Spacing between two successive nodes in a standing wave on a string is `x`. If frequency of the standing wave is kept unchanged but tension in the string is doubled, then new sapcing between successive nodes will become:

Options

A. `x//sqrt(2)`
B. `sqrt(2)x`
C. `x//2`
D. 2x

Answer

To solve the problem, we need to analyze how the spacing between two successive nodes in a standing wave changes when the tension in the string is doubled while keeping the frequency constant.

### Step-by-Step Solution:

1. **Understand the Relationship Between Velocity, Tension, and Linear Density**:
The velocity \( v \) of a wave on a string is given by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( T \) is the tension in the string and \( \mu \) is the linear density of the string.

2. **Relate Velocity to Wavelength and Frequency**:
The wave velocity is also related to the wavelength \( \lambda \) and frequency \( f \) by the equation:
\[ v = \lambda f \]
Since the frequency \( f \) is kept constant, any change in velocity will affect the wavelength.

3. **Determine the Effect of Doubling the Tension**:
If the tension \( T \) is doubled (i.e., \( T' = 2T \)), we can find the new velocity \( v' \):
\[ v' = \sqrt{\frac{T'}{\mu}} = \sqrt{\frac{2T}{\mu}} = \sqrt{2} \cdot \sqrt{\frac{T}{\mu}} = \sqrt{2} \cdot v \]
This means the new velocity \( v' \) is \( \sqrt{2} \) times the original velocity \( v \).

4. **Relate the New Velocity to the New Wavelength**:
Since the frequency remains constant, we can express the new wavelength \( \lambda' \) as:
\[ v' = \lambda' f \]
Substituting for \( v' \):
\[ \sqrt{2} v = \lambda' f \]
Since \( v = \lambda f \), we can substitute \( v \) into the equation:
\[ \sqrt{2} \lambda f = \lambda' f \]
Dividing both sides by \( f \) (since \( f \neq 0 \)):
\[ \lambda' = \sqrt{2} \lambda \]

5. **Calculate the New Spacing Between Nodes**:
The spacing between two successive nodes in a standing wave is given by:
\[ x = \frac{\lambda}{2} \]
Therefore, the new spacing \( x' \) will be:
\[ x' = \frac{\lambda'}{2} = \frac{\sqrt{2} \lambda}{2} = \frac{\sqrt{2}}{2} \lambda \]
Since the original spacing was \( x = \frac{\lambda}{2} \), we can express the new spacing in terms of \( x \):
\[ x' = \sqrt{2} x \]

### Final Answer:
The new spacing between successive nodes will become \( \sqrt{2} x \).

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