A particle at the edge of a ratating disc speeds up at a uniform angular acceleration `prop`. If the radius of the disc is `R`, find the angular distance covered by the particle till its acquires a total acceleration `a_0`.

Explanation

The angular displacement of the particle till it attains velocity `omega` is
`theta = (omega^2 - omega_0^2)/(2 prop)`
where `omega_0 = 0` (because the particle starts from rest)
Then, `theta = (omega^2 - omega_0^2)/(2 prop)`
Let us calculate `omega`
The total acceleration of the particle is
`a = sqrt(a_t^2 + a_r^2)`
where `a_t = R prop and a_r = R omega^2`
If the maximum acceleration of the particle is `a_0`, we have
`a_0 = sqrt(R^2 prop^2 + R^2 omega^4)` ltbrlt This gives `omega = ((a_0^2 - R^2 theta^2)/(2 R prop))^(1/4)`
Substituting `omega` from Eq. (ii) in Eq. (i), we have
`theta =sqrt(a_0^2 - R^2 prop^2)/(2 R prop)`.