# How To Calculate The Angular Velocity Formula

Angular Velocity Formula: n physics, angular velocity 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 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 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, {\displaystyle 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.

## Formula For Angular Velocity

In the simplest case of circular motion at radius {\displaystyle r}, with position given by the angular displacement {\displaystyle \phi (t)} from the x-axis, the orbital angular velocity is the rate of change of angle with respect to time: {\displaystyle \omega ={\tfrac {d\phi }{dt}}}. If {\displaystyle \phi } is measured in radians, the distance from the x-axis around the circle to the particle is {\displaystyle \ell =r\phi }, and the linear velocity is {\displaystyle v(t)={\tfrac {d\ell }{dt}}=r\omega (t)}, so that {\displaystyle \omega ={\tfrac {v}{r}}}.

In the general case of a particle moving in the plane, the orbital angular velocity is the rate at which the position vector relative to a chosen origin “sweeps out” angle. The diagram shows the position vector {\displaystyle \mathbf {r} } from the origin {\displaystyle O} to a particle {\displaystyle P}, with its polar coordinates {\displaystyle (r,\phi )}. (All variables are functions of time {\displaystyle t}.) The particle has linear velocity splitting as {\displaystyle \mathbf {v} =\mathbf {v} _{\|}+\mathbf {v} _{\perp }}, with the radial component {\displaystyle \mathbf {v} _{\|}} parallel to the radius, and the cross-radial (or tangential) component {\displaystyle \mathbf {v} _{\perp }} perpendicular to the radius. When there is no radial component, the particle moves around the origin in a circle; but when there is no cross-radial component, it moves in a straight line from the origin. Since radial motion leaves the angle unchanged, only the cross-radial component of linear velocity contributes to angular velocity.

The angular velocity ω is the rate of change of angular position with respect to time, which can be computed from the cross-radial velocity as:

{\displaystyle \omega ={\frac {d\phi }{dt}}={\frac {v_{\perp }}{r}}.}

Here the cross-radial speed {\displaystyle v_{\perp }} is the signed magnitude of {\displaystyle \mathbf {v} _{\perp }}, positive for counter-clockwise motion, negative for clockwise. Taking polar coordinates for the linear velocity {\displaystyle \mathbf {v} } gives magnitude {\displaystyle v} (linear speed) and angle {\displaystyle \theta } relative to the radius vector; in these terms, {\displaystyle v_{\perp }=v\sin(\theta )}, so that

{\displaystyle \omega ={\frac {v\sin(\theta )}{r}}.}

These formulas may be derived from {\displaystyle \mathbf {r} =(x(t),y(t))}{\displaystyle \mathbf {v} =(x'(t),y'(t))} and {\displaystyle \phi =\arctan(y(t)/x(t))}, together with the projection formula {\displaystyle v_{\perp }={\tfrac {\mathbf {r} ^{\perp }\!\!}{r}}\cdot \mathbf {v} }, where {\displaystyle \mathbf {r} ^{\perp }=(-y,x)}.

In two dimensions, angular velocity is a number with plus or minus sign indicating orientation, but not pointing in a direction. The sign is conventionally taken to be positive if the radius vector turns counter-clockwise, and negative if clockwise. Angular velocity then may be termed a pseudoscalar, a numerical quantity which changes sign under a parity inversion, such as inverting one axis or switching the two axes.

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.

To get our second formula for angular velocity, we recognize that theta is given in radians, and the definition of radian measure gives theta = s / r. Thus, we can plug theta = s / r into our first angular velocity formula. This gives w = (s / r) / t.

Angular velocity is rotational speed. Something is spinning. It is an abbreviation for Revolutions per minute. Other related units that express the same property are degrees per second and radians per second.

## Average Angular Velocity Formula

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:

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

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.

## Angular Velocity Formula Physics

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.

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.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.

We have one more thing to review before getting to angular velocity, and that is radians. When we deal with angular velocity, we use the radian measure of an angle, so it is important that we are familiar with radian measure. The technical definition of radian measure is the length of the arc subtended by the angle, divided by the radius of the circle the angle is a part of, where subtended means to be opposite of the angle and to extend from one point on the circle to the other, both marked off by the angle. This tells us that an angle theta = s / r radians, where s = length of the arc corresponding to theta, and r = radius of the circle theta is a part of.

Angular Velocity To Linear Velocity FormulaSince most of us are comfortable with the degree measurement of angles, it is convenient that we can easily convert degree measure to radian measure by multiplying the degree measure by pi / 180. For example, a 45 degree angle has a radian measure 45 (pi / 180), which is equal to pi / 4 radians.