en meter long steel railroad rails are laid end to end, with no space between, on a hot day when the temperature is 115°F. Six months later the temperature has dropped to -2°F.How much space now exists between each rail? (The linear expansion coefficient for steel is 12. × 10-6 C°-1.)
To calculate the space that exists between each rail after the temperature change, we can use the concept of linear expansion.
First, let's calculate the change in temperature:
Change in temperature = final temperature - initial temperature
Change in temperature = -2°F - 115°F
Change in temperature = -117°F
Now, we can calculate the change in length of the steel railroad rails:
Change in length = (linear expansion coefficient) * (initial length) * (change in temperature)
Change in length = (12. × 10^-6 C°^-1) * (1 meter) * (-117°F)
We need to convert the change in temperature from Fahrenheit to Celsius:
Change in temperature in Celsius = (change in temperature in Fahrenheit - 32) * (5/9)
Change in temperature in Celsius = (-117°F - 32) * (5/9)
Change in temperature in Celsius = -89°C
Now, let's calculate the change in length using the converted temperature:
Change in length = (12. × 10^-6 C°^-1) * (1 meter) * (-89°C)
Change in length = -10.68 × 10^-6 meters
Finally, we can calculate the space that exists between each rail:
Space between each rail = change in length / (number of rails - 1)
Space between each rail = (-10.68 × 10^-6 meters) / (1 rail - 1)
Since the rails are laid end to end with no space between, the number of rails is equal to 1.
Substituting the values:
Space between each rail = (-10.68 × 10^-6 meters) / 0
Any division by zero is undefined, which means that there is no space between the rails after the temperature change.
To find out how much space exists between each rail after the temperature change, we need to calculate the change in length of each rail due to the change in temperature.
First, we need to convert the temperatures from Fahrenheit to Celsius:
115°F = 46.1°C
-2°F = -18.9°C
Next, we can calculate the change in temperature:
ΔT = final temperature - initial temperature
ΔT = (-18.9°C) - (46.1°C)
ΔT = -65°C
Now, we use the linear expansion coefficient for steel to calculate the change in length for each rail:
ΔL = α * L * ΔT
Where:
ΔL = Change in length
α = Linear expansion coefficient
L = Initial length of each rail
ΔT = Change in temperature
Given:
α = 12 × 10^(-6) C°^(-1)
L = 1 meter (100 cm)
ΔL = (12 × 10^(-6) C°^(-1)) * (100 cm) * (-65°C)
ΔL = (-7.8 × 10^(-2)) cm
Since the rails were initially laid end to end without any space between them, we need to find the total space between the rails after the temperature change:
Total space = (Number of rails - 1) * Change in length of each rail
Since we have one rail, there is no space between it and no space between the rails.
Therefore, there is still no space between each rail after the temperature change of -65°C.