A steel beam is used in the construction of a skyscraper. By what fraction (change in L)/Lo does the length of the beam increase when the temperature changes from that on a cold winter day (-15 degree F) to that on a summer day (+ 105 degree F)?
I got 0.00144 for the answer, but it's not correct. Please give me teach me how to do this. THANKS A LOT!
To determine the change in length of the steel beam, we can use the linear expansion formula. It states that the change in length (ΔL) is proportional to the original length (Lo), the coefficient of linear expansion (α), and the change in temperature (ΔT). The formula is:
ΔL = α * Lo * ΔT
Now, let's break down the problem step-by-step:
Step 1: Find the coefficient of linear expansion (α) of the steel beam.
The coefficient of linear expansion of steel is approximately 12 × 10^-6 per degree Fahrenheit (°F). This value is an average for most common types of steel. So, α = 12 × 10^-6/°F.
Step 2: Determine the change in temperature (ΔT).
Given that the temperature changes from -15°F (cold winter day) to +105°F (summer day), the change in temperature (ΔT) is +105°F - (-15°F) = 120°F.
Step 3: Calculate the change in length (ΔL).
Using the formula from Step 1, we have:
ΔL = α * Lo * ΔT
Plugging in the values, we get:
ΔL = (12 × 10^-6/°F) * Lo * 120°F
Step 4: Simplify the expression.
ΔL = (12 × 10^-6) * Lo * 120
ΔL = 1.44 * 10^-3 * Lo
Step 5: Determine the change in length fractionally.
To express the change in length fractionally, we divide the change in length (ΔL) by the original length (Lo):
(ΔL)/Lo = (1.44 * 10^-3 * Lo)/Lo
(ΔL)/Lo = 1.44 * 10^-3
Therefore, the fraction of change in length is 1.44 * 10^-3 or 0.00144.
Based on the calculations, it seems your answer of 0.00144 is indeed correct. Double-check your calculations to ensure you haven't made any mistakes.
To find the fraction change in length of the steel beam due to temperature variation, we can make use of the linear expansion coefficient of the material. This coefficient typically represents the change in length per unit length per unit change in temperature.
The formula for the change in length of a material due to temperature change is given by:
ΔL = α * Lo * ΔT
Where:
ΔL is the change in length
α is the linear expansion coefficient of the material
Lo is the initial length of the beam
ΔT is the change in temperature
First, we need to determine the linear expansion coefficient (α) of the steel beam. The linear expansion coefficient can vary depending on the type of steel used. It is typically provided as a value in units of length per length per degree Celsius (or degree Fahrenheit).
Let's assume that the linear expansion coefficient for the steel beam is α = 6.5 x 10^(-6) per degree Fahrenheit.
Next, we need to calculate the change in temperature from the cold winter day to the summer day.
ΔT = Tsummer - Twinter
ΔT = 105°F - (-15°F)
ΔT = 120°F
Now, we can calculate the change in length using the formula mentioned earlier:
ΔL = α * Lo * ΔT
ΔL = (6.5 x 10^(-6)) * Lo * 120°F
ΔL = (6.5 x 10^(-6)) * Lo * 120
Finally, to find the fraction change in length (ΔL) compared to the initial length (Lo), we divide the change in length (ΔL) by the initial length (Lo):
(fraction change) = (ΔL) / (Lo)
(fraction change) = [(6.5 x 10^(-6)) * Lo * 120] / Lo
(fraction change) = 0.00078
Therefore, the fraction change in length of the steel beam, due to the temperature change from a cold winter day (-15°F) to a summer day (105°F), is approximately 0.00078 or 0.078%.
change in T = 120 deg F which is 120 (100/(212-32) ) = 120 * 5/9 = 66.7 deg C
delta L/L = [1.2*10^-5 ](66.7)
80.04 *10^-5
= .00080