# Angular Velocity Formula | Defination

In physics, 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 centre of rotation. Orbital angular velocity refers to how fast a rigid body’s centre of rotation 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 counter-clockwise rotation, while negative is clockwise.

For example, a geostationary satellite completes one orbit per day above the equator, or 360 degrees per 24 hours, and has angular velocity ω = 360 / 24 = 15 degrees per hour, or 2π / 24 ≈ 0.26 radians per hour. If angle is measured in radians, the linear velocity is the radius times the angular velocity, { v=r\omega }. With orbital radius 42,000 km from the earth’s center, the satellite’s speed through space is thus v = 42,000 × 0.26 ≈ 11,000 km/hr. The angular velocity is positive since the satellite travels eastward with the Earth’s rotation (counter-clockwise from above the north pole.)

In three dimensions, angular velocity is a pseudovector, with its magnitude measuring the rate at which an object rotates or revolves, and its direction pointing perpendicular to the instantaneous plane of rotation or angular displacement. The orientation of angular velocity is conventionally specified by the right-hand rule.

## Angular Velocity Formula RPM

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 that 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.

## Centripetal Acceleration Formula Angular Velocity

When an object follows a rotational path, it is said to move in an angular motion or the commonly known rotational motion. In the course of such motion, the velocity of the object is always changing. Velocity being a vector involves a movement of an object with speed that has direction. Now, since in a rotational motion, the particles tend to follow a circular path their direction at every point changes constantly. This change results in a change in velocity. This change in velocity with time gives us the acceleration of that object.

Angular acceleration is a non-constant velocity and is similar to linear acceleration of translational motion. Understanding linear displacement, velocity, and acceleration are easy and this is why when we intend to study rotational motion, we compare its vectors with translational motion. Like linear acceleration, angular acceleration (α) is the rate of change of angular velocity with time. Therefore, α = dω/ dt

Now since for rotation about a fixed axis the direction of angular velocity is fixed therefore the direction of angular momentum α is also fixed.  For such cases, the vector equation transforms into a scalar equation.

## Linear Velocity

Before we can get to angular velocity, we will first review linear velocity. Linear velocity applies to an object or particle that is moving in a straight line. It is the rate of change of the object’s position with respect to time.

One of the most common examples of linear velocity is your speed when you are driving down the road. Your speedometer gives your speed, or rate, in miles per hour. This is the rate of change of your position with respect to time, in other words, your speed is your linear velocity.

Linear velocity can be calculated using the formula v = s / t, where v = linear velocity, s = distance traveled, and t = time it takes to travel distance. For example, if I drove 120 miles in 2 hours, then to calculate my linear velocity, I’d plug s = 120 miles, and t = 2 hours into my linear velocity formula to get v = 120 / 2 = 60 miles per hour.

## Rotational Motion Equations

The angular acceleration formula is derived in the same essential way as the angular velocity formula: It is merely the linear acceleration in a direction perpendicular to a radius of the circle (equivalently, its acceleration along a tangent to the circular path at any point) divided by the radius of the circle or portion of a circle, which is:

α = at/r

This is also given by:

α = ω/t

because for circular motion, at = ωr/t = v/t.

α, as you probably know, is the Greek letter “alpha.” The subscript “t” here denotes “tangent.”

Curiously enough, however, rotational motion boasts another kind of acceleration, called centripetal (“center-seeking”) acceleration. This is given by the expression:

ac = v2/r

This acceleration is directed toward the point around which the object in question is rotating. This may seem strange, since the object is getting no closer to this central point since the radius r is fixed. Think of centripetal acceleration as a free-fall in which there is no danger of the object hitting the ground, because the force drawing the object toward it (usually gravity) is exactly offset by the tangential (linear) acceleration described by the first equation in this section. If ac were not equal to at, the object would either fly off into space or soon crash into the middle of the circle.