Final answer:
The angular acceleration of the fan is -1.41 rev/s², and during the 4.50 s interval, the fan makes approximately 28.575 revolutions. It takes an additional 2.24 seconds for the fan to come to rest if the angular acceleration remains constant.
Step-by-step explanation:
To calculate the angular acceleration, we can use the formula \( \alpha = \frac{\Delta \omega}{\Delta t} \), where \( \Delta \omega \) is the change in angular velocity and \( \Delta t \) is the change in time. First, convert the angular velocities from revolutions per minute (rev/min) to revolutions per second (rev/s) by dividing by 60. This gives us:
Initial angular velocity, \( \omega_i = \frac{570}{60} = 9.5 \) rev/s
Final angular velocity, \( \omega_f = \frac{190}{60} = 3.17 \) rev/s
Time interval, \( \Delta t = 4.50 \) s
Now apply the equation to find angular acceleration:
\( \alpha = \frac{3.17 - 9.5}{4.50} = -1.41 \) rev/s2
To find the number of revolutions made, use the equation \( \theta = \omega_i \cdot t + 0.5 \cdot \alpha \cdot t^2 \):
\( \theta = 9.5 \cdot 4.50 + 0.5 \cdot (-1.41) \cdot (4.50)^2 \) which calculates to \( \theta = 28.575 \) revolutions.
For the fan to come to rest, we need to find the time when angular velocity is zero (\( \omega_f = 0 \)). Use \( \omega_f = \omega_i + \alpha \cdot t \) and solve for \( t \):
\( 0 = 9.5 + (-1.41) \cdot t \) which gives \( t = \frac{9.5}{1.41} \approx 6.74 \) seconds. However, this is the total time from the initial velocity to rest. Because it already took 4.50 s to slow to 190 rev/min, the additional time required is \( 6.74 - 4.50 \approx 2.24 \) seconds.