Question

A copper rod and a steel rod maintain a difference in their lengths constant = 10 cm at all temperatures. If their coefficients of expansion are $1.6\times {{10}^{-5}}{{K}^{-1}}$ and $1.2\times {{10}^{-5}}{{K}^{-1}}$, then the length of the Cu rod is
A. 40 cm
B. 30 cm
C. 32 cm
D. 24 cm

Answer 1

Hint: We have the difference in length of copper rod and steel rod and it is said that the difference is constant in all temperatures. We are also given the coefficients of expansion of both the rods. By using the equation to find the length change due to temperature, we can solve this question.

Formula used:
Change in length,
$l'=l\left( 1+\alpha \Delta t \right)$

Complete step by step answer:
In the question we are given the difference in the length of a copper rod and a steel rod.
Let $'{{l}_{s}}'$ be the length of steel and $'{{l}_{c}}'$ be the length of copper rod.
Then, we have
${{l}_{s}}-{{l}_{c}}=10cm$
It is said that the difference of their length remains the same in all temperatures.
We are also given the coefficients of expansions of copper (${{\alpha }_{c}}$) and steel (${{\alpha }_{s}}$).
Coefficient of expansion of copper,
${{\alpha }_{c}}=1.6\times {{10}^{-5}}{{K}^{-1}}$
Coefficient of expansion of steel,
${{\alpha }_{s}}=1.2\times {{10}^{-5}}{{K}^{-1}}$
Length at different temperature is given by the equation,
\[l'=l\left( 1+\alpha \Delta t \right)\]
Here length at different temperature of copper will be,
${{l}_{c}}'={{l}_{c}}+{{l}_{c}}{{\alpha }_{c}}\Delta t$
Length at different temperature of steel will be,
${{l}_{s}}'={{l}_{s}}+{{l}_{s}}{{\alpha }_{s}}\Delta t$
Now, let us take the difference of these two equations, we get
${{l}_{s}}'-{{l}_{c}}'=\left( {{l}_{s}}-{{l}_{c}} \right)+\Delta t\left( {{l}_{s}}{{\alpha }_{s}}-{{l}_{c}}{{\alpha }_{c}} \right)$
Since it is said that the difference of length is always constant in all temperatures, the multiplying factor of $\Delta t$ will become zero.
Therefore, we can write
$\begin{align}
  & \dfrac{{{l}_{c}}}{{{l}_{s}}}=\left( \dfrac{{{\alpha }_{s}}}{{{\alpha }_{c}}} \right) \\
 & \dfrac{{{l}_{c}}}{{{l}_{s}}}=\dfrac{1.6\times {{10}^{-5}}{{K}^{-1}}}{1.2\times {{10}^{-5}}{{K}^{-1}}} \\
 & \dfrac{{{l}_{c}}}{{{l}_{s}}}=\dfrac{3}{4} \\
\end{align}$
Therefore,
$3{{l}_{s}}=4{{l}_{c}}$
We know that,
$\begin{align}
  & {{l}_{s}}-{{l}_{c}}=10cm \\
 & {{l}_{s}}=10+{{l}_{c}} \\
\end{align}$
Therefore,
$\begin{align}
  & 3\times \left( 10+{{l}_{c}} \right)=4{{l}_{c}} \\
 & 4{{l}_{c}}-3{{l}_{c}}=30 \\
 & {{l}_{c}}=30cm \\
\end{align}$
Therefore, we get the length of the copper rod as 30 cm.
Hence the correct answer is option B.

Note:
Don’t get confused with the use of change of length equation. Even though it is said that the length doesn’t change with temperature, this equation is valid.
The ratio of expansion of a material in accordance with the change in temperature is called the coefficient of thermal expansion.