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Question

Two wires of the same material and same length but diameters in the ratio 1:2 are stretched by the same force. The potential energy per unit volume of the two wires will be in the ratio:


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Solution

Step 1: Given Data

The ratio of diameters of the two materials: 1:2

d1d2=12

It is already given in the question that the two wires are made of the same materials, therefore the strain on both the materials will be the same.

Step 2: Formula Used

Strain=StressY

Stress=FA, A=πd24

U=12Stress×Strain

Uis the potential energy

F = Force applied
A = Area on the surface

where Y is Young's modulus

Stress is inversely proportional to the square of the diameter of the wire:

Stress1d2

Step 3: Relationship amongst Potential energy, Stress, and Young's Modulus:

U=12Stress×StrainU=12Stress×StressYU=12×(Stress)2Y=F22YA2U=8F2π2d4Y

{F and Y are the same for both wires}

Since Stress is inversely proportional to the square of the diameter of the wire, the potential energy will be inversely proportional to the diameter raised to its fourth power:

U1d4

Step 4: Relationship between Potential energy and the two diameters:

U1U2=d22d122=2414=16:1

Hence, the potential energy per unit volume of the two wires will be in the ratio 16:1.


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