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a) Suppose a bimetallic strip is constructed of copper and steel strips of thickness 1.1 mm and length 29 mm, and the temperature of the strip is reduced by 5.1 K. Determine the radius of curvature of the cooled strip (the radius of curvature of the interface between the two strips). (The linear expansion coefficients for copper ans steel are 1.70 10-5 °C−1 and 1.30 10-5 °C−1, respectively.)
b) If the strip is 29 mm long, how far is the maximum deviation of the strip from the straight orientation? (The deviation is measured from the straight orientation from the interface of the two strips.)
If the linear expansion coefficients for copper and steel are different (α(Cu) >α(st)), then at heating of the bimetallic strip the copper strip elongates greater than the steel strip , and the bimetallic strip will bend. If the length at the first temperature is L1 and at the second temperature is L2, then
L2(Cu) = L1(Cu) (1+ α(Cu) • Δt),
L2(st) = L1(st) (1+ α(st) • Δt).
If h is the thickness of each strip, R is the radius of curvature (pointed to the surface line between two strips), and φ is the angle between the ends of the bent bimetallic strip, then
L2(Cu) = φ (R+h),
L2(st) = φ (R-h).
Solving the system of these four equations we obtain
R = h [ (1+( α(Cu) +α(st)) • Δt)/( α(Cu) +α(st)) • Δt)].
b. The deviation (from geometry) is
d = R•sin(φ/2)•tan(φ/2).