# Angular Velocity Formula – Definition, Example & More

**Angular velocity formula** refers to how fast an object rotates or revolves relative to another point, i.e. how fast the angular position or orientation of an object changes with time. There are two types of angular velocity: orbital angular velocity and spin angular velocity. Spin angular velocity refers to how fast a rigid body rotates with respect to its center of rotation. Orbital angular velocity refers to how fast a point object revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. In general, angular velocity is measured in angle per unit time, e.g. radians per second. The SI unit of angular velocity is expressed as radians/sec with the radian having a dimensionless value of unity, thus the SI units of angular velocity are listed as 1/sec.

\Angular velocity is usually represented by the symbol omega (ω, sometimes Ω). By convention, positive angular velocity indicates counterclockwise rotation, while negative is clockwise. Angular velocity is the rate of velocity at which an object or a particle is rotating around a center or a specific point in a given time period. It is also known as rotational velocity. Angular velocity is measured in angle per unit time or radians per second (rad/s). The rate of change of angular velocity is angular acceleration. Let us learn in more detail about the relation between angular velocity and linear velocity, angular displacement, and angular acceleration.

**Angular Velocity Formula**

**What is Angular Velocity?**

Angular velocity is a vector quantity and is described as the rate of change of angular displacement which specifies the angular speed or rotational speed of an object and the axis about which the object is rotating. The amount of change of angular displacement of the particle at a given period of time is called angular velocity. The track of the angular velocity vector is vertical to the plane of rotation, in a direction which is usually indicated by the right-hand rule.

**Angular Velocity Equation**

First, when you are talking about “angular” anything, be it velocity or some other physical quantity, recognize that, because you are dealing with angles, you’re talking about traveling in circles or portions thereof. You may recall from geometry or trigonometry that the circumference of a circle is its diameter times the constant pi, or **πd**. (The value of pi is about 3.14159.) This is more commonly expressed in terms of the circle’s radius **r**, which is half the diameter, making the circumference **2πr**.

In addition, you have probably learned somewhere along the way that a circle consists of 360 degrees (360°). If you move a distance S along a circle than the angular displacement θ is equal to S/r. One full revolution, then, gives 2πr/r, which just leaves 2π. That means angles less than 360° can be expressed in terms of pi, or in other words, as radians.

Taking all of these pieces of information together, you can express angles, or portions of a circle, in units other than degrees:

360° = (2π)radians, or

1 radian = (360°/2π) = 57.3°,

Whereas linear velocity is expressed in length per unit time, angular velocity is measured in radians per unit time, usually per second.

If you know that a particle is moving in a circular path with a velocity **v** at a distance **r** from the center of the circle, with the direction of **v** always being perpendicular to the radius of the circle, then the angular velocity can be written

ω = v/r,

where **ω** is the Greek letter omega. Angular velocity units are radians per second; you can also treat this unit as “reciprocal seconds,” because v/r yields m/s divided by m, or s^{-1}, meaning that radians are technically a unitless quantity.

**Angular velocity formula in ****RPM**

**Revolutions Per Minute Uses**

Revolutions per minute are also used to express how fast a circular object such as a wheel spins. Because one revolution is equivalent to one complete rotation or spin about a center point, a wheel that makes one complete rotation about its center in a minute is said to rotate about its center at a rate of 1 revolution per minute or 1 rpm. Because the second hand of a clock takes 1 minute to make one complete revolution about its center, it has a rotation rate of 1 revolution per minute or 1 rpm.

**Angular Velocity to RPM Conversion**

Angular velocity in degrees per second can be converted to revolutions per minute by multiplying the angular velocity by 1/6 since one revolution is 360 degrees and there are 60 seconds per minute. If the angular velocity is given as 6 degrees per second, the rpm would be 1 revolution per minute, since 1/6 multiplied by 6 is 1.

**RPM to Angular Velocity Conversion**

Revolutions per minute can be converted to angular velocity in degrees per second by multiplying the rpm by 6 since one revolution is 360 degrees and there are 60 seconds per minute. If the rpm is 1 rpm, the angular velocity in degrees per second would be 6 degrees per second, since 6 multiplied by 1 is 6.