Question

Two bars A and B of circular cross section and of same volume of same material are subjected to tension. if the diameter of A is half that of B and if the force applied to both the rods is the same and it is in the elastic limit the ratio of extension of A t that of B will be

• A. 16
• B. 8
• C. 4
• D. 2

Answer 1

The correct option is A 16

The young's modulus of the material is given by

$Y=\frac{F/A}{\Delta l/l}$

$Y=\frac{F\times l}{A\times \Delta l}$

$Y=\frac{F\times l\times A}{A^2\times \Delta l}$ $(V=lA)$

$Y=\frac{FV}{A^2\times \Delta l}$. . . . . .(1)

From the given question, $F,V,Y$ are same.

From the equation (1),

$\Delta l\propto \frac{1}{A^2}$

$\Delta \propto \frac{1}{r^4}$ where $r=radius$

$\Delta \propto \frac{1}{d^4}$ where, d= diameter

The diameter of $A$ is half of the diameter of $B$

$d_A=\frac{d_B}{2}$

$d_B=2d_A$

Ratio, $\frac{\Delta l_A}{{\Delta }_B}=\frac{d_B^4}{d_A^4}=(\frac{2d_A}{d_A}{)}^4$

$\frac{\Delta l_A}{\Delta l_B}=16$

The correct option is A.