(a) ![2.79 rev/s^2](https://tex.z-dn.net/?f=2.79%20rev%2Fs%5E2)
The angular acceleration can be calculated by using the following equation:
![\omega_f^2 - \omega_i^2 = 2 \alpha \theta](https://tex.z-dn.net/?f=%5Comega_f%5E2%20-%20%5Comega_i%5E2%20%3D%202%20%5Calpha%20%5Ctheta)
where:
is the final angular speed
is the initial angular speed
is the angular acceleration
is the number of revolutions made by the disk while accelerating
Solving the equation for
, we find
![\alpha=\frac{\omega_f^2-\omega_i^2}{2d}=\frac{(20.0 rev/s)^2-(11.0 rev/s)^2}{2(50.0 rev)}=2.79 rev/s^2](https://tex.z-dn.net/?f=%5Calpha%3D%5Cfrac%7B%5Comega_f%5E2-%5Comega_i%5E2%7D%7B2d%7D%3D%5Cfrac%7B%2820.0%20rev%2Fs%29%5E2-%2811.0%20rev%2Fs%29%5E2%7D%7B2%2850.0%20rev%29%7D%3D2.79%20rev%2Fs%5E2)
(b) 3.23 s
The time needed to complete the 50.0 revolutions can be found by using the equation:
![\alpha = \frac{\omega_f-\omega_i}{t}](https://tex.z-dn.net/?f=%5Calpha%20%3D%20%5Cfrac%7B%5Comega_f-%5Comega_i%7D%7Bt%7D)
where
is the final angular speed
is the initial angular speed
is the angular acceleration
t is the time
Solving for t, we find
![t=\frac{\omega_f-\omega_i}{\alpha}=\frac{20.0 rev/s-11.0 rev/s}{2.79 rev/s^2}=3.23 s](https://tex.z-dn.net/?f=t%3D%5Cfrac%7B%5Comega_f-%5Comega_i%7D%7B%5Calpha%7D%3D%5Cfrac%7B20.0%20rev%2Fs-11.0%20rev%2Fs%7D%7B2.79%20rev%2Fs%5E2%7D%3D3.23%20s)
(c) 3.94 s
Assuming the disk always kept the same acceleration, then the time required to reach the 11.0 rev/s angular speed can be found again by using
![\alpha = \frac{\omega_f-\omega_i}{t}](https://tex.z-dn.net/?f=%5Calpha%20%3D%20%5Cfrac%7B%5Comega_f-%5Comega_i%7D%7Bt%7D)
where
is the final angular speed
is the initial angular speed
is the angular acceleration
t is the time
Solving for t, we find
![t=\frac{\omega_f-\omega_i}{\alpha}=\frac{11.0 rev/s-0 rev/s}{2.79 rev/s^2}=3.94 s](https://tex.z-dn.net/?f=t%3D%5Cfrac%7B%5Comega_f-%5Comega_i%7D%7B%5Calpha%7D%3D%5Cfrac%7B11.0%20rev%2Fs-0%20rev%2Fs%7D%7B2.79%20rev%2Fs%5E2%7D%3D3.94%20s)
(d) 21.7 revolutions
The number of revolutions made by the disk to reach the 11.0 rev/s angular speed can be found by using
![\omega_f^2 - \omega_i^2 = 2 \alpha \theta](https://tex.z-dn.net/?f=%5Comega_f%5E2%20-%20%5Comega_i%5E2%20%3D%202%20%5Calpha%20%5Ctheta)
where:
is the final angular speed
is the initial angular speed
is the angular acceleration
is the number of revolutions made by the disk while accelerating
Solving the equation for
, we find
![\theta=\frac{\omega_f^2-\omega_i^2}{2\alpha}=\frac{(11.0 rev/s)^2-0^2}{2(2.79 rev/s^2)}=21.7 rev](https://tex.z-dn.net/?f=%5Ctheta%3D%5Cfrac%7B%5Comega_f%5E2-%5Comega_i%5E2%7D%7B2%5Calpha%7D%3D%5Cfrac%7B%2811.0%20rev%2Fs%29%5E2-0%5E2%7D%7B2%282.79%20rev%2Fs%5E2%29%7D%3D21.7%20rev)