Question

A transverse wave is propagating along a string of length 1. When string is elongated by 1/20 cm, velocity of wave is v. What will be its velocity when it is elongated by 1/10?

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

Answer 1

Answer: (C)

$\sqrt{2}v$

Answer 2

The speed $v$ of a transverse wave on a string is proportional to the square root of the tension $T$ in the string:

$$v \propto \sqrt{T}$$

According to Hooke's Law, the tension $T$ is proportional to the extension $\Delta \ell$.

Therefore, we can write the ratio of the new velocity $v'$ to the original velocity $v$ as:

$$\frac{v'}{v} = \sqrt{\frac{\Delta \ell_2}{\Delta \ell_1}}$$

Given that the initial extension $\Delta \ell_1 = \frac{1}{20}$ cm and the final extension $\Delta \ell_2 = \frac{1}{10}$ cm, we have:

$$\frac{v'}{v} = \sqrt{\frac{1/10}{1/20}} = \sqrt{\frac{20}{10}} = \sqrt{2}$$

Thus, the new velocity $v'$ is:

$$v' = \sqrt{2} \, v$$