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`.
Tension in the rope is

Options

A. `60 N`
B. `100 N`
C. `120 N`
D. `240 N`

Explanation

To find the tension in the rope, we can follow these steps:

### Step 1: Calculate the mass per unit length (μ) of the rope
The mass of the rope is given as \(1.2 \, \text{kg}\) and the length of the rope is \(2.0 \, \text{m}\).

\[ \mu = \frac{\text{mass}}{\text{length}} = \frac{1.2 \, \text{kg}}{2.0 \, \text{m}} = 0.6 \, \text{kg/m} \]

### Step 2: Determine the wavelength (λ) of the standing wave
In the fundamental mode of a standing wave, the length of the rope (L) is equal to half the wavelength (λ/2). Therefore, we can express the wavelength as follows:

\[ L = \frac{\lambda}{2} \implies \lambda = 2L = 2 \times 2.0 \, \text{m} = 4.0 \, \text{m} \]

### Step 3: Calculate the wave velocity (v)
The wave velocity can be calculated using the formula:

\[ v = f \cdot \lambda \]

where \(f\) is the frequency. Given that the frequency \(f = 5.0 \, \text{Hz}\):

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

### Step 4: Relate wave velocity to tension (T) and mass per unit length (μ)
The relationship between wave velocity, tension, and mass per unit length is given by:

\[ v = \sqrt{\frac{T}{\mu}} \]

We can rearrange this equation to solve for tension \(T\):

\[ T = \mu v^2 \]

### Step 5: Substitute the values to find the tension
Now we can substitute the values of \(μ\) and \(v\) into the equation:

\[ T = 0.6 \, \text{kg/m} \cdot (20.0 \, \text{m/s})^2 \]

Calculating this gives:

\[ T = 0.6 \cdot 400 = 240 \, \text{N} \]

### Final Answer
The tension in the rope is \(240 \, \text{N}\).

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