A ceiling fan with 87-cm-diameter blades is turning at 70 rpm . Suppose the fan coasts to a stop 27 s after being turned off. Through how many revolutions does the fan turn while stopping?

To solve this problem, we need to determine the total number of revolutions the ceiling fan makes while it is coasting to a stop. This involves calculating the distance traveled by the fan in terms of the number of revolutions from the initial angular velocity to zero angular velocity over a given time period, assuming a constant angular deceleration.

### Step-by-Step Solution

1. **Convert the initial angular velocity to radians per second:**

   The fan is initially rotating at 70 revolutions per minute (rpm). We need to convert this to radians per second.

   \[
   \omega_0 = 70 \, \text{rpm} \times \left( \frac{2\pi \, \text{rad}}{1 \, \text{rev}} \right) \times \left( \frac{1 \, \text{min}}{60 \, \text{s}} \right)
   \]

   \[
   \omega_0 = 70 \times \frac{2\pi}{60} \, \text{rad/s}
   \]

   \[
   \omega_0 \approx 7.33 \, \text{rad/s}
   \]

2. **Calculate the angular acceleration:**

   The fan comes to a stop in 27 seconds. The final angular velocity, \(\omega_f\), is 0 rad/s. The angular acceleration, \(\alpha\), can be found using the formula:

   \[
   \omega_f = \omega_0 + \alpha t
   \]

   Solving for \(\alpha\):

   \[
   0 = 7.33 + \alpha \cdot 27
   \]

   \[
   \alpha = -\frac{7.33}{27} \, \text{rad/s}^2
   \]

   \[
   \alpha \approx -0.2715 \, \text{rad/s}^2
   \]

3. **Calculate the total angular displacement:**

   The total angular displacement, \(\theta\), in radians while the fan is stopping can be calculated using the kinematic equation:

   \[
   \theta = \omega_0 t + \frac{1}{2} \alpha t^2
   \]

   Substituting the values:

   \[
   \theta = (7.33) \times (27) + \frac{1}{2} \times (-0.2715) \times (27)^2
   \]

   \[
   \theta = 197.91 - 98.6
   \]

   \[
   \theta \approx 99.31 \, \text{rad}
   \]

4. **Convert the angular displacement to revolutions:**

   To find the number of revolutions, we convert radians to revolutions using the conversion \(1 \, \text{rev} = 2\pi \, \text{rad}\):

   \[
   \text{Number of revolutions} = \frac{\theta}{2\pi}
   \]

   \[
   \text{Number of revolutions} \approx \frac{99.31}{2\pi}
   \]

   \[
   \text{Number of revolutions} \approx 15.8
   \]

### Final Answer

The ceiling fan makes approximately **15.8 revolutions** while it coasts to a stop.

Would you like more details on any part of this solution, or do you have any other questions?

### Further Related Questions

1. What is the average angular velocity of the fan while it is stopping?
2. How long would it take for the fan to stop if it were rotating at a different speed?
3. How would the number of revolutions change if the stopping time were longer or shorter?
4. What is the total distance traveled by the tip of the fan blades while stopping?
5. How would changing the diameter of the fan blades affect the stopping distance?

### Tip

Remember, angular acceleration is constant if no external torques are acting on the object, allowing the use of uniform acceleration formulas from linear motion analogies.

A ceiling fan with 87-cm-diameter blades is turning at 70 rpm. Suppose the fan coasts to a stop 27 s after being turned off. Through how many revolutions does the fan turn while stopping?

Find out how to calculate the total revolutions a ceiling fan makes while coasting to a stop after turning off. This problem involves key concepts like angular velocity, angular acceleration, and kinematic equations, perfect for students in Grades 11-12.

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