A string of mass 2.50kg is under a tension os 200N. The length of the stretched string is 20.0m. If the transverse jerk is struck at one end of the string, how long does the disturbance take to reach the other end?

Options

A. 0.5 s
B. 1 s
C. 1.5 s
D. 2 s

Answer

To solve the problem, we need to determine how long it takes for a disturbance to travel along a stretched string. We will use the formula for the speed of a transverse wave on a string, which is given by:

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

where:
- \( v \) is the speed of the wave,
- \( T \) is the tension in the string,
- \( \mu \) is the mass per unit length of the string.

### Step 1: Calculate the mass per unit length (\( \mu \))

The mass per unit length (\( \mu \)) can be calculated using the formula:

\[ \mu = \frac{m}{L} \]

where:
- \( m \) is the mass of the string,
- \( L \) is the length of the string.

Given:
- \( m = 2.50 \, \text{kg} \)
- \( L = 20.0 \, \text{m} \)

Substituting the values:

\[ \mu = \frac{2.50 \, \text{kg}}{20.0 \, \text{m}} = 0.125 \, \text{kg/m} \]

### Step 2: Calculate the speed of the wave (\( v \))

Now, we can calculate the speed of the wave using the tension and the mass per unit length:

Given:
- \( T = 200 \, \text{N} \)

Substituting the values into the wave speed formula:

\[ v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{200 \, \text{N}}{0.125 \, \text{kg/m}}} \]

Calculating the denominator:

\[ v = \sqrt{200 \times 8} = \sqrt{1600} = 40 \, \text{m/s} \]

### Step 3: Calculate the time taken for the disturbance to travel the length of the string

The time taken (\( t \)) for the disturbance to travel the length of the string can be calculated using the formula:

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

Substituting the values:

\[ t = \frac{20.0 \, \text{m}}{40 \, \text{m/s}} = 0.5 \, \text{s} \]

### Final Answer

The disturbance takes **0.5 seconds** to reach the other end of the string.

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