Two wires are made of the same material and have the exact same volume. The first wire has a cross-s

Question

Two wires are made of the same material and have the exact same volume. The first wire has a cross-sectional area $A$ and the second wire has a cross-sectional area $2A$. If the first wire is stretched by an amount $\Delta l$ upon applying a force $F$, the force required to stretch the second wire by the same amount $\Delta l$ is

Professor PC Thomas Classes · Answer

(a) : Let the volume be $V$. For the first wire, length $L_1 = \frac{V}{A}$. For the second wire, length $L_2 = \frac{V}{2A} = \frac{L_1}{2}$. Using Young's Modulus $Y = \frac{FL}{A \Delta l}$, we have $\Delta l = \frac{F_1 L_1}{A_1 Y} = \frac{F L_1}{A Y}$. For the second wire, $\Delta l = \frac{F_2 L_2}{A_2 Y} = \frac{F_2 (L_1/2)}{(2A) Y} = \frac{F_2 L_1}{4 A Y}$. Equating the two expressions for $\Delta l$: $\frac{F L_1}{A Y} = \frac{F_2 L_1}{4 A Y}$. This yields $F_2 = 4F$.