha$ t2
$\omega^2$ = $\omega_o^2$ + 2 $\alpha$ $\theta$
Here $\theta$ is the angle change,
$\theta_o$ is the initial angular position,
$\omega$ is the angular velocity,
$\omega_o$ is initial angular velocity at t,
$\alpha$ is the angular acceleration.

Angular Projectile Motion

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Angular projectile motion is the angle that the projectile makes with the horizontal surface.
Projectile
Horizontal component in terms of $\theta$ is v cos $\theta$
Vertical component in terms of $\theta$ is v sin $\theta$.

Angular Motion with Constant Acceleration

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If a body is moving in a circular path maintains constant acceleration $\alpha$. The angular motion equations are
$\alpha$ = Constant
$\omega$ = $\omega_o$ + $\alpha$ t
$\theta$ = $\theta_o$ + $\omega_o$ t + $\frac{1}{2}$ $\alpha$ t2

Here $\theta$ is the angle change,
$\theta_o$ is the initial angular position,
$\omega$ is the angular velocity,
$\omega_o$ is initial angular velocity at t,
$\alpha$ is the angular acceleration.

Angular Acceleration

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Angular acceleration ($\alpha$) also known as rotational acceleration is the change in velocity of angular velocity with time. Its a vector quantity and is expressed in radians per second (rad s-2).
Angular Acceleration FormulaThat tells us that angular acceleration is a rate of change of angular velocity $\omega$.

Angular Velocity

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Angular velocity decides what rate the body is moving in a circular path. It is the rate of change of angular displacement with time. It is given as
Angular Velocity Formula$\omega$ is the angular velocity
$\theta_f$ is the final angular displacement
$\theta_i$ is the initial angular displacement
t is the time taken.
A body moves in a circular path displaces  from a point to another turns through an angle $\theta$ if seen from center. This is what is angular displacement is!
Angular Displacements
If S is the angular displacement and $\theta$ is change in angle. The angular displacement is
Angular Displacement Formula
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Angular speed ($\omega$) is that what decides how fast a body is changing its position in a circular path. Its the ratio of what's the change in angular position to how much time it takes. It is given by
Angular Speed FormulaHere $\theta$ is the angle change and t is the time taken.
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Examples of Angular Speed

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Here are given some simple examples of angular speed:

Solved Examples

Question 1: What is the angular speed of (a) Seconds hand and (b) Minute hand of a clock.
Solution:
 
Given : Angular displacement $\theta$ = 2 $\pi$ radians,
           Time taken t = 60 s

The angular speed of second hand is
$\omega$ = $\frac{\theta}{t}$
               = $\frac{2 \pi}{60}$
               = 0.1047 rad s-1.

The angular speed of minutes hand is
$\omega$ = $\frac{\theta}{t}$
               = $\frac{2 \pi}{3600}$
               = 0.001745 rad s-1.

 

Question 2: A wheel of radius 0.5 m rotates at 500 rpm. Calculate the angular velocity in rad s-1.
Solution:
 
Given: Radius of wheel = r = 0.5 m,
          Frequency of revolution f = 500 rpm = 8.33 rps

The angular velocity is
$\omega$ = 2 $\pi$ f
     = 2 $\times$ $\pi$ $\times$ 8.33 rps
     = 16.66 $\pi$ rad s-1.