If the ratio of lengths, radii and Young's modulus of steel and brass wires shown in the figure are a, b and c respectively, the ratio between the increase in lengths of brass and steel wires would be

Question

If the ratio of lengths, radii and Young's modulus of steel and brass wires shown in the figure are a, b and c respectively, the ratio between the increase in lengths of brass and steel wires would be <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/physics/81fb428e0a4f030db233183b27055e22-.png" /> (A) (b2 a/2 c) (B) (b c/2 a2) (C) (b a2/2 c) (D) (a/2 b2 c)

Tardigrade Admin · Accepted Answer

Free body diagram of the two blocks are

Given,

\frac{l _{1}}{l _{2}} = a, \frac{r _{1}}{r _{2}} = b, \frac{Y _{1}}{Y _{2}} = c

Let Young's modulus of steel is

Y_{1}

and of brass is

Y_{2}

∴ Y_{1} = \frac{F _{1} \cdot l _{1}}{A _{1} \cdot Δ _{1}}

......(i)
and

Y_{2} = \frac{F _{2} \cdot l _{2}}{A _{2} Δ l _{2}}

......(ii)
Dividing Eq. (i) by Eq. (ii), we get

\frac{Y _{1}}{Y _{2}} = \frac{\frac{F _{1} \cdot l _{1}}{A _{1} \cdot Δ _{1}}}{\frac{F _{2} \cdot l _{2}}{A _{2} \cdot Δ l _{2}}}

or

\frac{Y _{1}}{Y _{2}} = \frac{F _{1} \cdot A _{2} \cdot l _{1} \cdot Δ l _{2}}{F _{2} \cdot A _{1} \cdot l _{2} \cdot Δ _{1}}

......(iii)
Force on steel wire from free body diagram

T = F_{1} = (2 g)

newton
Force on brass wire from free body diagram

F_{2} = T^{'} = T + 2 g = (4 g)

newton
Now, putting the value of

F_{1}, F_{2}

, in Eq. (iii), we get

\frac{Y _{1}}{Y _{2}} = (\frac{2 g}{4 g}) \cdot (\frac{π r _{2}^{2}}{π r _{1}^{2}}) [\frac{l _{1}}{l _{2}}] \cdot (\frac{Δ l _{2}}{Δ l _{1}})

OR

c = \frac{1}{2} (\frac{1}{b ^{2}}) \cdot a (\frac{Δ l _{2}}{Δ _{1}})

OR

\frac{Δ l _{1}}{Δ l _{2}} = (\frac{a}{2 b ^{2} c})

Q. If the ratio of lengths, radii and Young's modulus of steel and brass wires shown in the figure are $a, b$ and $c$ respectively, the ratio between the increase in lengths of brass and steel wires would be

$\frac{b ^{2} a}{2 c}$

$\frac{b c}{2 a ^{2}}$

$\frac{b a ^{2}}{2 c}$

$\frac{a}{2 b ^{2} c}$

Q. If the ratio of lengths, radii and Young's modulus of steel and brass wires shown in the figure are a,b and c respectively, the ratio between the increase in lengths of brass and steel wires would be

2cb2a​

2a2bc​

2cba2​

2b2ca​

Q. If the ratio of lengths, radii and Young's modulus of steel and brass wires shown in the figure are $a, b$ and $c$ respectively, the ratio between the increase in lengths of brass and steel wires would be

$\frac{b ^{2} a}{2 c}$

$\frac{b c}{2 a ^{2}}$

$\frac{b a ^{2}}{2 c}$

$\frac{a}{2 b ^{2} c}$