The ratio of radius of two wires of the same metal is 2 : 1. If t… - Vidyadip

Question

The ratio of radius of two wires of the same metal is 2 : 1. If they are pulled by applying equal force, what will be the ratio of stresses?

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Answer

Two wires of same metal whose radii are in the ratio is 2 : 1.
If $r_1$ and $r_2$ are the radii of these wires then
$\frac{r_1}{r_2}=\frac{2}{1}$
These wires are pulled by applying equal force ( F ). Then, let $F _1$ be the stress of wire at radius $r_1$.
$
\therefore \quad F_1=\frac{F}{A_1}=\frac{F}{\pi r_1^2}
$
Let $F _2$ be the stress of wire at radius $r_2$.
$
\begin{aligned}
\therefore \quad F_2 & =\frac{F}{A_2}=\frac{F}{\pi r_2^2} \\
\frac{F_1}{F_2} & =\frac{\frac{F}{\frac{\pi r_1^2}{F}}}{\pi r_2^2} \\
\text { or } \quad \frac{F_1}{F_2} & =\frac{F}{\pi r_1^2} \times \frac{\pi r_2^2}{F}=\left(\frac{r_2}{r_1}\right)^2 \\
\frac{F_1}{F_2} & =\left(\frac{1}{2}\right)^2 \because \frac{r_1}{r_2}=\frac{2}{1}\\
\frac{F_1}{F_2} & =\frac{1}{4} \\
F_1: F_2 & =1: 4
\end{aligned}
$

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