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Straight Nickel

Physics - Radius of bimetallic strip?

A thin straight aluminium strip and a think straight nickel strip are rigidly linked.
The two strips are parallel to each other and there is a difference of 0.5mm between the 2 strips throughout.

When the temperature was raised by 50K, the straight bimetallic strip became bent and became an arc.
What is the radius of this arc?
Using the following coefficient of linear thermal expansions:
Aluminium : 23 * 10^-6K-1
Nickel : 13 *10^-6K-1

L = initial length of strips
t = separation between strips = 0.5mm
r = radius of arc into which the Ni strip bends
ka = coefficient of thermal expansion of Al
kn = coefficient of thermal expansion of Ni
a = angle subtended by the curved strip (measured in radians)
ca = length of arc of Al strip
cn = length of arc of Ni strip
deltaT = increase in temp

Assuming that r >> t we can argue as follows:

cn = r*a, ca = (r + t)*a

ca - cn = deltaT*L*(ka - kn) = t*a =~ t*L/r

r ~= t/(deltaT*(ka - kn))

I leave the arithmetic to you

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Code 100 Nickel Silver 3 Straight Snap Track HO Scale Atlas Trains
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