A `200cm` length of the wire weighs `0.6` gm. If the tension in the wire caused by a `500 gm` mass hanged from it. If the approximate wavelength of a wave of frequency `400Hz` sent down by it is `2^(n)xx10^(-2)m`. Find the value of `n` (`g=9.8m//s^(2)`)

Explanation

To solve the problem step by step, we will follow these calculations:

### Step 1: Convert the weight of the wire to mass per unit length (μ)
The weight of the wire is given as 0.6 grams, and the length of the wire is 200 cm.

1. Convert the weight to kilograms:
\[ \text{Weight} = 0.6 \text{ g} = 0.6 \times 10^{-3} \text{ kg} \]

2. Convert the length to meters:
\[ \text{Length} = 200 \text{ cm} = 2 \text{ m} \]

3. Calculate the mass per unit length (μ):
\[ \mu = \frac{\text{mass}}{\text{length}} = \frac{0.6 \times 10^{-3} \text{ kg}}{2 \text{ m}} = 3 \times 10^{-4} \text{ kg/m} \]

### Step 2: Calculate the tension (T) in the wire
The tension in the wire is caused by a mass of 500 g hanging from it.

1. Convert the mass to kilograms:
\[ \text{Mass} = 500 \text{ g} = 0.5 \text{ kg} \]

2. Use the formula for tension:
\[ T = mg = 0.5 \text{ kg} \times 9.8 \text{ m/s}^2 = 4.9 \text{ N} \]

### Step 3: Calculate the wave speed (v) in the wire
The wave speed can be calculated using the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]

Substituting the values:
\[ v = \sqrt{\frac{4.9 \text{ N}}{3 \times 10^{-4} \text{ kg/m}}} \]

Calculating this gives:
\[ v = \sqrt{16333.33} \approx 128 \text{ m/s} \]

### Step 4: Calculate the wavelength (λ) of the wave
The wavelength can be calculated using the formula:
\[ \lambda = \frac{v}{f} \]

Where \( f = 400 \text{ Hz} \):
\[ \lambda = \frac{128 \text{ m/s}}{400 \text{ Hz}} = 0.32 \text{ m} = 32 \times 10^{-2} \text{ m} \]

### Step 5: Express the wavelength in the form \( 2^n \times 10^{-2} \)
We need to express \( 32 \times 10^{-2} \) in the form \( 2^n \times 10^{-2} \).

1. Recognize that \( 32 = 2^5 \):
\[ 32 \times 10^{-2} = 2^5 \times 10^{-2} \]

2. Therefore, comparing with \( 2^n \times 10^{-2} \), we find:
\[ n = 5 \]

### Final Answer
The value of \( n \) is \( 5 \).

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