Properties of Matter 1 Question 3
3. Young’s moduli of two wires $A$ and $B$ are in the ratio $7: 4$. Wire $A$ is $2 m$ long and has radius $R$. Wire $B$ is $1.5 m$ long and has radius $2 mm$. If the two wires stretch by the same length for a given load, then the value of $R$ is close to
(2019 Main, 8 April II)
(a) $1.3 mm$
(b) $1.5 mm$
(c) $1.9 mm$
(d) $1.7 mm$
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Answer:
Correct Answer: 3. (d)
Solution:
- When a wire is stretched, then change in length of wire is $\Delta l=\frac{F l}{\pi r^{2} Y}$, where $Y$ is its Young’s modulus.
Here, for wires $A$ and $B$,
$$ \begin{aligned} l _A & =2 m, l _B=1.5 m \\ \frac{Y _A}{Y _B} & =\frac{7}{4}, r _B=2 mm=2 \times 10^{-3} m \text { and } \frac{F _A}{F _B}=1 \end{aligned} $$
As, it is given that $\Delta l _A=\Delta l _B$
$\Rightarrow \frac{F _A l _A}{\pi r _A^{2} Y _A}=\frac{F _B l _B}{\pi r _B^{2} Y _B}$
$\Rightarrow \quad r _A^{2}=\frac{F _A}{F _B} \cdot \frac{l _A}{l _B} \cdot \frac{Y _B}{Y _A} \cdot r _B^{2}$
$$ =1 \times \frac{2}{1.5} \times \frac{4}{7} \times 4 \times 10^{-6} m=3.04 \times 10^{-6} m $$
or $\quad r _A=1.7 \times 10^{-3} m$
or $\quad r _A=1.7 mm$
