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Find the equillibrium length for the sys...

Find the equillibrium length for the system after increasing temperature by `Delta T`.

Text Solution

Verified by Experts

here `l_(A) and l_(B)` are the natural length of the rod A and B after increase in temperature by `Delta T, and l_(0)` is actual length after temperature increase by `Delta T`.
So strain in A `=(l'_(0)-l'_(A))/(l'_(A))`

and in `B =(l'_(B)-l'_(0))/(l'_(B))`
Now force balance
Now `(F)/(A) = gamma_(A) =(l'_(0)-l'_(A))/(l'_(A))" ...(1)"`
and `(2F)/(A) = gamma_(B) (l'_(B)-l'_(0))/(l'_(B))" ....(2)"`

(1) ÷ (2)
`(1)/(2) =(gamma_(A))/(gamma_(B))([l'_(0)-l'_(0)(1+alpha_(A) Delta T)]l_(0)(1+alpha_(B) Delta T))/(l_(0)(1+alpha_(A) Delta T)[l_(0)(1+alpha_(B) Delta T)-l'_(0)])`
`l'_(0) =(l_(0) (gamma_(B) +2 gamma_(A))[1+(alpha_(B)+alpha_(A))Delta T))/(2l_(A)(1+alpha_(B) Delta T)+gamma_(B)(1+alpha_(A) Delta T))`
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