Question

A metal rod A of 25 cm length expands by 0.05 cm when heated through 100°C and metal rod B of length 40cm expands by 0.04cm when heated through same temperature. A third rod of length 50 cm, which is made of pieces of A and B placed end to end, expands by 0.06 cm for the same rise in temperature i.e. 100°C. Find the length of the each portion of the composite rod.

Solution

Let α₁ and α₂ be the temperature coefficients of linear expansion of two rods A and B respectively.
Therefore,

increment straight L subscript 1 space equals space straight L subscript 1 straight alpha subscript 1 straight theta

rightwards double arrow

straight alpha subscript 1 space equals space fraction numerator increment straight L subscript 1 over denominator straight L subscript 1 straight theta end fraction

and

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rightwards double arrow

straight alpha subscript 2 space equals space fraction numerator increment straight L subscript 2 over denominator straight L subscript 2 straight theta end fraction

Let the composite rod be made of

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length of A and

straight L apostrophe subscript 2

length of B.
∴

straight L equals straight L apostrophe subscript 1 plus straight L apostrophe subscript 2 equals 50 cm

...(1)
When this rod is heated through θ, then increase in the length is,

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space equals open parentheses straight L apostrophe subscript 1 fraction numerator increment straight L subscript 1 over denominator straight L subscript 1 straight theta end fraction plus straight L apostrophe subscript 2 fraction numerator increment straight L subscript 2 over denominator straight L subscript 2 straight theta end fraction close parentheses straight theta

equals straight L apostrophe subscript 1 fraction numerator increment straight L subscript 1 over denominator straight L subscript 1 end fraction plus straight L apostrophe subscript 2 fraction numerator increment straight L subscript 2 over denominator straight L subscript 2 end fraction

Here, we have

increment straight L subscript 1 space equals space 0.05 cm comma space space space increment straight L subscript 2 equals 0.04 cm comma space space increment straight L equals 0.06 cm

straight L apostrophe subscript 1 equals 25 space space cm comma space space space straight L apostrophe subscript 2 space equals space 40 space cm

∴

0.06 space equals space straight L apostrophe subscript 1 fraction numerator 0.05 over denominator 25 end fraction plus straight L apostrophe subscript 2 fraction numerator 0.04 over denominator 40 end fraction

rightwards double arrow

space space space space 2 straight L apostrophe subscript 1 plus straight L apostrophe subscript 2 equals 60

...(2)
From equation (1) and (2), we have

straight L apostrophe subscript 1 equals 10 cm comma space space space space space space space space space space space space space space space space space space straight L apostrophe subscript 2 equals 40 cm

L₁^' and L₂^' represent the length of each portion of the composite rod.

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