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What should be the lengths of a brass an...

What should be the lengths of a brass and an aluminium rod at `0^(@)`. if the difference between their lengths is to be maintained equal to 5 cm at any temperature ?
(For brass `alpha=18xx10^(-6)""^(@)C^(-1)` and for aluminium `alpha=24xx10^(-6)" "^(@)C^(-1)` )

Text Solution

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Suppose `l_(1)` and `l_(2)` are the lengths of aluminium and brass rods at `0^(@)C` respectively. At any temperature, differnce of their lengths remains the same. Hence, increase in their lengths with increase in temperature must also be the same.
`:.Deltal_(1)=Deltal_(2)`
`:.alpha_(1)l_(1)DeltaT=alpha_(2)l_(2)DeltaT`
`:.(l_(1))/(l_(2))=(alpha_(2))/(alpha_(2))=(24xx10^(-6))/(18xx10^(-6))=(4)/(2)` . . . (1)
Now according to the given condition,
`l_(1)-l_(2)=5` cm . .. . . (2)
From equations (1) and (2).
`(l_(1))/(l_(1)-5)=(4)/(3)`
`:.3l_(1)=4l_(1)-20`
`:.l_(1)=20` cm and `l_(2)=15` cm
Thus, at `0^(@)C` lengths of brass and aluminium rods should be 20 cm and 15 cm respectively.
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