Question 12: Properties of Matter

Question

The two wires $A$ and $B$ of equal cross-section but of different materials are joined together. The ratio of Young's modulus of wire $A$ and wire $B$ is 20/11. When the joined wire is kept under certain tension the elongations in the wires $A$ and $B$ are equal. If the length of wire $A$ is 2.2 m , then the length of wire $B$ is
$\_\_\_\_$ m.

Accepted Answer

Young's Modulus is defined as the ratio of Stress $(\sigma)$ to Strain $(\epsilon)$.
$$ Y=\frac{	ext { Stress }}{	ext { Strain }}=\frac{\sigma}{\epsilon} $$
Stress is force per unit area, $\sigma=\frac{	ext { Force (F) }}{	ext { Area (A) }}$
Strain is change in length per unit original length,
$$ \epsilon=\frac{	ext { Change in length }(\Delta \mathrm{L})}{	ext { Original length }(\mathrm{L})} $$
So, $Y=\frac{F / A}{\Delta L / L}$
$$ \Rightarrow \Delta \mathrm{L}=\frac{\mathrm{FL}}{\mathrm{AY}} $$
Since the wires are joined in series and kept under tension, the Force (F) acting on both wires is the same.
The area (A) is the same for both.
It is given that the elongation in both the wires is equal, $\Delta \mathrm{L}_{\mathrm{A}}=\Delta \mathrm{L}_{\mathrm{B}}$.
Since $\Delta \mathrm{L}_{\mathrm{A}}=\Delta \mathrm{L}_{\mathrm{B}}:$
$\frac{F L_A}{A Y_A}=\frac{F L_B}{A Y_B}$
$\Rightarrow \frac{\mathrm{L}_{\mathrm{A}}}{\mathrm{Y}_{\mathrm{A}}}=\frac{\mathrm{L}_{\mathrm{B}}}{\mathrm{Y}_{\mathrm{B}}}$
$$\Rightarrow $$ $$ \frac{\mathrm{L}_{\mathrm{A}}}{\mathrm{~L}_{\mathrm{B}}}=\frac{\mathrm{Y}_{\mathrm{A}}}{\mathrm{Y}_{\mathrm{B}}} $$
The ratio of Young's modulus is $\frac{Y_A}{Y_B}=\frac{20}{11}$ and the length of wire $A$ is $L_A=2.2 \mathrm{~m}$
$\frac{2.2}{\mathrm{~L}_{\mathrm{B}}}=\frac{20}{11}$
$$\Rightarrow $$ $\mathrm{L}_{\mathrm{B}}=2.2 	imes \frac{11}{20}$
$$\Rightarrow $$ $\mathrm{L}_{\mathrm{B}}=\frac{24.2}{20}$
$$\Rightarrow $$ $$ \mathrm{L}_{\mathrm{B}}=1.21 \mathrm{~m} $$
Therefore, the length of wire B is 1.21 m . Hence, the correct option is (C).

Answer

1.1

Answer

2.22

Answer

1.21

Answer

4.44

Properties of Matter

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