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Angular Speed

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Our world is on wheels. From pulley, bench riders, circular saws to the vehicle a person rides runs on wheels. At what rate the wheel turns is its angular speed. The angular speed is often measured in revolutions per second.
Angular Speed
In amusements park you may often take a ride. It could experienced that rotations taking place and in each rotations the swings in the outer ring travel at a greater distance and have the greater linear speed than the swings in the inner ring. Interestingly all of these swings complete the same revolutions during any given ride. So all these have same angular speed but Varied linear speed.

What is Angular Speed?

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Any rotating body rotates in a certain direction covers a distance in a given time. At what speed these bodies rotates is what angular Speed is! Thus it is nothing but a measure of rotational rate of any body. It is calculated by dividing the distance covered by the rotating body with the time it takes to rotate.The angular speed units are revolutions per seconds (rev/s) or radians per second (rad/s).

Angular Speed Formula

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Lets see the angular change using the figure:
Linear Speed

Angular speed is the rate of change of the angle (in radians) with time, and it has units rad/s given as
$\omega$ = $\frac{\theta}{t}$
Here $\theta$ is the angle $\theta$ in radians and t is time in seconds.

The time taken for complete rotation is the period T then angular speed is
$\omega$ = $\frac{2\pi}{T}$

If 2 $\pi$ is the angle covered in radians to get complete circle then angular speed is
$\omega$ = 2 $\pi$ f
Suppose you are going round in a circle of radius r at a linear speed v (ms-1). Then the distance covered in rotation is 2 $\pi$ r and time taken for one rotation is T. So the Linear speed is
Angular speed $\omega$ = $\frac{Distance\ moved}{time\ taken}$ = $\frac{2 \pi}{T}$ = $\frac{v}{r}$
Here $\omega$ is the angular frequency or angular speed
T is the period (measured in seconds)
f is the frequency (measured in hertz)
v is the tangential velocity of a point about the axis of rotation
r is the radius of rotation

The relationship between the angular speed and linear speed is
v = $\omega$ r
Here $\omega$ is angular speed and r is the radius of circular path.

Angular Speed of Earth

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Earth revolves the sun. It takes 365.25 days to complete the revolution around the sun.To convert the days to seconds T = 365.25 $\times$ 24 $\times$ 60 $\times$ 60 = 31557600 s.
The Angular speed is given by
$\omega$ = $\frac{2 \pi}{T}$
= $\frac{2 \times 3.142}{365.25 \times 24 \times 60 \times 60}$
= 1.998 $\times$ 10-7 radians.
Thus the angular speed of earth is 1.998 $\times$ 10-7 radians.

Finding Angular Speed

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Below are some angular speed problems you can go through it:

Solved Examples

Question 1: A wheels of 1 m rotates at the speed of 7 m/s. Calculate its angular speed.
Solution:
 
Given: Radius r = 1 m and Linear Speed v = 7 m/s
The Angular speed is given by
$\omega$ = $\frac{v}{r}$
               = $\frac{7 m/s}{1m}$
               = 7 rad/s
 

Question 2: A 2 kg mass rotates in a circular path of radius 6 m completes 1 revolution in 5 seconds. Calculate its angular speed.
Solution:
 
Given : Mass m = 2 kg, radius r = 6 m, time t = 5 s, v = ?
To find the angular speed lets find the linear speed v. It is given by
v = $\frac{2 \pi r}{t}$ = $\frac{2 \times 3.142 \times 6 m}{5s}$ 
  = $\frac{37.704}{5}$
  = 7.54 m/s
Angular speed $\omega$ = $\frac{v}{r}$
                                     = $\frac{7.54\ m/s}{6\ m}$
                                     = 1.2568 rad/s.