Thermodynamics - displacement

[helpme]

Two rods of different materials with coefficients of linear expansion α1
and α2 respectively; Young's moduli Y1 & Y2 respectively; of
initial lengths l1 and l2 respectively are joined at one end. the free
ends are fixed to two vertical walls as shown in the fig.

Temperature of the surrounding is then increased by T°C. Find out the displacement of the joint of the rods.







My work

Since my physics is not good so m not pretty sure about what i am applying..

but as i remember my friend guided me once to work on such questions..

I considered the displacement of the joint (say 'x') in any direction (right or left).

Now, x = Δl1 - Δl2 and,

k1Δl1 = k2Δl2

where, k = YA/L

but the two eqns. only are not proving helpful.

If someone could guide me on this question.

Answer is.

x = l1l2(-Y1α1 + Y2α2) / (Y2l1 + Y1l2)

[swansont]

What is relationship of the force each rod exerts on the other? What is the net force on the end of each rod?

[helpme]

What is relationship of the force each rod exerts on the other? What is the net force on the end of each rod?

probably F = kx the first rod will apply a force F = k x Δl₁

net force is F x (Δl₁ - Δl₂)

now if i m wrong please correct me.. and thanks for your reply.

[swansont]

If the system is in equilibrium, though, the net force is zero, i.e. any internal force of expansion is limited by the force from the other rod. You also know the force on the rods must be equal and opposite by Newton's third law.

[helpme]

If the system is in equilibrium, though, the net force is zero, i.e. any internal force of expansion is limited by the force from the other rod. You also know the force on the rods must be equal and opposite by Newton's third law.

ya but that is helping me no way...!!

to derive the result i've posted above...

can u just write the eqns. ????

or post a detailed "idea"..!!