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, let's break down the information given and apply the relevant physics concepts.

### Step 1: Understand the Problem
When the ceiling fan is switched on, it makes 10 revolutions in the first 4 seconds. We need to find out how many revolutions it makes in the next 4 seconds, assuming uniform angular acceleration.

### Step 2: Convert Revolutions to Radians
First, we need to convert the number of revolutions into radians since angular displacement is typically measured in 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 (\(\theta\)) under uniform angular acceleration is:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Where:
- \(\theta\) = angular displacement (in radians)
- \(\omega_0\) = initial angular velocity (in radians/second)
- \(\alpha\) = angular acceleration (in radians/second²)
- \(t\) = time (in seconds)

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

### Step 4: Plug in the Values for the First 4 Seconds
For the first 4 seconds:
- \(\theta = 20\pi\) radians
- \(t = 4\) seconds

Substituting these values into the equation:
\[ 20\pi = \frac{1}{2} \alpha (4^2) \]
\[ 20\pi = \frac{1}{2} \alpha (16) \]
\[ 20\pi = 8\alpha \]
Now, solve for \(\alpha\):
\[ \alpha = \frac{20\pi}{8} = \frac{5\pi}{2} \text{ radians/second}^2 \]

### Step 5: Calculate Total Displacement in 8 Seconds
Now we need to find the total angular displacement from 0 to 8 seconds using the same formula:
\[ \theta = \frac{1}{2} \alpha t^2 \]
For \(t = 8\) seconds:
\[ \theta = \frac{1}{2} \left(\frac{5\pi}{2}\right) (8^2) \]
\[ \theta = \frac{1}{2} \left(\frac{5\pi}{2}\right) (64) \]
\[ \theta = \frac{5\pi}{2} \times 32 = 80\pi \text{ radians} \]

### Step 6: Find the Displacement from 4 to 8 Seconds
To find the displacement during the next 4 seconds (from 4 to 8 seconds), we subtract the displacement from 0 to 4 seconds from the displacement from 0 to 8 seconds:
\[ \text{Displacement from 4 to 8 seconds} = \text{Displacement from 0 to 8 seconds} - \text{Displacement from 0 to 4 seconds} \]
\[ = 80\pi - 20\pi = 60\pi \text{ radians} \]

### Step 7: Convert Radians Back to Revolutions
Finally, to find the number of revolutions made in the next 4 seconds:
\[ \text{Number of revolutions} = \frac{\text{Displacement}}{2\pi} = \frac{60\pi}{2\pi} = 30 \]

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

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