A boy sitting on a swing which is moving to an angle of `30^@` from the vertical is blowing a whistle which has a frequency of 100 Hz. The whistle is at 2.0 m from the point of support of the swing. A girl stands in front of the swing. The maximum frequency she will hear is nearly `10^x` Hz (velocity of sound 330 m/s,g= `9.8 m//s^2`). Find x

Answer

To solve the problem step by step, we need to calculate the maximum frequency that the girl will hear using the Doppler effect. Here’s how we can approach it:

### Step 1: Identify the given values
- Frequency of the whistle, \( f = 100 \, \text{Hz} \)
- Distance from the swing's support to the whistle, \( l = 2.0 \, \text{m} \)
- Angle from the vertical, \( \theta = 30^\circ \)
- Velocity of sound, \( v = 330 \, \text{m/s} \)
- Acceleration due to gravity, \( g = 9.8 \, \text{m/s}^2 \)

### Step 2: Calculate the velocity of the source (swing)
The velocity of the source can be calculated using the formula:
\[ v_s = \sqrt{2 g l (1 - \cos \theta)} \]

Substituting the values:
\[ v_s = \sqrt{2 \times 9.8 \times 2 \times (1 - \cos 30^\circ)} \]

First, calculate \( \cos 30^\circ \):
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.866 \]

Now substitute:
\[ v_s = \sqrt{2 \times 9.8 \times 2 \times (1 - 0.866)} \]
\[ = \sqrt{2 \times 9.8 \times 2 \times 0.134} \]
\[ = \sqrt{5.25376} \approx 2.3 \, \text{m/s} \]

### Step 3: Calculate the maximum frequency heard by the girl
Using the Doppler effect formula for maximum frequency:
\[ f' = \frac{v}{v - v_s} \cdot f \]

Substituting the values:
\[ f' = \frac{330}{330 - 2.3} \cdot 100 \]
\[ = \frac{330}{327.7} \cdot 100 \]
\[ \approx 1.007 \cdot 100 \approx 100.7 \, \text{Hz} \]

### Step 4: Express the maximum frequency in the form \( 10^x \)
We need to express \( 100.7 \) in the form \( 10^x \):
\[ 100.7 \approx 10^2 \]

Thus, \( x = 2 \).

### Final Answer
The value of \( x \) is \( 2 \).

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