Question

The rod of length 2m is at temperature of 20c find the free expansion of the rod if the tempearture is increased to 50 c find stressnproduced when rod is fully prevented to expand

Answer 1

Coefficient of linear expansion α is defined as the fractional change in length per unit change in temperature.

Consider an element dx of the rod of length L at a distance x from one of its end. We are interested to find what change in length (shown by δ in the figure) of this element will take place due to change in temperature ΔT.

By definition of temperature coefficient of linear expansion, we have for this element,

α=δdxΔT

Or, δ=ΔTαdx

The total change in length (ΔL) in the rod can then be found by integrating this with proper limits

ΔL=∫L0ΔTαdx

If α=(ax+b) then

ΔL=∫L0ΔT(ax+b)dx=LΔT(aL2+b)

You can plug in your values to find ΔL. Add this to original length to get the final length.