Finding Revolutions from Angular Acceleration

The equation of motion for a constant acceleration:

(x(t)=x(0)+v(0)t+frac{1}{2}at^2)

has an angular equivalent:

(theta(t)=theta(0)+omega(0)t+frac{1}{2}alpha t^2)

For the uninitiated, θ(t) refers to the measurement of some angle at time ​t​ while θ(0) refers to the angle at time zero. ω(0) refers to the initial angular speed, at time zero. α is the constant angular acceleration.

An example of when you might want to find a revolution count after a certain time ​t​, given a constant angular acceleration, is when a constant torque is applied to a wheel.

Step 1

Suppose you want to find the number of revolutions of a wheel after 10 seconds. Suppose also that the torque applied to generate rotation is 0.5 radians per second-squared, and the initial angular velocity was zero.

Step 2

Plug these numbers into the formula in the introduction and solve for θ(t). Use θ(0)=0 as the starting point, without loss of generality. Therefore, the equation

(theta(t)=theta(0)+omega(0)t+frac{1}{2}alpha t^2)

becomes

(theta(10)=0+0+frac{1}{2}timesfrac{1}{2}times 10^2=25text{ radians})

Step 3

Divide θ(10) by 2π to convert the radians into revolutions. 25 radians / 2π = 39.79 revolutions.

Step 4

Multiply by the radius of the wheel, if you also want to determine how far the wheel traveled.

TL;DR (Too Long; Didn’t Read)

For nonconstant angular momentum, use calculus to integrate the formula for the angular acceleration twice with respect to time to get an equation for θ(t).