Consider three rods of length `L_(1), L_(2) and L_(3)` espectively joined in series. Each has same cross - sectional area with Young's moduli Y, 2Y and 3Y respectively and thermal coefficients of linear expansion `alpha, 2alpha and 3alpha` respectively. They are placed between two rigid fixed walls. The temperature of the whole system is increased and it is found that length of the middle rod does not change with temperature rise. Find the value of `(9L_(1))/(L_(3))`.

Answer

`5`

Explanation

Since the middle rod doesn.t expand at all the strain in the rod will be
strain `=2alpha DeltaT`
stress `=2Y(2alpha DeltaT)`
The stress is same in each rod
`4Y alpha Delta T=Y(L_(1)alpha DeltaT+x)/(L_(1))=3Y(L_(3)(3alpha)DeltaT+x)/(L_(3))`
On solving for x we get
`x=3L_(1)` prop `DeltaT=(5L_(3)alpha DeltaT)/3`
`implies(9L_(1))/(L_(3))=5`