Answer
The least time required for the rotation is $~~40~s$
Work Step by Step
We can find the maximum value of the angular speed:
$a_c = \omega^2~r$
$\omega^2 = \frac{a_c}{r}$
$\omega = \sqrt{\frac{a_c}{r}}$
$\omega = \sqrt{\frac{400~m/s^2}{0.25~m}}$
$\omega = 40~rad/s$
If the disk accelerates at a constant rate to $\omega = 40~rad/s$, then the average angular speed is $20~rad/s$
We can find the time to rotate through $400~rad$:
$t = \frac{\theta}{\omega_{ave}}$
$t = \frac{400~rad}{20~rad/s}$
$t = 20~s$
This time is the time required to complete half the rotation. By symmetry, the deceleration period also takes $20~s$
The minimum required time for the entire rotation is $~~40~s$
