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Angular Motion

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 Sub Topics 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.Vehicles in angular motion

Angular Motion Equations

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 $\alpha$.

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

Angular projectile motion is the angle that the projectile makes with the horizontal surface.

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

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

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).
That tells us that angular acceleration is a rate of change of angular velocity $\omega$.

Angular Velocity

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
$\omega$ is the angular velocity
$\theta_f$ is the final angular displacement
$\theta_i$ is the initial angular displacement
t is the time taken.

Angular Displacement

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!
If S is the angular displacement and $\theta$ is change in angle. The angular displacement is

Angular Speed

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
Here $\theta$ is the angle change and t is the time taken.

Examples of Angular Speed

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
$\omega$ = $\frac{\theta}{t}$
= $\frac{2 \pi}{60}$

The angular speed of minutes hand is
$\omega$ = $\frac{\theta}{t}$
= $\frac{2 \pi}{3600}$

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.