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Straight Nickel

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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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HO Scale Straight Train Track 25 pieces Code 100 Nickel Silver BRAND NEW
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Atlas 822 HO Scale Code 100 6 Straight Snap Track 4 pcs Nickel Silver
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lot ATLAS Nickel TRUE TRACK Straight Curve NEW 463 460 464
lot ATLAS Nickel TRUE TRACK Straight Curve NEW 463 460 464
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lot BACHMANN EZ Track Grey nickel silver curve straight
lot BACHMANN EZ Track Grey nickel silver curve straight
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HO Atlas Lot of 10 Pieces Nickel Silver 9 Straight Track Code 100
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HO Atlas Code 100 Nickel Silver Snap Track 9 Straight Section 821 6 pcs
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2 HO SCALE BACHMANN 44588 NICKEL SILVER 18 STRAIGHT EZ TRACK GRAY ROADBED
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10 Pieces of HO Scale Steel Nickel Rail Train Track 9 Curve  1 Straight Piece
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HO Atlas 823 Code 100 Snap Track Nickel Silver 3 Straight Section 4 pcs
HO Atlas 823 Code 100 Snap Track Nickel Silver 3 Straight Section 4 pcs
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Atlas  521HO Code 836 Straight 4 PCSNickel Silver New Mint On Card
Atlas 521HO Code 836 Straight 4 PCSNickel Silver New Mint On Card
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NEW Bachmann Nickel Silver E Z Track 9 Straight Terminal Rerailer HO 44410
NEW Bachmann Nickel Silver E Z Track 9 Straight Terminal Rerailer HO 44410
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Atlas HO Scale Code 100 Nickel Silver 9 Straight 6pcs 821 NIP
Atlas HO Scale Code 100 Nickel Silver 9 Straight 6pcs 821 NIP
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Atlas HO Scale Code 100 Nickel Silver 6 Straight Section 4pcs 822 NIP
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HO Scale 10 Atlas Code 100 Nickel Silver 25 Straight Sections NR 487
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HO Scale Nickel Silver Bachmann E Z Track 30 9 Track Sections STRAIGHT + CURVED
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ATLAS HO SCALE TRACKS 9STRAIGHT18RADIUS NICKEL SILVER 6PC EA LAY OUT B00K NOS
ATLAS HO SCALE TRACKS 9STRAIGHT18RADIUS NICKEL SILVER 6PC EA LAY OUT B00K NOS
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HO scale track Bachmann EZ Nickel Silver Curves Straights grey Roadbed Used
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Atlas Nickel Silver NOS HO Scale 6 sections 9 Straight Track 821 Group of 4
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Atlas CD100 Straight Track Asst Nickel Silver 12 HO 847
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Atlas Code 83 3 Straight Nickel Silver HO 522
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1 1 2 STRAIGHT 4 PACK OF NICKEL SILVER ATLAS TRUE TRACK CODE 83 ON ROADBED
1 1 2 STRAIGHT 4 PACK OF NICKEL SILVER ATLAS TRUE TRACK CODE 83 ON ROADBED
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Atlas 847 HO Scale Code 100 Straight Track Assortment 12 pieces Nickel Silver
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