Two rods of different materials with coeff. of thermal expansion a1 and a2 resp. Youngs moduli Y1 and Y2 resp; of length l1 and l2 resp. and of equal cross sectional area are joined at one end. The free ends are fixed to two vertical waals. Temp. of the surrounding is then increased by T degree C. Find out the displacement of the joint of the rods. Please give me the idea how to work this out...!!

its an excellent question,

even i got messed up with that heating part

see the tolat 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

=> Tolat extension is:-

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

now the summation for both the rods is zeroe (why?? read the 3rd line)

equate the expression to 0 and get the value of F!!

now the displace of the joint is given by :-

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

here you are

by my calculations the answer should be:-

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

aM I RIGHt??