Starting from rest, a wheel undergoes constant angular acceleration for a period of time T. At what time after the start of rotation does the wheel reach an angular speed equal to its average angular speed for this interval?

Let w be the angular speed of a rotating object, [tex]\alpha[/tex] its angular acceleration, and T the time the acceleration is acting upon the object. The basic formula for the angular motion is

[tex]w=w_o+\alpha t[/tex]

We are told the initial speed is zero, so

[tex]w=\alpha t[/tex]

The average angular speed from t=0 to t=T can be found by

[tex]\displaystyle \bar w=\frac{1}{T}\int_0^T{w.dt}[/tex]

[tex]\displaystyle \bar w=\frac{1}{T}\int_0^T{\alpha t.dt}[/tex]

[tex]\displaystyle \bar w=\frac{1}{T}\frac{\alpha T^2}{2}[/tex]

[tex]\displaystyle \bar w=\frac{\alpha T}{2}[/tex]

This value is reached at a certain time we need to compute, knowing that

[tex]w=\bar w[/tex]

Or equivalently

[tex]\displaystyle \alpha t=\frac{\alpha T}{2}[/tex]

Simplifying we have

[tex]\displaystyle t=\frac{T}{2}[/tex]

At t=T/2 the angular speed equals to its average angular speed