(a) [tex]2.79 rev/s^2[/tex]
The angular acceleration can be calculated by using the following equation:
[tex]\omega_f^2 - \omega_i^2 = 2 \alpha \theta[/tex]
where:
[tex]\omega_f = 20.0 rev/s[/tex] is the final angular speed
[tex]\omega_i = 11.0 rev/s[/tex] is the initial angular speed
[tex]\alpha[/tex] is the angular acceleration
[tex]\theta=50.0 rev[/tex] is the number of revolutions made by the disk while accelerating
Solving the equation for [tex]\alpha[/tex], we find
[tex]\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[/tex]
(b) 3.23 s
The time needed to complete the 50.0 revolutions can be found by using the equation:
[tex]\alpha = \frac{\omega_f-\omega_i}{t}[/tex]
where
[tex]\omega_f = 20.0 rev/s[/tex] is the final angular speed
[tex]\omega_i = 11.0 rev/s[/tex] is the initial angular speed
[tex]\alpha=2.79 rev/s^2[/tex] is the angular acceleration
t is the time
Solving for t, we find
[tex]t=\frac{\omega_f-\omega_i}{\alpha}=\frac{20.0 rev/s-11.0 rev/s}{2.79 rev/s^2}=3.23 s[/tex]
(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
[tex]\alpha = \frac{\omega_f-\omega_i}{t}[/tex]
where
[tex]\omega_f = 11.0 rev/s[/tex] is the final angular speed
[tex]\omega_i = 0 rev/s[/tex] is the initial angular speed
[tex]\alpha=2.79 rev/s^2[/tex] is the angular acceleration
t is the time
Solving for t, we find
[tex]t=\frac{\omega_f-\omega_i}{\alpha}=\frac{11.0 rev/s-0 rev/s}{2.79 rev/s^2}=3.94 s[/tex]
(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
[tex]\omega_f^2 - \omega_i^2 = 2 \alpha \theta[/tex]
where:
[tex]\omega_f = 11.0 rev/s[/tex] is the final angular speed
[tex]\omega_i = 0 rev/s[/tex] is the initial angular speed
[tex]\alpha=2.79 rev/s^2[/tex] is the angular acceleration
[tex]\theta=?[/tex] is the number of revolutions made by the disk while accelerating
Solving the equation for [tex]\theta[/tex], we find
[tex]\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[/tex]
