Question

The Young’s modulus of brass and steel are
respectively 10¹⁰ N/m² and 2 × 10¹⁰ N/m². A
brass wire and a steel wire of the same length are
extended by 1 mm under the same force, the radii
of brass and steel wires are RB and RS
respectively. Then
(a) Rs = ⎷2 Rв (b) Rs = Rв / ⎷2
(c) Rs = 4Rв (d) Rs = Rв / 4

Answer 1

$\Large\boxed{\sf{(b)\ R_s=\dfrac{R_b}{\sqrt2}}}$

We have,

$\longrightarrow\sf{Y=\dfrac{F\ L}{A\ \Delta L}}$

In case of same length, extension and force,

$\longrightarrow\sf{Y\propto\dfrac{1}{A}\quad\quad\dots(1)}$

But,

$\longrightarrow\sf{A=\pi r^2}$

$\Longrightarrow\sf{A\propto r^2}$

Then (1) becomes,

$\longrightarrow\sf{Y\propto\dfrac{1}{r^2}}$

$\longrightarrow\sf{r\propto\dfrac{1}{\sqrt Y}}$

Therefore,

$\longrightarrow\sf{\dfrac{R_s}{R_b}=\sqrt{\dfrac{Y_b}{Y_s}}}$

$\longrightarrow\sf{\dfrac{R_s}{R_b}=\sqrt{\dfrac{10^{10}}{2\times10^{10}}}}$

$\longrightarrow\sf{\dfrac{R_s}{R_b}=\dfrac{1}{\sqrt2}}$

$\longrightarrow\sf{\underline{\underline{R_s=\dfrac{R_b}{\sqrt2}}}}$