Question

Three rods of equal lengths are joined to form an equilateral triangle $\text{PQR}$ as shown in the figure. $\text{O}$ is the mid point of $\text{PQ}$ and distance $\text{OR}$ remains same for any small change in temperature. If the coefficient of linear expansion for $\text{PR}$ and $\text{RQ}$ is same i.e, ${\alpha }_2$, but that for $\text{PQ}$ is ${\alpha }_1$, then which of the following options correctly represents the relation between ${\alpha }_1$ and ${\alpha }_2$?

• A. ${\alpha }_2=3{\alpha }_1$
• B. ${\alpha }_2=4{\alpha }_1$
• C. ${\alpha }_1=3{\alpha }_2$
• D. ${\alpha }_1=4{\alpha }_2$

Answer 1

The correct option is D ${\alpha }_1=4{\alpha }_2$

Let the length of each rod be $l$.
Since, the combination of rods form an equilateral triangle, by using pythagoras theorem, we can say that,
$(OR{)}^2=(PR{)}^2-(PO{)}^2=l^2-{\left(\frac{l}{2}\right)}^2=\frac{3l^2}{4}......(1)$
A small change in temperature brings a change in the length of the rods. We can understand this with the use of the formula
new length, $l^{{}^{\prime }}=l(1+\alpha t)$
Given that,
Coefficient of linear expansion for rod $PR$ is ${\alpha }_2$
Coefficient of linear expansion for rod $RQ$ is ${\alpha }_2$
and
Coefficient of linear expansion for rod $PQ$ is ${\alpha }_1$

Given that $OR$ doesn't change.
For small change in temperature,
$P^{\prime }{R}^{\prime }=PR(1+{\alpha }_{2}t)=l(1+{\alpha }_{2}t)$
& $P^{\prime }{O}^{\prime }=PO(1+{\alpha }_{1}t)=\frac{l}{2}(1+{\alpha }_{1}t)$

Using $\left(1\right)$, we get
$\frac{3l^2}{4}=\left[l\right(1+{\alpha }_{2}t){]}^2-{\left[\frac{l}{2}\right(1+{\alpha }_{1}t\left)\right]}^2$
$\frac{3l^2}{4}=l^2(1+{\alpha }_2^2{t}^2+2{\alpha }_{2}t)-\frac{l^2}{4}(1+{\alpha }_1^2{t}^2+2{\alpha }_{t}t)$
Neglecting the terms, ${\alpha }_2^2{t}^2$ and ${\alpha }_1^2{t}^2$ and solving, we get [because terms are very small]
$0=l^2\left(2{\alpha }_{2}t\right)-\frac{l^2}{4}\left(2{\alpha }_{1}t\right)$
$\Rightarrow 2{\alpha }_2=\frac{2{\alpha }_1}{4}$
$\Rightarrow {\alpha }_1=4{\alpha }_2$
Hence, option (d) is the correct answer.