Find the displacement of an air particle 3.5 m from the origin of disturbance at t=0.05 s, when a wave of amplitude 0.2 mm and frequency 500 Hz travels along it with a velocity `350 ms^(-1)`.

Answer

To find the displacement of an air particle at a distance of 3.5 m from the origin of disturbance at time t = 0.05 s, we can follow these steps:

### Step 1: Identify the given values
- Amplitude (A) = 0.2 mm = 0.2 × 10^(-3) m
- Frequency (f) = 500 Hz
- Velocity (v) = 350 m/s
- Distance from the origin (x) = 3.5 m
- Time (t) = 0.05 s

### Step 2: Calculate the angular frequency (ω)
The angular frequency (ω) is given by the formula:
\[ \omega = 2\pi f \]
Substituting the value of frequency:
\[ \omega = 2\pi \times 500 = 1000\pi \text{ rad/s} \]

### Step 3: Calculate the wave number (k)
The wave number (k) can be calculated using the formula:
\[ k = \frac{\omega}{v} \]
Substituting the values of ω and v:
\[ k = \frac{1000\pi}{350} \]
Simplifying this gives:
\[ k = \frac{1000\pi}{350} = \frac{20\pi}{7} \text{ rad/m} \]

### Step 4: Write the wave equation
The displacement of a particle in a wave can be described by the equation:
\[ y = A \sin(kx - \omega t) \]
Substituting the values of A, k, x, and ω, we have:
\[ y = 0.2 \times 10^{-3} \sin\left(\frac{20\pi}{7} \times 3.5 - 1000\pi \times 0.05\right) \]

### Step 5: Calculate the argument of the sine function
First, calculate \( kx \):
\[ kx = \frac{20\pi}{7} \times 3.5 = 10\pi \text{ rad} \]
Next, calculate \( \omega t \):
\[ \omega t = 1000\pi \times 0.05 = 50\pi \text{ rad} \]

Now, substitute these into the sine function:
\[ y = 0.2 \times 10^{-3} \sin(10\pi - 50\pi) \]
\[ y = 0.2 \times 10^{-3} \sin(-40\pi) \]

### Step 6: Simplify the sine function
Since \(\sin(-\theta) = -\sin(\theta)\) and \(\sin(n\pi) = 0\) for any integer n:
\[ \sin(-40\pi) = 0 \]

### Step 7: Calculate the displacement
Thus, the displacement \(y\) is:
\[ y = 0.2 \times 10^{-3} \times 0 = 0 \]

### Final Answer
The displacement of the air particle at 3.5 m from the origin at t = 0.05 s is **0 m**.

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