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Two rods of different materials are plac...

Two rods of different materials are placed between massive walls as shown in figure. The cross section of the rods is A, their moduil of elastricity are `E_(1)` and `E_(2)` respectively. If rods are heated by t degrees, then (coefficients of liner expansion of material of rods are `alpha_(1)` and `alpha_(2)` respectively)

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KEY IDEAS
When the rods are heated in a free state, their total length will increase by
`Deltal=Deltal_1 +Deltal_2=(alpha_1l_1+alpha_2l_2) DeltaT`
Compression by the same amount `Deltal` will reduce the length of the rods by `Deltal_1^(.) and Deltal_2^(.)` where

`Deltal_1^(.) + Deltal2^(.) = Deltal`
This requires the force

`F=(E_1A)/(l_1)Deltal_1^(.)=(E_2ADeltal_1^(.))/l_2`
Calculation : Upon solving this system of equations , we find that
`F=(a_1l_1+a_2l_2)/(l_1/E_1+l_2/E_2)ADeltaT`
The rods will act upon each other with this force.
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