In a standing wave experiment , a `1.2 - kg` horizontal rope is fixed in place at its two ends `( x = 0 and x = 2.0 m)` and made to oscillate up and down in the fundamental mode , at frequency of `5.0 Hz`. At `t = 0` , the point at `x = 1.0 m` has zero displacement and is moving upward in the positive direction of `y - axis` with a transverse velocity `3.14 m//s`.
What is the correct expression of the standing wave equation ?

Options

A. `(0.1) sin ( pi//2) x sin (10 pi) t`
B. `(0.1) sin (pi) x sin (10 pi) t`
C. `(0.05) sin (pi//2) x cos (10 pi) t`
D. `(0.04) sin (pi//2) x sin (10 pi) t`

Explanation

To derive the standing wave equation for the given problem, we will follow these steps:

### Step 1: Determine the mass per unit length (μ) of the rope.
Given:
- Mass of the rope (m) = 1.2 kg
- Length of the rope (L) = 2.0 m

The mass per unit length (μ) is calculated as:
\[ \mu = \frac{m}{L} = \frac{1.2 \, \text{kg}}{2.0 \, \text{m}} = 0.6 \, \text{kg/m} \]

### Step 2: Find the wavelength (λ) in the fundamental mode.
In the fundamental mode of a fixed string, the wavelength is given by:
\[ \frac{\lambda}{2} = L \implies \lambda = 2L = 2 \times 2.0 \, \text{m} = 4.0 \, \text{m} \]

### Step 3: Calculate the wave speed (v).
The wave speed (v) can be calculated using the formula:
\[ v = f \cdot \lambda \]
Where:
- Frequency (f) = 5.0 Hz
- Wavelength (λ) = 4.0 m

Thus,
\[ v = 5.0 \, \text{Hz} \times 4.0 \, \text{m} = 20.0 \, \text{m/s} \]

### Step 4: Calculate the tension (T) in the rope.
Using the relationship between wave speed, tension, and mass per unit length:
\[ v = \sqrt{\frac{T}{\mu}} \implies T = \mu v^2 \]
Substituting the values:
\[ T = 0.6 \, \text{kg/m} \times (20.0 \, \text{m/s})^2 = 0.6 \times 400 = 240 \, \text{N} \]

### Step 5: Determine the amplitude (A) of the wave.
At \(t = 0\), the transverse velocity (\(\frac{\partial y}{\partial t}\)) is given as 3.14 m/s. The maximum transverse velocity is given by:
\[ \frac{\partial y}{\partial t} = \omega A \]
Where \(\omega = 2\pi f\):
\[ \omega = 2\pi \times 5.0 \, \text{Hz} = 10\pi \, \text{rad/s} \]
Thus:
\[ 3.14 = 10\pi A \implies A = \frac{3.14}{10\pi} \approx 0.1 \, \text{m} \]

### Step 6: Write the standing wave equation.
The general form of the standing wave equation is:
\[ y(x, t) = A \sin(kx) \cos(\omega t) \]
Where:
- \(k = \frac{2\pi}{\lambda} = \frac{2\pi}{4} = \frac{\pi}{2} \, \text{rad/m}\)

Substituting the values of \(A\), \(k\), and \(\omega\):
\[ y(x, t) = 0.1 \sin\left(\frac{\pi}{2} x\right) \cos(10\pi t) \]

### Final Answer:
The correct expression for the standing wave equation is:
\[ y(x, t) = 0.1 \sin\left(\frac{\pi}{2} x\right) \cos(10\pi t) \]

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