Question

Consider a wire with density $\rho$ and stress $\sigma$. For the same density, if the stress increases 2 times, the speed of the transverse waves along the wire changes by

• A. $\sqrt{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. 2
• D. $\frac{1}{2}$
• E. 4

Hint:

Wave speed in a wire is independent of the wire's cross-sectional area as long as the material (density) and the applied stress remain the same.

Answer 1

The Correct Option is A

Concept:
The speed ($v$) of a transverse wave in a stretched string/wire is: \[ v = \sqrt{\frac{T}{\mu}} \] Where $T$ is the tension and $\mu$ is the mass per unit length.

Step 1: Relate the formula to stress ($\sigma$) and density ($\rho$).
[itemsep=6pt]
• Tension ($T$): $\text{Stress} \times \text{Area} = \sigma A$.
• Linear Density ($\mu$): $\text{Density} \times \text{Area} = \rho A$. Substituting these into the velocity formula: \[ v = \sqrt{\frac{\sigma A}{\rho A}} = \sqrt{\frac{\sigma}{\rho}} \]

Step 2: Calculate the change in speed.
Since $\rho$ is constant, $v \propto \sqrt{\sigma}$. If stress increases 2 times ($\sigma' = 2\sigma$): \[ v' = \sqrt{\frac{2\sigma}{\rho}} = \sqrt{2} \times v \] The speed changes by a factor of $\sqrt{2}$.