A steel beam is used in the construction of a skyscraper. By what fraction īL/Lo does the length of the beam increase when the temperature changes from that on a cold winter day (-15.0oF) to that on a summer day (+105.0oF)? Take: Ąsteel= 12x10-6/Co
To find the fraction by which the length of the steel beam increases, we need to use the formula for linear thermal expansion:
ΔL = αLΔT
Where:
ΔL is the change in length of the beam.
α is the coefficient of linear expansion for the material (in this case, steel), which is given as 12x10^(-6)/°C.
L is the original length of the beam.
ΔT is the change in temperature.
Before calculating the fraction, we need to convert the given temperatures from Fahrenheit to Celsius since the coefficient of linear expansion is given in terms of Celsius.
To convert -15.0°F to Celsius:
(°F - 32) × 5/9 = ( -15 - 32) × 5/9 = -26.11°C
To convert +105.0°F to Celsius:
(°F - 32) × 5/9 = (105 - 32) × 5/9 = 40.56°C
Now, let's calculate the change in length:
ΔL = αLΔT
ΔL = (12x10^(-6)/°C) * L * (40.56°C - (-26.11°C))
Now, we can calculate the fraction ƒ´L/Lo:
ƒ´L/Lo = ΔL / L
ƒ´L/Lo = (ΔL) / L
ƒ´L/Lo = [(12x10^(-6)/°C) * L * (40.56°C - (-26.11°C))] / L
Simplifying the equation, the L in the numerator cancels out with the L in the denominator:
ƒ´L/Lo = (12x10^(-6)/°C) * (40.56 - (-26.11))
Now, calculate the fraction:
ƒ´L/Lo = (12x10^(-6)/°C) * (40.56 + 26.11)
ƒ´L/Lo = (12x10^(-6)/°C) * (66.67)
ƒ´L/Lo = 7.9992 x 10^(-6) (approximately)
Therefore, the length of the steel beam increases by approximately 7.9992 x 10^(-6) times its original length when the temperature changes from -15.0°F to +105.0°F.