Question

Two rods of different materials with coefficients of thermal conductivity 'alpha 1' and 'alpha 2' ; Young's modulii Y_{​​​​​​1} and Y_{​​​​​​2} respectively; of initial lengths l_{​​​​​​1} and l_{​​​​​​2} respectively of equal cross-sectional areas are joined end at one end.The free ends are fixed to two vertical walls. Tenperature of the surrounding is then increased by T°C. Find out the displacement of the joint of the rods.

Answer 1

see the total length of the rod ie. fwom one wall to the other remains constant

now each rod will have two forces acting on it

Tensile force and the other due to the heating effect

each rod shall expand by:-

(a₁l₁T) - ((Fl₁)/AY₁))

where everything has its usual meaning

F is the thermal stress developed and A is cross section area

=> Total extension is:-

(a₁l₁T) - ((Fl₁)/AY₁)) + (a2l2T) - ((Fl2)/AY2)

now the displace of he joint is given by :-

delta l = l1 + ((a₁l₁T) - ((Fl₁)/AY₁)) [ put the value of F here]!!

here you are

(((l₁l₂(Y₂a₂ - Y₁a₁)T))) /((Y₂l₁ + Y₁l₂))