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.
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To determine the minimum spacing required around the automobile windshield to prevent it from breaking due to temperature changes, we need to calculate the change in dimension caused by the temperature change and multiply it by the linear expansion coefficient of glass.
First, let's convert the temperature change from Fahrenheit (°F) to Celsius (°C):
ΔT = 146°F = (146°F - 32°F) × 5/9 = 80°C
Next, let's calculate the change in dimension for each side of the windshield by using the formula:
ΔL = αL₀ΔT
where ΔL is the change in length, α is the linear expansion coefficient, L₀ is the initial length, and ΔT is the change in temperature.
For the length (390 cm):
ΔL_length = (9 × 10^-6 °C^-1) × (390 cm) × (80°C) = 0.2808 cm
For the width (59 cm):
ΔL_width = (9 × 10^-6 °C^-1) × (59 cm) × (80°C) = 0.04248 cm
Now, we need to convert the change in dimension from centimeters to millimeters (since the answer should be in mm):
ΔL_length_mm = 0.2808 cm × 10 mm/cm = 2.808 mm
ΔL_width_mm = 0.04248 cm × 10 mm/cm = 0.4248 mm
To determine the minimum spacing needed around the windshield, we take the maximum change in dimension from both length and width and double it (since there are two sides):
Minimum spacing = 2 × max(ΔL_length_mm, ΔL_width_mm)
= 2 × max(2.808 mm, 0.4248 mm)
= 2 × 2.808 mm
= 5.616 mm
Therefore, the minimum spacing required around the automobile windshield to prevent it from breaking, if the temperature changes by 146°F, is 5.616 mm.