In Melde's experiment, the tuning fork was arranged in parallel position and the vibrating length of string was 0.8m. Upon setting the tuning fork into vibration, four loops were formed along the string. If the linear density of the string is 0.5 mg/cm and the frequency of the tuning fork is 96Hz, then tension in the string will be

Options

A. `0.1843xx10^(-2)N`
B. `9.215xx10^(-2)N`
C. `0.9215xx10^(-2)N`
D. `1.843xx10^(-2)N`

Explanation

To find the tension in the string during Melde's experiment, we can use the formula that relates frequency, tension, linear density, and the number of loops formed. Here’s a step-by-step solution:

### Step 1: Understand the given data
- Vibrating length of the string (L) = 0.8 m
- Number of loops (p) = 4
- Frequency of the tuning fork (ν) = 96 Hz
- Linear density of the string (ρ) = 0.5 mg/cm

### Step 2: Convert linear density to standard units
The linear density is given in mg/cm. We need to convert it to kg/m:
\[ \rho = 0.5 \, \text{mg/cm} = 0.5 \times 10^{-3} \, \text{g/cm} = 0.5 \times 10^{-3} \times 10^{-2} \, \text{kg/m} = 0.5 \times 10^{-5} \, \text{kg/m} = 0.5 \times 10^{-4} \, \text{kg/m} \]

### Step 3: Use the formula for tension
The formula relating tension (T), frequency (ν), length (L), number of loops (p), and linear density (ρ) is:
\[ T = \frac{\nu^2 \cdot L^2 \cdot \rho}{p^2} \]

### Step 4: Substitute the values into the formula
Substituting the known values into the formula:
\[ T = \frac{(96)^2 \cdot (0.8)^2 \cdot (0.5 \times 10^{-4})}{(4)^2} \]

### Step 5: Calculate each component
- Calculate \(ν^2\):
\[ ν^2 = 96^2 = 9216 \]
- Calculate \(L^2\):
\[ L^2 = (0.8)^2 = 0.64 \]
- Calculate \(p^2\):
\[ p^2 = 4^2 = 16 \]

### Step 6: Substitute and calculate T
Now substitute these values back into the equation:
\[ T = \frac{9216 \cdot 0.64 \cdot (0.5 \times 10^{-4})}{16} \]
Calculating the numerator:
\[ 9216 \cdot 0.64 = 5898.24 \]
Now, substituting this into the tension formula:
\[ T = \frac{5898.24 \cdot (0.5 \times 10^{-4})}{16} \]
Calculating:
\[ T = \frac{2949.12 \times 10^{-4}}{16} = 1.843 \times 10^{-2} \, \text{N} \]

### Final Answer
The tension in the string is:
\[ T \approx 1.843 \times 10^{-2} \, \text{N} \]