In an experiment to determine the Young's modulus, steel wires of five different lengths (1,2,3,4 , and 5 ~m) but of same cross section (2 ~mm^2) were taken and curves between extension and load were obtained. The slope (extension/load) of the curves were plotted with the wire length and the following graph is obtained. If the Young's modulus of given steel wires is x \times 10^11 \,Nm^-2 , then the value of x is \underline\hspace2cm .

Properties of Matter PYQ — JEE Main 2022 · 27 Jul · Shift 2 · Q56 with Solution

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Correct answer: 2

\Delta L = \frac{F L}{A Y}

where:

\frac{\Delta L}{F} = \frac{L}{A Y}

So, if we plot $\left(\frac{\text{extension}}{\text{load}}\right)$ versus $L$ , we get a straight line with slope

m = \frac{1}{A Y}

From the given graph, the line relating $\frac{\text{extension}}{\text{load}}$ and $L$ has slope

m = 2.5 \times 10^{-6}\ \text{N}^{-1}

(per metre of length, as read from the graph).

A = 2\,\text{mm}^2 = 2 \times 10^{-6}\,\text{m}^2

m = \frac{1}{A Y}

we get

Y = \frac{1}{A m}

Substitute values:

Y = \frac{1}{(2 \times 10^{-6})(2.5 \times 10^{-6})}

Y = \frac{1}{5 \times 10^{-12}} = 2 \times 10^{11}\,\text{N m}^{-2}

Y = x \times 10^{11}\,\text{N m}^{-2}

then

x = 2