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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

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B

0.61v

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C

1.50v

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D

0.75v

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Solution

The correct option is A

1.22v


Step 1: Given data

Initial extension in string x1=x

Increased extension in string x2=1.5x

Speed of sound in the stretched string v1=v

Let v2 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=Tμ where v=speedofsound,T=tensionandμ=massperlength

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

σε

where σ stress, ε strain

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

σεTAxl

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

Step3: Compute the speed of sound

From the formula v=Tμ, 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, v1x11

when the extension in the string is increased to 1.5x

v2x22

On dividing equation (2) by equation (1)

v2v1=x2x1v2v1=1.5xxv2=1.22v

So, the speed of sound will be 1.22v.

Hence, option A is the correct answer.


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