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Rotational Speed

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You could observe across rotating or spinning bodies like a motor runs, a ball spins, merry goes round etc. Each undergo movement as rotations what we call rotational motion and the speed associated is called is known as rotational speed. Lets learn about this.


Rotational Speed Definition

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The rotational speed is speed gained in rotations. It judges about the number of rotations per unit time and gives idea about how many revolutions will be covered by any body in one second of time.The rotational speed and angular speed are many times expressed as synonym of each other. The rotational speed units are radians/sec or degrees/sec. But in common it is expressed as revolutions per minute (rpm) or rotations per second (rps).

Rotational Speed Formula

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Rotational Speed known as speed of revolution is the number of complete rotations or revolutions per unit time. It is a cyclic frequency expressed in Hertz (Hz), is given by the symbol $\omega$. It varies from the tangential speed as it speed will be constant.

The rotational speed is the angle of rotation or revolution for the time taken. It is measured in units such as degree per second or revolutions per minute.
$\omega$ = $\frac{Revoltuions}{time\ taken}$
Here $\theta$ is angular displacement and t is time taken

The rotational speed in terms of linear speed is
$\omega$ = $\frac{v}{r}$
where v is the tangential speed and r is radial distance.

Rotational Speed of Earth

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The rotational speed is that time the earth needs to rotate around its own axis.The rotational period of earth changes also depends on the position of observer on earth. If the observer is on the equator the rotational speed will be 1673km/hr.
There's a way to calculate it manually
If Earth's radius r is 6.4 x 106m and angular velocity v = 7.3 x 10-5rad/s. Then the speed of any point on the earth's surface at the equator due to the rotation of earth is
V = r $\omega$ = 6.4 x 106 x 7.3 x 10-5 m/s = 470 m/s = 1692 km/hr.

Rotational Speed to Linear Speed

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The rotational speed is the speed of the object when it is in a circular motion and the motion is called the circular or rotational or the angular motion. The linear speed of any object is that speed a body gains when its in a linear path.
If the body is moving in a circular path. It also undergoes linear path v having radius r from the center. Then relationship of linear speed and angular speed is given by
v = $\omega$ r
Where v is the linear speed,
$\omega$ is the rotational speed,
and r is the radius of the body.

Calculate Rotational Speed

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Here are some problems on rotational speed you can go through it:

Solved Examples

Question 1: A wheel rotates has radius of 0.3 m at 500 rpm. Calculate its angular speed in rad s-1.
Solution:
 
Given: Radius of the wheel = r = 0.3 m, Frequency of revolution f = 500 rpm = $\frac{500}{60}$ = 8.33 rps
The angular velocity is
$\omega$ = 2 $\pi$ f
               = 2 $\times$ 3.142 $\times$ 8.33
               = 52.367 rad s-1.
 

Question 2: Calculate the angular velocity of
(a) Seconds hand
(b) Minute hand of a clock.
Solution:
 
(a) Angular displacement = d $\theta$ = 2 $\pi$ rad, time taken = dt = 60 s
Angular velocity of seconds hand $\omega$ = $\frac{d \theta}{dt}$ = $\frac{2 \pi}{60}$ = 0.1047 rad s-1

(b) Angular displacement = d $\theta$ = 2 $\pi$ rad, time taken dt = 3600 s
Angular velocity of seconds hand $\omega$ = $\frac{d \theta}{dt}$ = $\frac{2 \pi}{3600}$ = 1.746 $\times$ 10-3 rad s-1