To solve the problem step by step, we will follow these steps:
### Step 1: Determine the wavelength of the sound wave
The frequency \( f \) of the sound wave is given as 512 Hz, and the speed of sound \( v \) is given as 340 m/s. We can use the formula for the speed of a wave:
\[
v = f \cdot \lambda
\]
Where \( \lambda \) is the wavelength. Rearranging the formula gives:
\[
\lambda = \frac{v}{f}
\]
Substituting the values:
\[
\lambda = \frac{340 \, \text{m/s}}{512 \, \text{Hz}} \approx 0.664 \, \text{m}
\]
### Step 2: Relate the wavelength to the length of the barrel
The pop-gun acts like a closed-end pipe, where the fundamental frequency corresponds to a quarter of the wavelength fitting into the length of the barrel. Thus,
\[
L = \frac{\lambda}{4}
\]
Substituting the value of \( \lambda \):
\[
L = \frac{0.664 \, \text{m}}{4} \approx 0.166 \, \text{m} \approx 16.6 \, \text{cm}
\]
### Step 3: Calculate the initial pressure
The initial distance between the cork and the piston is given as 25 cm. The pressure inside the barrel when the cork is ejected can be calculated using the ideal gas law. For an isothermal process, the pressure is inversely proportional to the volume.
Let \( P_1 \) be the initial pressure and \( P_2 \) be the pressure when the cork is ejected. The volume changes from \( V_1 \) to \( V_2 \):
\[
P_1 \cdot V_1 = P_2 \cdot V_2
\]
Where:
- \( V_1 = A \cdot L_1 = 3 \, \text{cm}^2 \cdot 25 \, \text{cm} = 75 \, \text{cm}^3 \)
- \( V_2 = A \cdot L_2 = 3 \, \text{cm}^2 \cdot 16.6 \, \text{cm} = 49.8 \, \text{cm}^3 \)
Using the atmospheric pressure \( P_0 = 1 \, \text{kg wt/cm}^2 \):
\[
P_1 \cdot 75 = P_2 \cdot 49.8
\]
Assuming \( P_2 \) is the atmospheric pressure when the cork is ejected:
\[
P_2 = P_0 = 1 \, \text{kg wt/cm}^2
\]
Thus, we can find \( P_1 \):
\[
P_1 = \frac{P_2 \cdot V_2}{V_1} = \frac{1 \cdot 49.8}{75} \approx 0.664 \, \text{kg wt/cm}^2
\]
### Step 4: Calculate the force required to eject the cork
The force \( F \) required to eject the cork can be calculated using the formula:
\[
F = P_1 \cdot A
\]
Substituting the values:
\[
F = 0.664 \, \text{kg wt/cm}^2 \cdot 3 \, \text{cm}^2 = 1.992 \, \text{kg wt}
\]
### Step 5: Relate the force to the given expression
We know that the force required to eject the cork is given as \( 0.75 \times x \, \text{kg wt} \):
\[
1.992 = 0.75 \times x
\]
Solving for \( x \):
\[
x = \frac{1.992}{0.75} \approx 2.656
\]
### Final Answer
The value of \( x \) is approximately 2.656.
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