A unifrom rope of mass 0.1kg and length 2.45 m hangs from a rigid support. The time taken by the transverse wave formed in the rope to travel through the full length of the rope is (Assume g=`=9.8 m//s^2`)

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the mass per unit length (μ) of the rope.
Given:
- Mass of the rope (m) = 0.1 kg
- Length of the rope (l) = 2.45 m

The mass per unit length (μ) is calculated as:
\[ \mu = \frac{m}{l} = \frac{0.1 \, \text{kg}}{2.45 \, \text{m}} \approx 0.04082 \, \text{kg/m} \]

### Step 2: Calculate the tension (T) in the rope.
The tension in the rope due to its weight is given by:
\[ T = m \cdot g \]
Where:
- \( g = 9.8 \, \text{m/s}^2 \)

Substituting the values:
\[ T = 0.1 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 0.98 \, \text{N} \]

### Step 3: Calculate the speed (v) of the wave in the rope.
The speed of the wave is given by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]

Substituting the values of T and μ:
\[ v = \sqrt{\frac{0.98 \, \text{N}}{0.04082 \, \text{kg/m}}} \approx \sqrt{24.06} \approx 4.90 \, \text{m/s} \]

### Step 4: Calculate the time taken (t) for the wave to travel the length of the rope.
The time taken for the wave to travel the full length of the rope is given by:
\[ t = \frac{l}{v} \]

Substituting the values:
\[ t = \frac{2.45 \, \text{m}}{4.90 \, \text{m/s}} \approx 0.5 \, \text{s} \]

### Final Answer:
The time taken by the transverse wave to travel through the full length of the rope is approximately **0.5 seconds**.

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