Consider a wire with density (d) and stress `(sigma)` . For the same density . if the stress increases 2 times , the speed of the transverse waves along the wire change by

Options

A. `sqrt2`
B. `1/sqrt2`
C. 2
D. `1/2`

Answer

To solve the problem, we need to determine how the speed of transverse waves in a wire changes when the stress is doubled while keeping the density constant.

### Step-by-Step Solution:

1. **Understanding Wave Speed in a Wire**:
The speed of transverse waves \( v \) in a wire can be expressed using the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( T \) is the tension in the wire and \( \mu \) is the mass per unit length (linear density) of the wire.

2. **Expressing Linear Density**:
The linear density \( \mu \) can be defined as:
\[ \mu = d \cdot A \]
where \( d \) is the density of the material and \( A \) is the cross-sectional area of the wire.

3. **Relating Tension and Stress**:
The stress \( \sigma \) in the wire is defined as:
\[ \sigma = \frac{F}{A} \]
where \( F \) is the force applied perpendicular to the cross-section. Rearranging gives:
\[ F = \sigma \cdot A \]
Thus, we can express the tension \( T \) as:
\[ T = \sigma \cdot A \]

4. **Substituting into the Wave Speed Formula**:
Substituting \( T \) and \( \mu \) into the wave speed formula gives:
\[ v = \sqrt{\frac{\sigma \cdot A}{d \cdot A}} = \sqrt{\frac{\sigma}{d}} \]

5. **Analyzing the Change in Stress**:
According to the problem, the stress is increased to \( 2\sigma \). Therefore, the new speed \( v' \) can be calculated as:
\[ v' = \sqrt{\frac{2\sigma}{d}} \]

6. **Finding the Ratio of New Speed to Original Speed**:
To find how the speed changes, we take the ratio of the new speed to the original speed:
\[ \frac{v'}{v} = \frac{\sqrt{\frac{2\sigma}{d}}}{\sqrt{\frac{\sigma}{d}}} = \sqrt{2} \]

7. **Conclusion**:
Therefore, when the stress is doubled, the speed of the transverse waves increases by a factor of \( \sqrt{2} \).

### Final Answer:
The speed of the transverse waves along the wire changes by a factor of \( \sqrt{2} \).

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