A rod of metal - 1 of length 50.0 cm elongates by 0.10 cm when it is heated from `0^(@)C` to `100^(@)C`. Another rod of metal - 2 of length 80.0. cm elongates by 0.08 cm for the same rise in temperature. A third rod of length 50.0 cm, made by welding pieces of rod 1 rad and 2 rad placed end to end, elongates by 0.03 cm when it is heated from `0^(@)C` to `50^(@)C`. Then what is the length of metal - 1 in the third rod at `0^(@)C`?

Explanation

`"For meta - 1, For metal - 2,"`
`alpha_(1)=(Deltal)/(l Deltat)=(0.10)/(50.0xx100)" "alpha_(2)=(Deltal)/(lDeltat)=(0.08)/(80.0xx100)`
`alpha_(1)=2.0xx10^(-5)//^(@)C" "alpha_(2)=1.0xx10^(-5)//^(@)C`
Let the lengths of metal - 1 and metal - 2 in the third rod at `0^(@)C` be `l_(1) and l_(2)`, respectively.
Then `" "l_(1)+l_(2)=50.0`
When this rod is heated to `50^(@)C`, then
`(l._(1)=l_(1)(l+alpha_(1)50), l._(2)=l_(2)(l+alpha_(2)50)`
`and" "l._(1)+l._(2)=l_(1)+l_(2)+(l_(1)alpha_(1)+l_(2)alpha_(2))50`
`50.03=50.0+(2l_(1)+l_(2))xx50xx10^(-5)`
`2l_(1)+l_(2)=60" ...(2)"`
`"Solving (1) and (2), "l_(1)=10.0cm, l_(2)=40.0cm`