Question

What should be the lengths of steel and copper rod so that the length of steel rod is $5cm$ longer then the copper rod at all the temperatures? Coefficients of linear expansion for copper and steel are 1.7 and 1.1 :

• A. $2.17,14.17cm$
• B. $9.17,14.17cm$
• C. $9.17,18.17cm$
• D. $3.17,5.17cm$

Answer 1

Answer: (B)

It is given that the difference in length of the two rods is always $5cm$. Thus the expansion in both the rods must be same for all temperatures. Thus we can say that at all temperature differences, we have
$\Delta L_{Cu}=\Delta L_{\text{steel }}$
or ${\alpha }_{Cu}{\ell }_1\Delta t={\alpha }_{st}{\ell }_2\Delta t$
$[$ If ${\ell }_1$ and ${\ell }_2$ are the initial lengths of $Cu$ and steel rods]
or ${\alpha }_{Cu}{\ell }_1={\alpha }_{st}{\ell }_2$
or $1.7{\ell }_1=1.1{\ell }_2\dots .(1)$
It is given that ${\ell }_2-{\ell }_1=5cm\dots$ (2)
$\left(\frac{1.7}{1.1}-1\right){\ell }_1=5cm$
or ${\ell }_1=\frac{5\times 1.1}{0.6}=9.17cm$
Now from equation (2) ${\ell }_2=14.17cm$