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

Options

A. 0.44
B. 0.4
C. 0.2
D. 0.1

Answer

To solve the problem step by step, we will follow the outlined process to find the percentage increase in tension required to raise the wave speed by 20%.

### Step 1: Determine the new speed (V2)
The initial speed (V1) of the wave is given as 150 m/s. We need to increase this speed by 20%.

\[ \text{New Speed (V2)} = V1 + 20\% \text{ of } V1 \]

Calculating 20% of V1:

\[ 20\% \text{ of } 150 = \frac{20}{100} \times 150 = 30 \]

Thus, the new speed V2 is:

\[ V2 = 150 + 30 = 180 \text{ m/s} \]

### Step 2: Relate the speeds and tensions
The relationship between the speed of a wave on a string and the tension is given by:

\[ \frac{V2}{V1} = \sqrt{\frac{T2}{T1}} \]

Where:
- \( V1 = 150 \, \text{m/s} \)
- \( V2 = 180 \, \text{m/s} \)
- \( T1 = 120 \, \text{N} \)
- \( T2 \) is the new tension we need to find.

### Step 3: Square both sides
Squaring both sides of the equation gives:

\[ \left(\frac{V2}{V1}\right)^2 = \frac{T2}{T1} \]

Substituting the known values:

\[ \left(\frac{180}{150}\right)^2 = \frac{T2}{120} \]

### Step 4: Calculate the left side
Calculating \( \frac{180}{150} \):

\[ \frac{180}{150} = 1.2 \]

Now squaring it:

\[ (1.2)^2 = 1.44 \]

### Step 5: Solve for T2
Now substituting back into the equation:

\[ 1.44 = \frac{T2}{120} \]

Multiplying both sides by 120 gives:

\[ T2 = 1.44 \times 120 = 172.8 \, \text{N} \]

### Step 6: Calculate the percentage increase in tension
To find the percentage increase in tension, we use the formula:

\[ \text{Percentage Increase} = \frac{T2 - T1}{T1} \times 100\% \]

Substituting the values:

\[ \text{Percentage Increase} = \frac{172.8 - 120}{120} \times 100\% \]

Calculating the difference:

\[ 172.8 - 120 = 52.8 \]

Now substituting this back into the percentage formula:

\[ \text{Percentage Increase} = \frac{52.8}{120} \times 100\% = 44\% \]

### Final Answer
The percentage increase in tension required to raise the wave speed by 20% is **44%**.

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