Young's moduli of the material of wires A and B are in the ratio… — Solution | JEE Physics

MCQEasyJEE 2023Young's Modulus, Bulk & Rigidity Modulus

Young's moduli of the material of wires A and B are in the ratio $1:4$ , while its area of cross sections are in the ratio $1:3$ . If the same amount of load is applied to both the wires, the amount of elongation produced in the wires A and B will be in the ratio of: [Assume length of wires A and B are same]

A
$12 : 1$
B
$1 : 36$
C
$1 : 12$
D
$36 : 1$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: Young's modulus ratio for wires A and B is $1:4$ . Cross-sectional area ratio is $1:3$ . The same load is applied and the lengths are equal.

Find: The ratio of elongations $\Delta \ell_1 : \Delta \ell_2$ .

Using the relation from the solution:

\frac{W}{A} = Y \cdot \frac{\Delta \ell_1}{\ell}

and

\frac{W}{3A} = 4Y \cdot \frac{\Delta \ell_2}{\ell}

Extracted Working

From the given working:

\Delta \ell_1 = \frac{W\ell}{AY}

\Delta \ell_2 = \frac{W\ell}{12AY}

Therefore,

\frac{\Delta \ell_1}{\Delta \ell_2} = \frac{12}{1}

So the elongation ratio is $12:1$ . The solution marks option D, but this disagrees with both the extracted calculation and the listed options. The defensible correct option is A.

Common mistakes

Using elongation as directly proportional to Young's modulus is incorrect. Since $\Delta \ell = \frac{W\ell}{AY}$ , elongation is inversely proportional to both area and Young's modulus. Always place $A$ and $Y$ in the denominator.
Ignoring the change in cross-sectional area for wire B leads to a wrong ratio. Wire B has area $3A$ , not $A$ . Substitute the area ratio carefully before comparing elongations.
Confusing the option label with the computed value is a provided-page error here. The working gives $12:1$ , so match the numerical result with the listed options instead of blindly following the printed option letter.

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Answer

Step-by-step solution

Common mistakes

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