How to Calculate the Angular Velocity

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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.

The Basics of Motion

Things move through three-dimensional physical space in two main ways: translation and rotation. 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.

Describing these two kinds of motion are treated as separate physics problems; that is, when calculating the distance the ball travels through the air based on things like its initial launch angle and the speed with which it leaves the bat, you can ignore its rotation, and when calculating its rotation you can treat it as sitting in one place for present purposes.

The 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 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.

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.

Related Quantities and Expressions

Although angular velocity is usually expressed, as noted, in radians per second, there may be instances in which it is preferable or necessary to use degrees per second instead, or conversely, to convert from degrees to radians before solving a problem.

Say you were told that a light source rotates through 90° every second at a constant velocity. What is its angular velocity in radians?

First, remember that 2π radians = 360°, and set up a proportion:

360/2π = 90/x

360x = 180π

x = ω = π/2

The answer is one-half pi radians per second.

If you were further told that the light beam has a range of 10 meters, what would be the tip of the beam's linear velocity v, its angular acceleration α and its centripetal acceleration ac?

To solve for v, from above, v = ωr, where ω = π/2 and r = 10m:

(π/2)(10) = 5π rad/s = 15.7 m/s

To solve for α, simply add another time unit to the denominator:

α = 5π rad/s2

(Note that this only works for problems in which the angular velocity is constant.)

Finally, also from above, ac = v2/r = (15.7)2/10 = 24.65 m/s2.

Angular Velocity vs. Linear Velocity

Building on the previous problem, imagine yourself on a very large merry-go-round, one with an unlikely radius of 10 kilometers (10,000 meters). This merry-go-round makes one complete revolution every 1 minute and 40 seconds, or every 100 seconds.

One consequence of the difference between angular velocity, which is independent of the distance from axis of rotation, and linear circular velocity, which is not, is that two people experiencing the same ω may be undergoing vastly different physical experienced. If you happen to be 1 meter from the center if this putative, massive merry-go-round, your linear (tangential) velocity is:

ωr = (2π rad/100 s)(1 m) = 0.0628 m/s, or 6.29 cm (less than 3 inches) per second.

But if you're on the rim of this monster, your linear velocity is:

ωr = (2π rad/100 s)(10,000 m) = 628 m/s. That's about 1,406 miles per hour, faster than a bullet. Hang on!

References

About the Author

Kevin Beck holds a bachelor's degree in physics with minors in math and chemistry from the University of Vermont. Formerly with ScienceBlogs.com and the editor of "Run Strong," he has written for Runner's World, Men's Fitness, Competitor, and a variety of other publications. More about Kevin and links to his professional work can be found at www.kemibe.com.

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