Question

The speed of a wave on a string is 150 m/s when the tension is 120 N. The percentage increase in the tension in order to raise the wave speed by 20% is

• A. 44%
• B. 40%
• C. 20%
• D. 10%

Hint:

\(v \propto \sqrt{T}\). For 20% increase in speed, tension increases by 44%.

Answer 1

The Correct Option is A

To solve this problem, we need to understand the relationship between the speed of a wave on a string, the tension in the string, and the mass per unit length of the string. The formula that relates these quantities is:

\(v = \sqrt{\frac{T}{\mu}}\)

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.

We are given that initially, the speed of the wave on the string is \(150 \, \text{m/s}\) when the tension is \(120 \, \text{N}\). We need to find the percentage increase in tension required to raise the wave speed by 20%.

Step 1: Calculate the New Speed

The new speed of the wave after a 20% increase is:

\(v_{\text{new}} = 150 \times 1.20 = 180 \, \text{m/s}\)

Step 2: Establish the Relationship for New Speed

The new speed \(v_{\text{new}}\) with new tension \(T_{\text{new}}\) can be expressed as:

\(v_{\text{new}} = \sqrt{\frac{T_{\text{new}}}{\mu}}\)

We have:

• \(v = 150 \, \text{m/s}\), \(T = 120 \, \text{N}\)
• \(v_{\text{new}} = 180 \, \text{m/s}\)

Step 3: Derive Formula for Tension Comparison

Using the wave speed formula for both initial and new conditions:

• Initial: \(150 = \sqrt{\frac{120}{\mu}}\)
• New: \(180 = \sqrt{\frac{T_{\text{new}}}{\mu}}\)

Squaring both sides, the equations become:

• \(150^2 = \frac{120}{\mu}\)
• \(180^2 = \frac{T_{\text{new}}}{\mu}\)

Divide the second equation by the first to solve for \(T_{\text{new}}\):

\(\frac{180^2}{150^2} = \frac{T_{\text{new}}}{120}\)

\(\left(\frac{180}{150}\right)^2 = \frac{T_{\text{new}}}{120}\)

\(\frac{9}{6.25} = \frac{T_{\text{new}}}{120}\)

Solving this gives:

\(T_{\text{new}} = 120 \times \frac{36}{25} = 172.8 \, \text{N}\)

Step 4: Calculate the Percentage Increase in Tension

The increase in tension is:

 

Therefore, the percentage increase in the tension required to raise the wave speed by 20% is 44%, which matches the correct option provided.