is small compared to l . So α ( . ) which gives α v = α l ( . ) What happens by preventing the thermal expansion of a rod by fixing its ends rigidly?
Clearly, the rod acquires a compressive strain due to the external forces provided by the rigid support at the ends. The corresponding stress set up in the rod is called thermal stress . For example, consider a steel rail of length m and area of cross-section cm that is prevented from expanding while the temperature rises by ° C. The coefficient of linear expansion of steel is α l(steel) = .
Youngs modulus of steel is Y (steel) = × N m – . Therefore, the thermal stress developed is F A Y steel = . × N m – , which corresponds to an external force of ∆ F = AY steel ∆ l = . × × × – j N.
If two such steel rails, fixed at their outer ends, are in contact at their inner ends, a force of this magnitude can easily bend the rails. Example . Show that the coefficient of area expansion, ( ∆ A/A )/ ∆ T , of a rectangular sheet of the solid is twice its linear expansivity, α l . Answer Fig.
. Consider a rectangular sheet of the solid material of length a and breadth b (Fig. . ).
When the temperature increases by ∆ T , a increases by ∆ a = α l a ∆ T and b increases by ∆ b = α l b ∆ T . From Fig. . , the increase in area ∆ A = ∆ A + ∆ A + ∆ A ∆ A = a ∆