Question

What should be lengths of steel and copper rod, so that the length of steel rod is $5cm$ longer than the copper rod at all temperature? $(\alpha$ for copper $=1.7\times {10}^{-5}{}^{\circ }{C}^{-1}$ and $\alpha$ for steel $=1.1\times {10}^{-5}{}^{\circ }{C}^{-1})$

Answer 1

Let the length of copper be $L_c$ and for steel be $Ls$.

For copper ${\alpha }_c=1.7\times {10}^{-5}{}^o{C}^{-1}$,for steel ${\alpha }_s=1.1\times {10}^{-5}{}^o{C}^{-1}$

Now the change in copper length is $\Delta L_c=L_c{\alpha }_c\Delta T$

Now the change in copper length is $\Delta L_s=L_s{\alpha }_s\Delta T$

Since given that difference in length should be always 5 cm, It means $\Delta L_s=\Delta L_c$

So $L_s{\alpha }_s\Delta T=L_c{\alpha }_c\Delta T$

$L_s\times 1.1=L_c\times 1.7$ ...(1)

Also

$L_s-L_c=5$ ...2

On solving both equation (1) and (2),

we get $L_s=14.17cm$ and $L_c=9.17cm$