Question

The extension in a string obeying Hooke's law is $x$. The speed of sound in the stretched string is $v$. If the extension in the string is increased to $1.5x$, the speed of sound will be

• A. $1.22v$
• B. $0.61v$
• C. $1.50v$
• D. $0.75v$

Answer 1

The correct option is A

$1.22v$

Step 1: Given data

Initial extension in string $x_1=x$

Increased extension in string $x_2=1.5x$

Speed of sound in the stretched string $v_1=v$

Let $v_2$ be the speed of sound when the extension in the string is increased to $1.5x$

Extension in a string obeying Hooke's law

Step 2: Formula used

The speed of sound on a string is determined using the formula $v=\sqrt{\frac{T}{\mu }}$ where $v=speedofsound,T=tensionand\mu =massperlength$

From Hooke's law, stress is directly proportional to strain

$\sigma \propto \epsilon$

where $\sigma$ stress, $\epsilon$ strain

Stress is defined as internal force per unit area. and strain is the ratio of change in internal length to the actual length.

$$\begin{gathered} \sigma \propto \epsilon \\ \frac{T}{A}\propto \frac{x}{l} \end{gathered}$$

Thus the tension in the string is directly proportional to the extension

Step3: Compute the speed of sound

From the formula $v=\sqrt{\frac{T}{\mu }}$, it is clear that the speed of sound is directly proportional to the square root of the tension or we can say that it extension from Hooke's law. So,

Initially, $v_1\propto \sqrt{x_1}\dots \dots \dots \dots \dots \dots \dots \left(1\right)$

when the extension in the string is increased to $1.5x$

$v_2\propto \sqrt{x_2}\dots \dots \dots \dots \dots \dots \left(2\right)$

On dividing equation (2) by equation (1)

$$\begin{gathered} \frac{v_2}{v_1}=\sqrt{\frac{x_2}{x_1}} \\ \frac{v_2}{v_1}=\sqrt{\frac{1.5x}{x}} \\ {v}_2=1.22v \end{gathered}$$

So, the speed of sound will be $1.22v$.

Hence, option A is the correct answer.