What Is Angular Acceleration?

David Isaac Rudel

Earth completes one full revolution around the Sun, 360 degrees (2π radians), every 365.24 days. This means the angle formed by an imaginary line connecting Earth to the Sun changes by a little less than 1 degree (π/180 radians) per day. Scientists use the term angular velocity to describe the motion of such an imaginary line. The angular acceleration of an object equals the rate at which this velocity changes.

Scientist with beakers
Scientist with beakers

Angular acceleration depends on the reference point chosen. An imaginary line connecting Earth to the Sun changes its angular velocity much more slowly than an imaginary line connecting Earth to the center of the galaxy. When discussing angular acceleration, there is no requirement that the object in question travel in a complete path around the reference point. One can discuss the changing angular velocity of one car with respect to another or of a vibrating hydrogen atom relative to the larger oxygen atom in a water molecule.

In the jargon of physics, acceleration is always a vector quantity regardless of whether it is linear or angular. If a car moving right at a rate of 33 feet/second (10 m/s) slams on the brakes to stop after 2 seconds, a scientist would describe the car’s average linear acceleration as ft/s2 ( m/s2). When describing angular acceleration, counterclockwise motion is considered positive and clockwise rotation is negative.

Scientists use the Greek letter alpha, α, to denote angular acceleration. By convention, vectors are bolded and their scalar values are denoted using non-bolded font. Thus, α refers to its magnitude. Angular acceleration can be written out in components as a, b, c>, where a is the angular acceleration around the x-axis, b is the acceleration around the y-axis, and c is the acceleration around the z-axis.

All the linear quantities used to describe objects or systems in Newtonian mechanics have angular analogs. The angular version of Newton’s famous F=ma is τ = Iα, where τ is torque and I is the moment of inertia for the system. These latter two quantities are the angular equivalents of force and mass, respectively.

In certain settings, the angular acceleration of a system around an axis is related to the linear acceleration of the system through space. For example, the distance a ball rolls in a given time is related to how quickly its outer surface rotates about its center, so long as one assumes the ball is not skidding or slipping. Thus the linear speed of the ball, s, must be related to the angular speed ω by the formula s=ωr, where r is the radius of the ball. Hence, the size of the linear acceleration must be related to α by a= αr.

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