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The system shown is figure consists of 3...

The system shown is figure consists of `3` springs and two rods. If the temperature of the rod is increased by `DeltaT`, then the total energy stored in three springs is `beta xx (99)/(484)kL^(2)alpha^(2)(DeltaT)^(2)`. Determine the value of `beta`. The spring are initially relaxed and there is no friction anywhere. For rod the coefficient of linear expansion is `alpha`

Text Solution

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Let `x_(1)`, `x_(2)` and `x_(3)` be the compressions in the three springs, then
`x_(1)+x_(2)+x_(3)=LalphaDeltaT+(L)/(2)alphaDeltaT=(3)/(2)LalphaDeltaT` ………..`(1)`
Considering the F.B.D.s of the rod requires rods

Equilibrium of the rod requires that
`x_(1)=2x_(2)==3x_(3)`........`(2)`
`:. x_(1)+(x_(1))/(2)+(x_(3))/(3)=(3)/(2)LalphaDeltaT`
`rArr x_(1)=(9)/(11)LalphaDeltaT`
`:.` Energy stored in spring of spring constant `k`.
`E_(1)=(1)/(2)kx_(1)^(2)=(81)/(242)kL^(2)alpha^(2)DeltaT^(2)`
Similarly
`E_(2)=(81)/(484)kL^(2)alpha^(2)DeltaT^(2)` and `E_(3)=(27)/(242)kL^(2)alpha^(2)DeltaT^(2)`
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