The length of stretched string is 2 m and its mass is `8xx10^(-3)` kg. If a tension of 2 kg f is applied to the wire, how long will a transverse wave take in reaching from one end of the wire to the other end ?

Answer

To solve the problem step by step, we need to calculate the time it takes for a transverse wave to travel from one end of a stretched string to the other. We will follow these steps:

### Step 1: Calculate the Linear Mass Density (μ)
The linear mass density (μ) is defined as the mass per unit length of the string.

Given:
- Mass of the string (m) = \(8 \times 10^{-3}\) kg
- Length of the string (L) = 2 m

\[ \mu = \frac{m}{L} = \frac{8 \times 10^{-3} \text{ kg}}{2 \text{ m}} = 4 \times 10^{-3} \text{ kg/m} \]

### Step 2: Convert Tension to Newtons
The tension (T) is given in kgf (kilogram-force). To convert this to Newtons, we use the conversion factor \(1 \text{ kgf} = 9.81 \text{ N}\).

Given:
- Tension = 2 kgf

\[ T = 2 \text{ kgf} \times 9.81 \text{ N/kgf} = 19.62 \text{ N} \]

### Step 3: Calculate the Velocity of the Wave (v)
The velocity of a wave on a string is given by the formula:

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

Substituting the values we have:

\[ v = \sqrt{\frac{19.62 \text{ N}}{4 \times 10^{-3} \text{ kg/m}}} \]

Calculating this gives:

\[ v = \sqrt{4905} \approx 70 \text{ m/s} \]

### Step 4: Calculate the Time Taken (t)
The time taken for the wave to travel the length of the string can be calculated using the formula:

\[ t = \frac{L}{v} \]

Substituting the values:

\[ t = \frac{2 \text{ m}}{70 \text{ m/s}} \approx 0.02857 \text{ s} \]

### Final Answer
The time taken for the transverse wave to travel from one end of the wire to the other end is approximately \(0.02857\) seconds.

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