Question 19: Properties of Matter

Question

Two wires A and B are made of same material having ratio of lengths $\frac{L_A}{L_B}=\frac{1}{3}$ and their diameters ratio $\frac{d_A}{d_B}=2$. If both the wires are stretched using same force, what would be the ratio of their respective elongations?

Accepted Answer

Given:
The ratio of lengths: $\frac{L_A}{L_B} = \frac{1}{3}$
The ratio of diameters: $\frac{d_A}{d_B} = 2$
Both wires are subject to the same stretching force, and since they are made of the same material, their Young's modulus ($Y$) is the same.
The elongation ($\Delta L$) of a wire subject to a force is given by:
$ \Delta L = \frac{F \cdot L}{A \cdot Y} $
where $F$ is the force applied, $L$ is the original length, $A$ is the cross-sectional area, and $Y$ is Young's modulus.
For wires $A$ and $B$:
$\Delta L_A = \frac{F_A \cdot L_A}{A_A \cdot Y_A}$
$\Delta L_B = \frac{F_B \cdot L_B}{A_B \cdot Y_B}$
Since $F_A = F_B$ and $Y_A = Y_B$, the ratio of their elongations becomes:
$ \frac{\Delta L_A}{\Delta L_B} = \frac{L_A \cdot A_B}{L_B \cdot A_A} $
The cross-sectional area $A$ is a function of diameter, $A = \frac{\pi}{4} d^2$. Therefore,
$ A_A = \frac{\pi}{4} d_A^2 \quad 	ext{and} \quad A_B = \frac{\pi}{4} d_B^2 $
Substituting these into the elongation ratio:
$ \frac{\Delta L_A}{\Delta L_B} = \left(\frac{L_A}{L_B}ight) \left(\frac{\frac{\pi}{4} d_B^2}{\frac{\pi}{4} d_A^2}ight) = \left(\frac{L_A}{L_B}ight) \left(\frac{d_B}{d_A}ight)^2 $
Substitute the given ratios:
$ \frac{\Delta L_A}{\Delta L_B} = \left(\frac{1}{3}ight)\left(\frac{1}{2}ight)^2 = \left(\frac{1}{3}ight)\left(\frac{1}{4}ight) = \frac{1}{12} $
Thus, the ratio of the elongations of wires $A$ and $B$ is $\frac{1}{12}$.

Answer

$3: 4$

Answer

$1: 12$

Answer

$1: 3$

Answer

$1: 6$

Properties of Matter

Two wires A and B are made of same material having ratio of lengths \frac{L_A}{L_B}=\frac{1}{3} and their diameters ratio \frac{d_A}{d_B}=2. If both the wires are stretched using same force, what would be the ratio of their respective elongations?

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