Mechanical Properties of Fluids

Two rods of length L 1 and L 2 and coefficient of expansions α 1 and α 2 are heated. What should be the relation between L 1 and L 2 so that difference in the length of two rods is independent of temperature? - Wired Faculty

Question

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Two rods of length L₁ and L₂ and coefficient of expansions α₁ and α₂ are heated. What should be the relation between L₁ and L₂ so that difference in the length of two rods is independent of temperature?

Solution

On increasing the temperature, the length of both the rods will increase.
If the change in length of both the rods is same irrespective of the rise in temperature, then the difference in the lengths of two rods can be independent of temperature.
Increase in length of the first rod is,

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

Increase in length of the second rod is,

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The required condition is,

space space straight L subscript 1 straight alpha subscript 1 straight theta space equals space straight L subscript 2 straight alpha subscript 2 straight theta

rightwards double arrow

straight L subscript 1 over straight L subscript 2 equals straight alpha subscript 2 over straight alpha subscript 1

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