A disk, with a radius of 0.25 m, is to be rotated like a merry-go-round through 800 rad, starting from rest, gaining angular speed at the constant rate `alpha_(1)` through the first 400 rad and then losing angular speed at the constant rate - `alpha_(1)` until it is again at rest. The magnitude of the centripetal acceleration of any portion of the disk is not to exceed `400m//s^(2)`. (a) What is the least time required for the rotation? (b) What is the corresponding value of `alpha_(1)`?

Explanation

To solve the problem step by step, we will break it down into parts (a) and (b) as requested.

### Given Data:
- Radius of the disk, \( r = 0.25 \, \text{m} \)
- Total angle of rotation, \( \theta_t = 800 \, \text{rad} \)
- First part angle, \( \theta_1 = 400 \, \text{rad} \)
- Initial angular speed, \( \omega_i = 0 \, \text{rad/s} \)
- Final angular speed after the first part, \( \omega_{\text{max}} \)
...