Angular Velocity

Angular velocity is a measurement of the rate of change of angular position of an object over a period of time. The symbol used for angular velocity is usually a lower case Greek symbol omega, ω. Angular velocity is represented in units of radians per time or degrees per time (usually radians in physics), with relatively straightforward conversions allowing the scientist or student to use radians per second or degrees per minute or whatever configuration is needed in a given rotational situation, whether it be a large ferris wheel or a yo-yo. (See our article on dimensional analysis for some tips on performing this sort of conversion.)

Calculating Angular Velocity

Calculating angular velocity requires understanding the rotational motion of an object, θ. The average angular velocity of a rotating object can be calculated by knowing the initial angular position, θ₁, at a certain time t₁, and a final angular position, θ₂, at a certain time t₂. The result is that the total change in angular velocity divided by the total change in time yields the average angular velocity, which can be written in terms of the changes in this form (where Δ conventionally is a symbol that stands for "change in"):

• ω_av: Average angular velocity
• θ₁: Initial angular position (in degrees or radians)
• θ₂: Final angular position (in degrees or radians)
• Δθ = θ₂ - θ₁: Change in angular position (in degrees or radians)
• t₁: Initial time
• t₂: Final time
• Δt = t₂ - t₁: Change in time

Average Angular Velocity:
ω _av = ( θ ₂ - θ ₁) / ( t ₂ - t ₁) = Δ θ / Δ t

The attentive reader will notice a similarity to the way you can calculate standard average velocity from the known starting and ending position of an object. In the same way, you can continue to take smaller and smaller Δt measurements above, which gets closer and closer to the instantaneous angular velocity. The instantaneous angular velocity ω is determined as the mathematical limit of this value, which can be expressed using calculus as:

Instantaneous Angular Velocity:
ω = Limit as Δ t approaches 0 of Δ θ / Δ t = dθ / dt

Those familiar with calculus will see that the result of these mathematical reformulations is that the instantaneous angular velocity, ω, is the derivative of θ (angular position) with respect to t (time) ... which is precisely what our initial definition of angular velocity was, so everything works out as expected.

Also Known As: average angular velocity, instantaneous angular velocity