Question

the extension in the string, obeying Hooke’s law, is $x$ . The speed of the sound in the stretched string is $v$ . if the extension in the string is increase to $1.5x$ , the speed of sound will be:
A. $1.22\,v$
B. $0.61\,v$
C. $1.50\,v$
D. $0.75$

Answer 1

it is given in the question that the extension in the string obeys Hooke’s law. Hooke’s law is the law that states that the stress applied on a solid substance is directly proportional to the strain such that the stress applied is less than the elastic limit of a substance. Therefore, the tension in the string will be equal to extension in the string.

FORMULA USED:
The formula used for the speed of the sound is given by
$v = \sqrt {\dfrac{T}{\mu }}$

Here, $v$ is the speed, $T$ is the elastic property of the string and $\mu$ is the inertial property.

COMPLETE STEP BY STEP ANSWER:
It is given in the question that the extension in the string is obeying Hooke’s law. Hooke’s law is the law that states that the stress applied on a solid substance is directly proportional to the strain such that the stress applied is less than the elastic limit of a substance.

Now, the speed of the sound can be calculated by using the following formula
$v = \sqrt {\dfrac{T}{\mu }}$

Now, let $v'$ is the speed of the sound, when the extension produced in the string will be increased, therefore, the speed $v'$ is given below
$v' = \sqrt {\dfrac{{T'}}{\mu }}$

Now, dividing $v$ and $v'$ , we get
$\dfrac{v}{{v'}} = \sqrt {\dfrac{T}{{T'}}}$

Now, putting the value of extensions, we get
$\dfrac{v}{{v'}} = \sqrt {\dfrac{x}{{1.5\,x}}}$
$\Rightarrow \,\dfrac{v}{{v'}} = \sqrt {\dfrac{1}{{1.5}}}$
$\Rightarrow \dfrac{{v'}}{v} = \sqrt {\dfrac{{1.5}}{1}}$
$\Rightarrow \,v' = v\sqrt {1.5}$
$\Rightarrow \,v' = 1.22\,v$

Therefore, if the extension in the string is increased to $1.5\,x$ , then the speed of the sound will be $1.22v$ .

Hence, option (A) is the correct option.

NOTE: Here, $v$ is the speed of sound before extension of the string is increased and $v'$ is the speed of sound after the extension is increased.
Now, according to Hooke’s law, $F \propto x$

Where, $F$ is the force and $x$ is the extension in the string.
Now, we know that tension is the same as force.

Therefore, from Hooke’s law, we can say that $T \propto x$ , here $T$ is the elastic property of the string and $x$ is the extension in the string. That is why, in the above example, we have put the values of extension in place of tension.