When a ceiling fan is switched on, it makes 10 rotations in the first 3 seconds. Assuming a uniform angular acceleration, how many rotation it will make in the next 3 seconds?

Options

A. 10
B. 20
C. 40
D. 40

Answer

To solve the problem step by step, we will use the equations of rotational motion. Here’s how we can approach it:

### Step 1: Understand the Problem
The ceiling fan makes 10 rotations in the first 3 seconds. We need to find out how many rotations it will make in the next 3 seconds, assuming uniform angular acceleration.

### Step 2: Convert Rotations to Radians
Since the equations of motion use radians, we need to convert the rotations into radians.
1 rotation = \(2\pi\) radians.
Thus, 10 rotations = \(10 \times 2\pi = 20\pi\) radians.

### Step 3: Use the Angular Displacement Formula
We will use the second equation of rotational motion:
\[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]
Where:
- \(\theta\) = angular displacement (in radians)
- \(\omega_i\) = initial angular velocity (in radians/second)
- \(\alpha\) = angular acceleration (in radians/second²)
- \(t\) = time (in seconds)

Given:
- \(\theta = 20\pi\) radians
- \(\omega_i = 0\) (the fan starts from rest)
- \(t = 3\) seconds

### Step 4: Substitute Known Values
Substituting the known values into the equation:
\[ 20\pi = 0 \cdot 3 + \frac{1}{2} \alpha (3^2) \]
This simplifies to:
\[ 20\pi = \frac{1}{2} \alpha \cdot 9 \]
\[ 20\pi = \frac{9}{2} \alpha \]

### Step 5: Solve for Angular Acceleration \(\alpha\)
Rearranging the equation to solve for \(\alpha\):
\[ \alpha = \frac{20\pi \cdot 2}{9} = \frac{40\pi}{9} \text{ radians/second}^2 \]

### Step 6: Calculate Angular Displacement in the Next 3 Seconds
Now we need to find the angular displacement for the next 3 seconds (from \(t = 3\) to \(t = 6\) seconds). The total time for this phase is 6 seconds.

Using the same formula:
\[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]
Now, \(\omega_i\) at \(t = 3\) seconds is not 0. We need to find it first using:
\[ \omega_f = \omega_i + \alpha t \]
Where \(\omega_f\) is the final angular velocity after the first 3 seconds:
\[ \omega_f = 0 + \left(\frac{40\pi}{9}\right) \cdot 3 = \frac{120\pi}{9} \text{ radians/second} \]

### Step 7: Find Angular Displacement from \(t = 3\) to \(t = 6\)
Now, substituting into the angular displacement formula for \(t = 3\) seconds (initial) to \(t = 6\) seconds (final):
\[ \theta = \left(\frac{120\pi}{9}\right) \cdot 3 + \frac{1}{2} \left(\frac{40\pi}{9}\right) (3^2) \]
Calculating:
\[ \theta = \left(\frac{120\pi}{9}\right) \cdot 3 + \frac{1}{2} \left(\frac{40\pi}{9}\right) \cdot 9 \]
\[ \theta = \frac{360\pi}{9} + \frac{180\pi}{9} = \frac{540\pi}{9} = 60\pi \text{ radians} \]

### Step 8: Calculate the Total Number of Rotations
We need to find the number of rotations in the next 3 seconds:
\[ \text{Number of rotations} = \frac{\text{Angular displacement}}{2\pi} = \frac{60\pi}{2\pi} = 30 \]

### Final Answer
The ceiling fan will make **30 rotations** in the next 3 seconds.

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