Question

Two wires $A$ and $B$ are of the same materials. Their lengths are in the ratio $1:2$ and diameters are in the ratio $2:1$ when stretched by force $F_A$ and $F_B$ respectively, they get equal increase in their lengths. Then the ratio $F_A$ /$F_B$ should be

• A. $1:1$
• B. $8:1$
• C. $2:1$
• D. $1:2$

Answer 1

The correct option is B $8:1$

Given,
Stretching force for wire $A=F_A$
Stretching force for wire $B=F_B$
Ratio of length, $L_A:L_B=1:2$
Ratio of diameters, $D_A:D_B=2:1$
Young’s modulus for the wire,
$Y=\frac{\frac{F}{A}}{\frac{\Delta L}{L}}$
$F=Y\frac{\pi D^2}{4}\frac{l}{L}$
As we know, young modulus and 𝑙 will be constant for object, therefore
$\frac{F_A}{F_B}=\frac{D_A^2}{D_B^2}\times \frac{L_B}{L_A}$
$\frac{F_A}{F_B}=\frac{4}{1}\times \frac{2}{1}=\frac{8}{1}$