Suppose a bimetallic strip is constructed of copper and steel strips of thickness 1.23 mm and length 22.3 mm, and the temperature of the strip is reduced by 6.30 K. Determine the radius of curvature of the cooled strip (the radius of curvature of the interface between the two strips).If the strip is 22.3 mm long, how far is the maximum deviation of the strip from the straight orientation?

To determine the radius of curvature of the cooled strip (the interface between the two strips), we can use the formula for radius of curvature:

R = (t * d^2) / (6 * L)

Where:
R = Radius of curvature
t = Thickness of the strip (in this case, the thickness of the copper strip)
d = Change in temperature
L = Length of the strip (in this case, the length of the copper strip)

Given:
Thickness of the copper strip (t) = 1.23 mm
Change in temperature (d) = -6.30 K (negative because the temperature is reduced)
Length of the copper strip (L) = 22.3 mm

Plugging these values into the formula:

R = (1.23 * (-6.3)^2) / (6 * 22.3)
R = (1.23 * 39.69) / 133.8
R = 48.8607 / 133.8
R ≈ 0.364 mm

So the radius of curvature of the cooled strip is approximately 0.364 mm.

To determine the maximum deviation of the strip from the straight orientation, we can use the formula for sagitta (maximum deviation):

S = (R - sqrt(R^2 - (L/2)^2))

Where:
S = Maximum deviation (sagitta)
R = Radius of curvature
L = Length of the strip (in this case, the length of the copper strip)

Plugging in the values:

S = (0.364 - sqrt(0.364^2 - (22.3/2)^2))
S = (0.364 - sqrt(0.364^2 - 5.58^2))
S = (0.364 - sqrt(0.132496 - 31.2164))
S = (0.364 - sqrt(-31.083904))
(Note: Since the value inside the square root is negative, it means the strip does not deviate from the straight orientation.)

Therefore, the maximum deviation of the strip from the straight orientation is zero (0).

To find the radius of curvature of the cooled strip, we can use the formula for the radius of curvature of a bimetallic strip:

R = (α_Cu - α_Steel) * t^2 / (6 * ΔT)

where R is the radius of curvature, α_Cu is the coefficient of linear expansion for copper, α_Steel is the coefficient of linear expansion for steel, t is the thickness of the strip, and ΔT is the change in temperature.

Given that the thickness of the strip is 1.23 mm and the change in temperature is 6.30 K, we need to find the coefficients of linear expansion for copper and steel.

The coefficient of linear expansion for copper is approximately 16.6 x 10^(-6) K^(-1), and for steel, it is approximately 12 x 10^(-6) K^(-1).

Now we can substitute the values and solve for the radius of curvature:

R = (16.6 x 10^(-6) - 12 x 10^(-6)) * (1.23 x 10^(-3))^2 / (6 * 6.30)

Calculating this expression gives us the radius of curvature of the cooled strip.

To determine the maximum deviation of the strip from the straight orientation, we can use a trigonometric relationship. The maximum deviation occurs at the center of the strip and can be calculated using the formula:

d = R * sin(θ)

where d is the maximum deviation, R is the radius of curvature, and θ is the angle of deviation.

To find θ, we can use the formula:

θ = L / (2 * R)

where L is the length of the strip.

Given that the length of the strip is 22.3 mm, we can substitute the values into the equation and solve for θ. Once we have θ, we can calculate the maximum deviation using the formula mentioned earlier.

Thus, by performing these calculations, we can determine the radius of curvature of the cooled strip and the maximum deviation of the strip from the straight orientation.