# Angular Speed

Angular speed is the measure of how fast the central angle of a rotating body changes with respect to time. In this short piece of article, let us learn more angular speed formula, the relationship between angular speed and linear speed along with a few angular speed problems.

## What is Angular Speed?

Angular speed is defined as the rate of change of angular displacement and is given by the expression

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

where $\theta$ is the angular displacement

t is the time

$\omega$ is the angular speed

The unit of angular speed is radian per second. Both angular speed and angular velocity are represented by the same formula. But it should be noted that angular velocity is different from the angular speed. Angular velocity is a vector quantity that expresses both direction and magnitude while angular speed expresses the magnitude only.

### Angular Speed Formula

A scalar measure of rotation rate is known as Angular Speed (ω). In one complete rotation, angular distance travelled is 2π and time is time period (T) then, the angular speed is given by,
$Angular\,Speed=\frac{2\pi}{T}$.
From the above equation, we can concur that ω is equivalent to 2πf, where 1/T is equivalent to f (frequency).
Thus, the rotation rate is also articulated as an angular frequency.

## Relationship Between Angular Speed and Linear Speed

Let the object be travelling in a round path of radius r and angular displacement be θ then we have, angle, θ = arc/radius

We know that linear speed, V =S/t,
where S is a linear displacement of arc, and
θ = S/r

Thus, linear speed V =(θ.r)/t
= r. (θ/t)

V = r ω

Hence, Angular speed,

ω = V/r
Where V is equivalent to the linear speed

This is the relation amongst angular speed, linear speed, and radius of the circular path.
From this relation, one can compute this speed.

### Angular Speed of Earth

Our Earth takes about 365.25 days to finish one revolution around the Sun, now translate days into seconds,
T = 365.25 x 24 x 60 x 60 = 31557600 seconds
Angular speed = 2π/T
Therefore,
Hence,
ω = 1.99 x 10−7 radians /seconds.

### Unit of Angular Speed

Angular Speed is articulated as,
ω = V/r
Where,
linear speed is equivalent to V
the radius of the circular path is equivalent to r
We get linear speed,
V = r. (θ/t)
∴ ω = (θ/t)