Problem concerning about thermal physics

AI Thread Summary
A metal rod fixed at both ends bends into a circular arc when heated, due to thermal expansion. The problem requires finding the radius of curvature using the Taylor expansion of the sine function, while also considering the effects of buckling under compressional load. The discussion highlights that the rod's ends are effectively hinged, allowing for bending rather than simple compression. It emphasizes the need to calculate the new length of the rod after heating, assuming it could expand freely. Understanding the constraints and properties of the rod is crucial for solving the problem accurately.
Richardbryant
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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
 
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Are you familiar with the phenomenon of column buckling under compressional load?
 
Chestermiller said:
Are you familiar with the phenomenon of column buckling under compressional load?

I am sorry i don't familiar with this phenomenon
 
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 ?
 
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