How To Find 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

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

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.