# Angular Velocity Formula Definition

**Angular Velocity Formula: **Laura received her Master’s degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions. Angular velocity applies to objects that move along a circular path. We will explore the definition of angular velocity and learn three different formulas we can use to calculate this type of velocity.

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.

## Formula For Angular Velocity

In 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 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.

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 the 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.

## Angular Velocity Formula Rpm

In everyday discourse, “speed” and “velocity” are often used interchangeably. In physics, however, these terms have specific and distinct meanings. “Speed” is the rate of displacement of an object in space, and it is given only by a number with specific units (often in meters per second or miles per hour). Velocity, on the other hand, is is a speed coupled to a direction. Speed, then, is called a scalar quantity, whereas velocity is a vector quantity.

When a car is zipping along a highway or a baseball is whizzing through the air, the speed of these objects is measured in reference to the ground, whereas the velocity incorporates more information. For example, if you’re in a car traveling at 70 miles per hour on Interstate 95 on the East Coast of the United States, it’s also helpful to know whether it is headed northeast toward Boston or south toward Florida. With the baseball, you might want to know if its y-coordinate is changing more rapidly than its x-coordinate (a fly ball) or if the reverse is true (a line drive). But what about the spinning of the tires or the rotation (spin) of the baseball as the car and the ball move toward their ultimate destination? For these kinds of questions, physics offers the concept of **angular velocity**.

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Things move through three-dimensional physical space in two main ways: translation and rotation. The translation is the displacement of the entire object from one location to another, like a car driving from New York City to Los Angeles. Rotation, on the other hand, is the cyclical motion of an object around a fixed point. Many objects, such as the baseball in the above example, exhibit both types of movement at the same time; as a fly ball moved through the air from home plate toward the outfield fence, it also spins at a given rate around its own center.

## Centripetal Acceleration Formula Angular 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.

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.

### Radians

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 VelocitySince 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.

**Angular velocity** is less common than linear velocity because it only applies to objects that are moving along a circular path. For instance, a racecar on a circular track, a roulette ball on a roulette wheel, or a Ferris wheel, all have an angular velocity.

Angular Velocity FormulasFor example, a Ferris wheel may be rotating pi / 6 radians every minute. Therefore, the Ferris wheel’s angular velocity would be pi / 6 radians per minute. The angular velocity of an object is the object’s angular displacement with respect to time. When an object is traveling along a circular path, the central angle corresponding to the object’s position on the circle is changing. The angular velocity, represented by *w*, is the rate of change of this angle with respect to time.

There are **three formulas** we can use to find angular velocity.

**Option 1**

The first comes straight from the definition. Angular velocity is the rate of change of the position angle of an object with respect to time, so *w* = theta / *t*, where *w* = angular velocity, theta = position angle, and *t* = time.

## Angular Velocity Formula Physics

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 then 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.

### 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 = a_{t}/r

This is also given by:

α = ω/t

because for circular motion, a_{t} = 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:

a_{c} = v^{2}/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 **a _{c}** were not equal to

**a**, the object would either fly off into space or soon crash into the middle of the circle.

_{t}## What is the unit of angular velocity?

The SI **unit of angular velocity** is radians per second. But it may be measured in other **units** as well (such as degrees per second, degrees per hour, etc.). **Angular velocity** is usually represented by the symbol omega (Ω or ω).

## What is the formula of Omega?

Angular frequency (ω), also known as radial or circular frequency, measures angular displacement per unit time. Its units are therefore degrees (or radians) per second. Hence, 1 Hz ≈ 6.28 rad/sec. Since 2π radians = 360°, 1 radian ≈ 57.3°.

## Are angular speed and angular velocity the same?

**Angular speed**is the rate at which an object changes its angle (measured) in radians, in a given time period.

**Angular speed and angular velocity**use the

**same**formula; the difference between the two is that

**Angular speed**is a scalar quantity, while

**angular velocity**is a vector quantity.