One way for pilots to train for the physical demands of flying at high speeds is with a device called the "human centrifuge." It involves having the pilots travel in circles at high speeds so that they can experience forces greater than their own weight. The diameter of the NASA device is 17.8 m. (a) Suppose a pilot starts at rest and accelerates at a constant rate so that he undergoes 30 rev in 2 min. What is his angular acceleration (in rad/s2)? (b) What is his angular velocity (in rad/s) at the end of that time? (c) After the 2-min period, the centrifuge moves at a constant speed. The g-force experienced is the centripetal force keeping the pilot moving along a circular path. What is the g-force experienced by the pilot? ( 1 g = mass × 9 .80 m/s 2 ) (d) The pilot can tolerate 12 g’s in the horizontal direction. How long would it take the centrifuge to reach that state if it starts at the angular speed found in part (c) and accelerates at the rate found in part (a)?

[tex]\theta=\omega_it+\frac{1}{2}\alpha t^2\\\Rightarrow 30\times 2\pi=0\times t+\frac{1}{2}\times \alpha\times (2\times 60)^2\\\Rightarrow \alpha=\frac{30\times 2\pi \times 2}{(2\times 60)^2}\\\Rightarrow \alpha=0.02617\ rad/s^2[/tex]

The angular acceleration is 0.026167 rad/s²

[tex]\omega_f=\omega_i+\alpha t\\\Rightarrow \omega_f=0+0.02617\times 2\times 60\\\Rightarrow \omega_f=3.1404\ rad/s[/tex]

The angular velocity at the end of that time is 3.1404 rad/s

Centripetal acceleration

[tex]a_c=\omega^2r\\\Rightarrow a_c=3.1404^2\times \frac{17.8}{2}\\\Rightarrow a_c=87.773\ m/s^2[/tex]

Dividing centripetal acceleration by g

[tex]\frac{a_c}{g}=\frac{87.773}{9.8}\\\Rightarrow a_c=8.95642g[/tex]

The centripetal acceleration is [tex]a_c=8.95642g[/tex]

When Centripetal acceleration is 12 g

[tex]a_c=12\times 9.8=117.6\ m/s^2[/tex]

[tex]a_c=\omega^2r\\\Rightarrow \omega=\sqrt{\frac{117.6}{\frac{17.8}{2}}}\\\Rightarrow \omega=3.63503\ rad/s[/tex]

[tex]\omega_f=\omega_i+\alpha t\\\Rightarrow t=\frac{\omega_f-\omega_i}{\alpha}\\\Rightarrow t=\frac{3.63503-3.1404}{0.02617}\\\Rightarrow t=18.90064\ s[/tex]

The time taken to reach that state if it starts at the angular speed found in part is 18.90064 seconds