(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) To determine the radius of curvature of the cooled strip, we can use the formula:
1/R = (α1 - α2) * ΔT
where R is the radius of curvature, α1 and α2 are the linear expansion coefficients of copper and steel respectively, and ΔT is the temperature change.
Given:
α1 (copper) = 1.70 x 10^-5 °C^-1
α2 (steel) = 1.30 x 10^-5 °C^-1
ΔT = -5.1 K (negative because the temperature is reduced)
Plugging in the values, we have:
1/R = (1.70 x 10^-5 °C^-1 - 1.30 x 10^-5 °C^-1) * (-5.1 K)
= (0.40 x 10^-5 °C^-1) * (-5.1 K)
Simplifying further:
1/R = -2.04 x 10^-5 °C^-1 K
R = -1 / (2.04 x 10^-5 °C^-1 K)
Therefore, the radius of curvature of the cooled strip is approximately -4.9 x 10^4 mm or 4.9 cm.
(b) To find the maximum deviation of the strip from the straight orientation, we can use the formula:
d = R * (1 - cos(θ))
where d is the maximum deviation, R is the radius of curvature, and θ is the angle formed by the deviation.
In this case, θ will be the maximum possible angle, which can be calculated using the formula:
θ = (L / R)
Given:
L = 29 mm (length of the strip)
Plugging in the values, we have:
θ = (29 mm) / (4.9 cm)
Converting mm to cm:
θ = (2.9 cm) / (4.9 cm)
θ ≈ 0.59 radians
Now we can calculate the maximum deviation:
d = (4.9 cm) * (1 - cos(0.59))
d ≈ 0.75 cm
Therefore, the maximum deviation of the strip from the straight orientation is approximately 0.75 cm.
To determine the radius of curvature of the cooled strip, we need to use the formula for the strain (change in length) in a bimetallic strip caused by a temperature change.
The strain, ε, of a bimetallic strip is given by the equation:
ε = α1 * ΔT * L1 - α2 * ΔT * L2
Where:
α1 and α2 are the linear expansion coefficients of copper and steel respectively,
ΔT is the change in temperature, and
L1 and L2 are the lengths of the copper and steel strips respectively.
First, let's calculate the strain using the given values:
α1 = 1.70 x 10^(-5) °C^(-1)
α2 = 1.30 x 10^(-5) °C^(-1)
ΔT = -5.1 K (negative because the temperature is reduced)
L1 = L2 = 29 mm = 29 x 10^(-3) m
Now, substituting the values into the equation:
ε = (1.70 x 10^(-5) °C^(-1)) * (-5.1 K) * (29 x 10^(-3) m)
- (1.30 x 10^(-5) °C^(-1)) * (-5.1 K) * (29 x 10^(-3) m)
ε = -0.000062063
Next, we can use the following relation for radius of curvature, R, and strain, ε:
R = (t1 * t2) / (2 * ε * (t1 + t2))
Where:
t1 and t2 are the thicknesses of the copper and steel strips, respectively.
Given:
t1 = t2 = 1.1 mm = 1.1 x 10^(-3) m
Now, substituting the values into the equation:
R = (1.1 x 10^(-3) m * 1.1 x 10^(-3) m) / (2 * -0.000062063 * (1.1 x 10^(-3) m + 1.1 x 10^(-3) m))
R ≈ 8.14 m
So, the radius of curvature of the cooled strip (the interface between the two strips) is approximately 8.14 meters.
Now, let's move on to part (b) of the question.
To determine the maximum deviation of the strip from the straight orientation, we can use the formula for deflection of a bimetallic strip:
deflection = ε * (L1 + L2) * (t1 + t2)
Using the previously calculated values for ε, L1, L2, t1, and t2:
deflection = -0.000062063 * (29 x 10^(-3) m + 29 x 10^(-3) m) * (1.1 x 10^(-3) m + 1.1 x 10^(-3) m)
deflection ≈ -2.44 x 10^(-6) m
The negative sign indicates that the strip is deflected in the opposite direction.
So, the maximum deviation of the strip from the straight orientation is approximately 2.44 x 10^(-6) meters.