Explanation:
In each problem, use the linear or volumetric thermal expansion equation.
Linear thermal expansion:
ΔL = α L₀ ΔT
where ΔL is the change in length,
α is the linear thermal expansion coefficient,
L₀ is the original length,
and ΔT is the change in temperature.
Volumetric thermal expansion:
ΔV = β V₀ ΔT
where ΔL is the change in volume,
β is the volumetric thermal expansion coefficient (β = 3α),
V₀ is the original volume,
and ΔT is the change in temperature.
"What is the length of the same railroad track on a cold winter day when the temperature is 0°F?
"
I assume the railroad track is made of steel, and is originally 30 m at a temperature of 0°C. Let's first find the change in temperature by converting Fahrenheit to Celsius.
T = 5/9 (0 − 32)
T = -17.8°C
Now we can find the change in length:
ΔL = (11×10⁻⁶ /°C) (30 m) (-17.8
°C − 0°C)
ΔL = -0.006 m
The track shrinks by 0.006 m, so the final length is 24.994 m.
"Estimate the fractional change in the volume of Earth's oceans due to an average temperature change of 1°C."
ΔV = β V₀ ΔT
ΔV / V₀ = β ΔT
ΔV / V₀ = (2.0×10⁻⁴ /°C) (1 °C)
ΔV / V₀ = 2.0×10⁻⁴
"Use the fact that the average depth of the ocean is 4.00×10³ m to estimate the change in depth."
ΔL = α L₀ ΔT
ΔL = (β/3) L₀ ΔT
ΔL = (2.0×10⁻⁴ /°C /3) (4.00×10³ m) (
1 °C)
ΔL = 0.27 m
"A cylindrical brass sleeve is to be shrink fitted over a brass shaft whose diameter is 3.212 cm at 0°C. The diameter of the sleeve is 3.196 cm at 0°C. To what temperature must the sleeve be heated before it will slip over the shaft?"
ΔL = α L₀ ΔT
(3.212 cm − 3.196 cm) = (1.9×10⁻⁵ /°C) (3.196 cm) (T − 0°C)
T = 263 °C
"Alternatively, to what temperature must the shaft be cooled before it will slip into the sleeve?"
ΔL = α L₀ ΔT
(3.196 cm − 3.212 cm) = (1.9×10⁻⁵ /°C) (3.212 cm) (T − 0°C)
T = -262 °C
