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.)
a.
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).
a) To determine the radius of curvature of the cooled strip, we can use the concept of thermal expansion. The difference in length between the copper and steel strips when they are heated or cooled causes the bimetallic strip to bend.
The formula for the radius of curvature, R, of a thin strip is given by:
1 / R = (α2 - α1) * ΔT / (L * α2 * α1)
Where:
R = radius of curvature of the strip
α1 = linear expansion coefficient of copper
α2 = linear expansion coefficient of steel
ΔT = change in temperature
L = length of the strip
Now, let's plug in the given values:
α1 = 1.70 * 10^(-5) °C^(-1) (linear expansion coefficient of copper)
α2 = 1.30 * 10^(-5) °C^(-1) (linear expansion coefficient of steel)
ΔT = -5.1 K (change in temperature, negative because it is reduced)
L = 29 mm = 0.029 m (length of the strip)
1 / R = (1.30 * 10^(-5) - 1.70 * 10^(-5)) * (-5.1) / (0.029 * 1.30 * 10^(-5) * 1.70 * 10^(-5))
Now, calculate the value of 1 / R, and then take its reciprocal to find R.
b) To calculate the maximum deviation of the strip from the straight orientation, we can use the formula for the displacement, d, of the end of the strip:
d = L^2 / (2R)
Where:
d = maximum deviation from the straight orientation
L = length of the strip
R = radius of curvature of the strip (obtained from part a)
Let's plug in the given values:
L = 29 mm = 0.029 m (length of the strip)
R = obtained from part a
Now, substitute the values into the formula and calculate the maximum deviation, d.
Please note that if the radius of curvature, R, obtained in part a is negative, it means that the strip will bend in the opposite direction.