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Question

Speed of transverse wave on a straight wire (mass 6·0gm, length 60cm and area of cross-section 1·0mm2) is 90ms. If the Young’s modulus of wire is 16×1011Nm-2, the extension of wire over its natural length is


A

0·03mm

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B

0·04mm

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C

0·02mm

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D

0·01mm

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Solution

The correct option is A

0·03mm


Step 1: Given data:

Mass of wire m=6·0gm=6·0×10-3kg

Length of wirel=60cm=60×10-2m

Cross-section areaA=1mm2=1·0×10-6m2

Young's Modulus of wireY=16×1011Nm-2

The speed of the transverse wavev=90ms

Step2: Formula used:

From the relation of the speed of wave and mass per length

Speedofwavev=TensionTMassperlengthμ

Mass per unit length is the linear density given as-

massperlength(μ)=mass(m)length(l)

Young's modulus will be calculated by-

stressstrain=Y

Step 3: Compute the tension in the wire

From the formula,

Speedofwavev=TensionTMassperlengthμ

Speedofwavev=TensionTMassofwiremLengthofwirel90=T6·0×10-360×10-290=T1×10-28100=T1×10-2T=81N

Thus, the tension will be 81N.

Step4: Compute extension in the wire

We know that,

stressstrain=Y

Strain will be-ll

Stress will be-forcearea=Tensionarea

On putting the given values,

stressY=llTAY=lll=T×lA×Yl=81×60×10-21×10-616×1011l=3.0375×10-5ml=0·03mm

Thus, the extension of wire over its natural length is 0.03mm.

Hence, option A is the correct answer.


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