when a ceiling fan is switched on it makes 10 revolutions in the first 4 second. assuming a uniform angular acceleration, how many revolution it will makes in the next 4 seconds?

Options

A. 10
B. 20
C. 30
D. 40

Answer

To solve the problem step by step, we will follow the principles of angular motion under uniform angular acceleration.

### Step 1: Understand the given data
- The ceiling fan makes 10 revolutions in the first 4 seconds.
- We need to find out how many revolutions it makes in the next 4 seconds.

### Step 2: Convert revolutions to radians
Since angular motion equations typically use radians, we convert revolutions to radians.
- 1 revolution = \( 2\pi \) radians
- Therefore, 10 revolutions = \( 10 \times 2\pi = 20\pi \) radians.

### Step 3: Use the angular displacement formula
The formula for angular displacement under uniform angular acceleration is:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Where:
- \( \theta \) is the angular displacement (in radians),
- \( \omega_0 \) is the initial angular velocity (in radians/second),
- \( \alpha \) is the angular acceleration (in radians/second²),
- \( t \) is the time (in seconds).

Since the fan starts from rest, \( \omega_0 = 0 \). Thus, the formula simplifies to:
\[ \theta = \frac{1}{2} \alpha t^2 \]

### Step 4: Calculate angular acceleration
For the first 4 seconds:
\[ \theta = 20\pi = \frac{1}{2} \alpha (4^2) \]
\[ 20\pi = \frac{1}{2} \alpha (16) \]
\[ 20\pi = 8\alpha \]
\[ \alpha = \frac{20\pi}{8} = \frac{5\pi}{2} \text{ radians/second}^2 \]

### Step 5: Calculate the angular displacement for the next 4 seconds
Now, we need to find the angular displacement for the next 4 seconds (from \( t = 4 \) seconds to \( t = 8 \) seconds).

Using the formula:
\[ \theta_{4 \text{ to } 8} = \omega_0 t + \frac{1}{2} \alpha t^2 \]
First, we need to find the angular velocity at \( t = 4 \) seconds:
\[ \omega = \omega_0 + \alpha t = 0 + \left(\frac{5\pi}{2}\right)(4) = 10\pi \text{ radians/second} \]

Now, for the next 4 seconds (from \( t = 4 \) to \( t = 8 \)):
\[ \theta_{4 \text{ to } 8} = (10\pi)(4) + \frac{1}{2} \left(\frac{5\pi}{2}\right)(4^2) \]
Calculating each term:
1. \( (10\pi)(4) = 40\pi \)
2. \( \frac{1}{2} \left(\frac{5\pi}{2}\right)(16) = \frac{5\pi}{2} \times 8 = 20\pi \)

Adding these:
\[ \theta_{4 \text{ to } 8} = 40\pi + 20\pi = 60\pi \text{ radians} \]

### Step 6: Convert angular displacement back to revolutions
To find the number of revolutions:
\[ \text{Number of revolutions} = \frac{\theta_{4 \text{ to } 8}}{2\pi} = \frac{60\pi}{2\pi} = 30 \text{ revolutions} \]

### Final Answer
The ceiling fan will make **30 revolutions** in the next 4 seconds.

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