Mechanical Properties of Solids - Result Question 7

8. Two wires A and B are of the same material. Their lengths are in the ratio $1: 2$ and the diameter are in the ratio $2: 1$. If they are pulled

by the same force, then increase in length will be in the ratio

(a) $2: 1$

(b) $1: 4$

(c) $1: 8$

(d) $8: 1$

[1988]

Topic 2: Bulk and Rigidity Modulus & Work Done in Stretching a Wire

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Solution:

  1. (c) We know that Young’s modulus

$Y=\frac{F}{\pi r^{2}} \times \frac{L}{\ell}$

Since Y, F are same for both the wires, we have, $\ell \propto \frac{L}{r^{2}}$

or, $\frac{\ell_1}{\ell_2}=\frac{r_2^{2} \times L_1}{r_1^{2} \times L_2}=\frac{(D_2 / 2)^{2} \times L_1}{(D_1 / 2)^{2} \times L_2}$

or, $\frac{\ell_1}{\ell_2}=\frac{D_2^{2} \times L_1}{D_1^{2} \times L_2}=\frac{D_2^{2}}{(2 D_2)^{2}} \times \frac{L_2}{2 L_2}=\frac{1}{8}$

So, $\quad \ell_1: \ell_2=1: 8$