A 5.5 m length of string has a mass of 0.035 kg. If the tension in the string is 77 N the speed of a wave on the string is

Options

A. `110 ms^(-1)`
B. `165 ms^(-1)`
C. `77 ms ^(-1)`
D. ` 102ms^(-1)`

Answer

To find the speed of a wave on a string, we can use the formula:

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

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:
- \( m = 0.035 \, \text{kg} \)
- \( l = 5.5 \, \text{m} \)

Substituting the values:

\[ \mu = \frac{0.035 \, \text{kg}}{5.5 \, \text{m}} \]

Calculating \( \mu \):

\[ \mu = 0.00636 \, \text{kg/m} \]

**Step 2: Substitute the values into the wave speed formula**

Now that we have \( \mu \), we can substitute it into the wave speed formula along with the tension \( T \).

Given:
- \( T = 77 \, \text{N} \)

Substituting the values into the formula:

\[ v = \sqrt{\frac{77 \, \text{N}}{0.00636 \, \text{kg/m}}} \]

**Step 3: Calculate the speed of the wave**

Now we perform the calculation:

\[ v = \sqrt{12159.69} \]

Calculating the square root:

\[ v \approx 110.25 \, \text{m/s} \]

Rounding to two decimal places, we find:

\[ v \approx 110 \, \text{m/s} \]

Thus, the speed of the wave on the string is approximately **110 m/s**.

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