Problem concerning about thermal physics

[Richardbryant]

Homework Statement

A metal rod of length length L, linear coefficient of expansion a, is fixed at both ends to the walls. When the temperature is increased byΔT, the rod bends into a circular arc due to thermal expansion.

2 Relevant equations
a)Find the radius of curvature R of the rod by considering the taylor expansion of sine function.
b) Find the value of R if L=4m, a=1.2x10^-6 k^-1 and ΔT=20k

The Attempt at a Solution

Sorry that i don't really have much idea about this question, here is some of my guessing work.

As the temperature increased, the metal rod increased from L to L+ΔL[/B]
sinθ for small angle, the taylor expansion should expand to the 1st order, which is θ=L/2R

 

[Chestermiller]

Are you familiar with the phenomenon of column buckling under compressional load?

 

[Richardbryant]

Are you familiar with the phenomenon of column buckling under compressional load?

I am sorry i don't familiar with this phenomenon

 

[Nidum]

If you have a long slender rod and apply a force acting axially along it's length then one of two things can happen as the force is made larger and larger .

The rod could be simply crushed or the rod could bend sideways into a curve . When it bends sideways into a curve it is said to have 'buckled'

Buckling is a fascinating subject to study . Fortunately though you do not actually need to know much about the details of buckling to be able to solve the given problem .

The problem statement is incomplete so we are left to infer or assume some information that we need :

The metal rod buckles into an arc . That tells us that the ends of the rod are hinged rather than fully fixed to the walls .

No material properties are given for the rod so changes of rod length due to compression can be neglected .

I'll give you two hints now about how to start solving this problem :

Work out what the new length of the rod would be after heating if it was free to expand axially .

Use the information that the ends of the rod are in actual fact constrained to be the same distance apart before and after heating .

OK ?