Two steel rods and an aluminium rod of equal length l, and equal cross-section are joined rigidly at their ends as shown in the figure bellow. All the rods are in a state of zero tension at 0°C. Find the length of the system when the temperature is raised to θ. Coefficient of linear expansion of aluminium and steel are α_a and α_s respectively. Young’s modulus of aluminium is Y_α and steel is Y_y.

Question

Two steel rods and an aluminium rod of equal length l, and equal cross-section are joined rigidly at their ends as shown in the figure bellow. All the rods are in a state of zero tension at 0°C. Find the length of the system when the temperature is raised to θ. Coefficient of linear expansion of aluminium and steel are α_a and α_s respectively. Young’s modulus of aluminium is Y_α and steel is Y_y.

Philoid · Accepted Answer

Given:
At T₁ = 0° C
Length of two steel rods and one aluminium rod joined rigidly at T₁= l
Coefficient of linear expansion of aluminium : α_aCoefficient of linear expansion of steel are : α_sYoung’s modulus of aluminium : Y_αYoung’s modulus of steel : Y_sIncreased Temperature : T₂ = θ
Change in temperature : Δ T= θ – 0 = θ
Now, since these rods are rigidly joined to each other when the temperature is increased these rods will undergo thermal expansion. The rate of thermal expansion for steel and aluminium will be different as they has different values of Coefficient of thermal expanion .
While expanding the rods will exert force on each other.
Hence ,due to thermal expansion the length of the rods will change , we will name the final length as l’ at temperature θ .
Thus Change in Length: Δ l= l’-l
Formulae used:
We know that:

Now,we know that :
The Total Young’s Modulus of whole system:
Now,

Now,

Hence by 3^rd Formula and equation (2),(3)

Hence, the length of the system when temperature is increase to θ
is