The difference in length of an iron rod and a copper rod at `50^(@)C` is 2 cm. This difference remains the same also at `450^(@)C`. What are the lengths of the rods at `0^(@)C`? Given, `alpha` for iron `=12 times 10^(-6@)C^(-1)` and `alpha` for copper `=17 times 10^(-6@)C^(-1)`.

Explanation

Let x and y be the lengths of the iron and the copper rods at `50^(@)C` respectively.
Since the difference in lengths of the two rods remains the same for any rise in temperature, both the rods will have the same expansion.
Increase in length of the iron rod
`" " =x times 12 times 10^(-6)(450-50)=x times 12 times 10^(-6) times 400`
Increase in length of the copper rod
`" "=y times 17 times 10^(-6)(450-50)=y times 17 times 10^(-6) times 400`
According to the question,
`" " x times 12 times 10^(-6) times 400=y times 17 times 10^(-6) times 400`
or, `" " 12x = 17y` or, `x=17/12y`
`because " " x gt y`
`therefore " " x-y=2 " or, " x=2+y`
or, `" " 2+y=17/12y " or, " 5y=24 " or, " y=4.8cm`
`therefore " " x=2+4.8=6.8cm`
Now, suppose the lengths of the iron and the copper rods are `x_(0) " and " y_(0)` respectively at `0^(@)C.`
`therefore " " 6.8=x_(0){1+12 times 10^(-6) times 50} " or, " x_(0)=6.796cm`
`therefore " " 4.8=y_(0)(1+17 times 10^(-6) times 50} " or, " y_(0)=4.796cm`
Hence, lengths of the iron and the copper rods at `0^(@)C` are 6.796 cm and 4.796 cm respectively.