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 solve the problem, we need to find the tension in the rope based on the given information about the standing wave. Let's break down the solution step by step:

### Step 1: Understand the Setup
We have a horizontal rope of mass \( m = 1.2 \, \text{kg} \) fixed at both ends (at \( x = 0 \) and \( x = 2.0 \, \text{m} \)). The rope is vibrating in the fundamental mode at a frequency of \( f = 5.0 \, \text{Hz} \).

### Step 2: Determine the Length of the Rope
The length \( L \) of the rope is given as:
\[ L = 2.0 \, \text{m} \]

### Step 3: Calculate the Wavelength
In the fundamental mode of a standing wave, the wavelength \( \lambda \) is related to the length of the rope by:
\[ L = \frac{\lambda}{2} \implies \lambda = 2L \]
Substituting the value of \( L \):
\[ \lambda = 2 \times 2.0 \, \text{m} = 4.0 \, \text{m} \]

### Step 4: Calculate the Wave Speed
The speed \( v \) of the wave on the rope can be calculated using the formula:
\[ v = f \cdot \lambda \]
Substituting the values of \( f \) and \( \lambda \):
\[ v = 5.0 \, \text{Hz} \times 4.0 \, \text{m} = 20.0 \, \text{m/s} \]

### Step 5: Calculate the Mass per Unit Length
The mass per unit length \( \mu \) of the rope is given by:
\[ \mu = \frac{m}{L} \]
Substituting the values of \( m \) and \( L \):
\[ \mu = \frac{1.2 \, \text{kg}}{2.0 \, \text{m}} = 0.6 \, \text{kg/m} \]

### Step 6: Relate Wave Speed to Tension
The wave speed is also related to the tension \( T \) in the rope and the mass per unit length \( \mu \) by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
Squaring both sides gives:
\[ v^2 = \frac{T}{\mu} \implies T = v^2 \cdot \mu \]
Substituting the values of \( v \) and \( \mu \):
\[ T = (20.0 \, \text{m/s})^2 \cdot 0.6 \, \text{kg/m} = 400 \cdot 0.6 = 240 \, \text{N} \]

### Final Answer
The tension in the rope is:
\[ \boxed{240 \, \text{N}} \]

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