A steel beam is used in the construction of a skyscraper. By what fraction L/L0 does the length of the beam increase when the temperature changes from that on a cold winter day (-20°F) to that on a summer day (+108°F)?

I will be happy to critique your thinking or work. This s a standard formula.

To find the fraction L/L0 by which the length of the steel beam increases, we can use the Coefficient of Linear Expansion (α) of the material. The formula for calculating the change in length (ΔL) due to temperature change is as follows:

ΔL = α * L0 * ΔT

Where:
ΔL = Change in length
α = Coefficient of Linear Expansion
L0 = Initial length of the beam
ΔT = Change in temperature

To calculate the change in length, we need to know the coefficient of linear expansion for the steel used in the beam. Since this information is not provided, we will assume an average value. The coefficient of linear expansion for most steels is approximately 11.7 x 10^-6 per degree Celsius or 6.5 x 10^-6 per degree Fahrenheit.

Using the given temperatures (-20°F to +108°F), we need to convert them to degrees Celsius to use the coefficient of linear expansion.

Converting:
-20°F to C: (C = (F - 32) * 5/9)
= (-20 - 32) * 5/9
= -28.89°C

+108°F to C:
= (+108 - 32) * 5/9
= +42.22°C

Now we can calculate the change in length:

ΔT = +42.22 - (-28.89)
= +71.11°C

Assuming the coefficient of linear expansion (α) as 11.7 x 10^-6 per degree Celsius, we can calculate the change in length:

ΔL = (11.7 x 10^-6) * L0 * (+71.11)
= (11.7 x 10^-6) * L0 * 71.11
= 8.33 x 10^-4 * L0

Therefore, the length of the steel beam increases by approximately 8.33 x 10^-4 times its initial length (L/L0).

To determine the fraction by which the length of the steel beam increases with temperature change, we need to use the coefficient of linear expansion. The coefficient of linear expansion represents the change in length per unit length per degree of temperature change.

The formula to calculate the change in length of the steel beam is given by:

ΔL = α * L0 * ΔT

Where:
ΔL is the change in length
α is the coefficient of linear expansion
L0 is the original length of the beam
ΔT is the temperature change

First, we need to find the value of the coefficient of linear expansion for steel. The coefficient of linear expansion for steel varies depending on the specific type of steel, but a commonly used value is approximately 12 x 10^-6 per degree Celsius.

Since the given temperature change is in Fahrenheit, we need to convert it to Celsius. We can use the conversion formula:

°C = (°F - 32) * 5/9

Given temperatures:
T1 (winter) = -20°F
T2 (summer) = 108°F

Converting the given temperatures to Celsius:
T1 = (-20 - 32) * 5/9 = -28.9°C
T2 = (108 - 32) * 5/9 = 42.2°C

Now we can calculate the change in length:

ΔL = α * L0 * ΔT
ΔL = (12 x 10^-6 /°C) * L0 * (42.2 - (-28.9))°C

Simplifying the equation:
ΔL = (12 x 10^-6 /°C) * L0 * 71.1°C

Finally, we can calculate the fraction by dividing the change in length by the original length:

Fraction L/L0 = ΔL / L0
Fraction L/L0 = (12 x 10^-6 /°C) * L0 * 71.1°C / L0

Simplifying further:
Fraction L/L0 = 12 x 10^-6 * 71.1°C

Therefore, the fraction by which the length of the steel beam increases when the temperature changes from -20°F to 108°F is 12 x 10^-6 multiplied by 71.1°C.