If we want to increases the frequency of transverse oscillations of stretched string by 50%, the tenstion must be increases by

Options

A. `100%`
B. `125%`
C. `150%`
D. `50%`

Explanation

To solve the problem of how much the tension must be increased in a stretched string to increase the frequency of transverse oscillations by 50%, we can follow these steps:

### Step 1: Understand the relationship between frequency and tension
The frequency \( f \) of a transverse wave on a string is given by the formula:
\[ f = \frac{1}{2L} \sqrt{\frac{T}{\mu}} \]
where:
- \( L \) is the length of the string,
- \( T \) is the tension in the string,
- \( \mu \) is the mass per unit length of the string.

From this formula, we can see that frequency is directly proportional to the square root of the tension:
\[ f \propto \sqrt{T} \]

### Step 2: Set up the initial and final frequencies
Let the initial frequency be \( f_1 \) and the final frequency after a 50% increase be \( f_2 \):
\[ f_2 = f_1 + 0.5 f_1 = 1.5 f_1 \]

### Step 3: Relate the frequencies to the tensions
Using the relationship between frequency and tension, we can express the initial and final tensions as:
\[ \frac{f_1}{f_2} = \frac{\sqrt{T_1}}{\sqrt{T_2}} \]
Substituting \( f_2 \):
\[ \frac{f_1}{1.5 f_1} = \frac{\sqrt{T_1}}{\sqrt{T_2}} \]
This simplifies to:
\[ \frac{1}{1.5} = \frac{\sqrt{T_1}}{\sqrt{T_2}} \]
or
\[ \frac{2}{3} = \frac{\sqrt{T_1}}{\sqrt{T_2}} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \left(\frac{2}{3}\right)^2 = \frac{T_1}{T_2} \]
This results in:
\[ \frac{4}{9} = \frac{T_1}{T_2} \]

### Step 5: Rearrange to find \( T_2 \)
Rearranging gives:
\[ T_2 = \frac{9}{4} T_1 \]

### Step 6: Calculate the increase in tension
Let’s assume the initial tension \( T_1 = 100 \, \text{N} \) (this is an arbitrary choice for calculation). Then:
\[ T_2 = \frac{9}{4} \times 100 = 225 \, \text{N} \]

### Step 7: Find the increase in tension
The increase in tension is:
\[ \Delta T = T_2 - T_1 = 225 - 100 = 125 \, \text{N} \]

### Step 8: Calculate the percentage increase in tension
To find the percentage increase:
\[ \text{Percentage Increase} = \left(\frac{\Delta T}{T_1}\right) \times 100 = \left(\frac{125}{100}\right) \times 100 = 125\% \]

### Conclusion
Thus, to increase the frequency of transverse oscillations of a stretched string by 50%, the tension must be increased by **125%**.

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