# How to Find Revolutions From Angular Acceleration

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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.

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.

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}

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

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

#### Tips

• 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).

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