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Two Aluminium rods and a steel rod of eq...

Two Aluminium rods and a steel rod of equal cross-sectional area and equal length `l_(0)` are joined rigidly side by side as shown in figure. Initially the rods are at `0^(@)C`. Find the length of the rod at the temperature `theta` if young's modulus of elasticity of the aluminium and steel are `Y_(a)` and `Y_(s)` respectively and coefficient of linear expansion of aluminum and steel are `alpha_(a)` and `alpha_(s)` respectively.
`|{:("Aluminium"),("Steel"),("Aluminium"):}|`

Text Solution

Verified by Experts

The correct Answer is:
`l_(0) [1+(2Y_(a)alpha_(a)+Y_(s)alpha_(s))/(2Y_(a)+Y_(s))theta]`

If rods are free to expand.
`l_(s) = l_(0) (1+ alpha_(s) theta)`
`l_(a) = l_(0) (1+ alpha_(a) theta)`
`Y = (F//A)/(x//l)`
`x = (F xx l)/(AY) if alpha_(s) gt alpha_(s)`
for steel (compression) `x = (2F xx l_(0))/(AY_(s))`
`= l_(0) (1+ alpha_(s) Delta theta) ...(ii)`
by solving (i) and (ii)
we get
then `l = l_(0) [1+(2Y_(a)alpha_(a)+Y_(s)alpha_(s) theta)/(2Y_(a)+Y_(s))]`
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