Angular Speed

Angular speed measures how fast the central angle of a rotating body changes with respect to time. In this article, we will be familiarising ourselves with angular speed.

What Is Angular Speed?

Angular speed is defined as the rate of change of angular displacement, and it is expressed as follows:

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where θ is the angular displacement, t is the time and ω is the angular speed

Angular Speed Unit

The unit of angular speed is radian per second. The same formula represents both angular speed and angular velocity. Angular velocity is a vector quantity that expresses both direction and magnitude, while angular speed describes magnitude only.

Angular Speed Formula

Angular Speed (ω) is the scalar measure of rotation rate. In one complete rotation, angular distance travelled is 2π and time is time period (T). Angular speed is given by the following formula:

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From the above equation, we can conclude that ω is equivalent to 2πf, where 1/T is equivalent to f (frequency).

Angular Speed of Earth

Our Earth takes about 365.25 days to finish one revolution around the Sun. 

Converting days into seconds, we get

T = 365.25 x 24 x 60 x 60 = 31557600 seconds

We know that angular speed = 2π/T, hence

ω = 1.99 x 10-7 radians /seconds.

The angular speed of Earth is 1.99 x 10-7 radians /seconds.

Angular Speed Examples

The Earth rotates once on its axis once every 24 hours. What is its angular speed?

Solution:

The angle traversed in one revolution is 2π. The time taken for this rotation is 24 hours.

Converting hours into second we get,

t = 24 hr x 60 min/hr x 60 sec/min = 86400 sec

We know that angular speed is given by the formula, ω = θ /t.

Substituting the values in the equation, we get

ω = 2π/86400 sec

Solving, we get

ω = 0.0000726 radians/sec = 7.26 x 10-5 rad/sec

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.

Linear speed is given by the following formula:

v = s/t

Where s is the linear displacement of arc and θ = S/r. 

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

Therefore, V = r ω

Rearranging, we get,

ω = V/r

Where V is equivalent to the linear speed.

The equation gives the relation between angular speed, linear speed, and radius of the circular path.