Given: the linear expansion coefficient of glass is 9 x 10^-6 ('C)^-1.
An automobile windshield has dimensions of 59 cm by 390 cm. What minimum spacing around the windshield is needed to prevent the windshield from breaking if the temp. changes by 146' F?
answer should be in units of mm.
To find the minimum spacing around the windshield needed to prevent it from breaking due to temperature changes, we need to calculate the change in size of the windshield based on its dimensions and the temperature change.
First, let's convert the temperature change from Fahrenheit (°F) to Celsius (°C). The formula to convert from Fahrenheit to Celsius is:
°C = (°F - 32) * (5/9)
Given that the temperature change is 146 °F, we can calculate the equivalent temperature change in Celsius as follows:
ΔT(°C) = (146 - 32) * (5/9) = 78.888 °C
Next, we can use the linear expansion coefficient of glass to calculate the change in size of the windshield. The formula for linear expansion is:
ΔL = L0 * α * ΔT
Where:
ΔL is the change in length,
L0 is the initial length,
α is the linear expansion coefficient, and
ΔT is the change in temperature.
Since the windshield's dimensions are given in centimeters, let's convert them to meters to match the units of the linear expansion coefficient:
Length (L0) = 59 cm = 0.59 m
Width (W0) = 390 cm = 3.90 m
Now, we can calculate the change in length (ΔL) and width (ΔW) as follows:
ΔL = L0 * α * ΔT
ΔW = W0 * α * ΔT
ΔL = 0.59 m * (9 x 10^-6 °C^-1) * 78.888 °C = 0.051 mm (approximately)
ΔW = 3.90 m * (9 x 10^-6 °C^-1) * 78.888 °C = 0.334 mm (approximately)
To prevent the windshield from breaking, we need to provide a minimum spacing around the windshield that accommodates these changes. Since we have the change in length and width, we can add them to the initial dimensions to get the final dimensions.
Final Length = L0 + ΔL = 0.59 m + 0.051 mm [converted to meters]
Final Width = W0 + ΔW = 3.90 m + 0.334 mm [converted to meters]
Finally, we convert the final dimensions back to millimeters to get the minimum spacing required around the windshield:
Minimum Spacing = (Final Length - Initial Length) + (Final Width - Initial Width) in mm
Please substitute the calculated values into the above equation to find the minimum spacing required to prevent the windshield from breaking.