Answer:
a) [tex]T_{min} = 80\,s[/tex]
Explanation:
a) Let consider that disk accelerates and decelerates at constant rate. The expression for angular acceleration and deceleration are, respectively:
Acceleration
[tex]\alpha_{1} = \frac{\omega_{max}^{2}}{2\cdot (400\,rad)}[/tex]
Deceleration
[tex]\alpha_{2} = -\frac{\omega_{max}^{2}}{2\cdot (400\,rad)}[/tex]
Since angular acceleration and deceleration have same magnitude but opposite sign. Let is find the maximum allowed angular speed from maximum allowed centripetal acceleration:
[tex]a_{r,max} = \omega_{max}^{2}\cdot r[/tex]
[tex]\omega_{max} = \sqrt{\frac{a_{r,max}}{r} }[/tex]
[tex]\omega_{max} = \sqrt{\frac{100\,\frac{m}{s^{2}} }{0.25\,m} }[/tex]
[tex]\omega_{max} = 20\,\frac{rad}{s}[/tex]
Maximum magnitude of acceleration/deceleration is:
[tex]\alpha = 0.5\,\frac{rad}{s^{2}}[/tex]
The least time require for rotation is:
[tex]T_{min} = 2\cdot \left(\frac{20\,\frac{rad}{s} }{0.5\,\frac{rad}{s^{2}} } \right)[/tex]
[tex]T_{min} = 80\,s[/tex]
