001) 233.15 K
002) 3. a difference of 9 on the Fahrenheit scale.
003) 0.0153 m
004) 9.9×10⁷ N/m²
005) 0.199 cm
006) 3.07 m
007) 24.8 psi
Explanation:
001
Converting Fahrenheit to Celsius:
T(°F) = 1.8 T(°C) + 32
The temperature at which they are numerically equal is:
T = 1.8 T + 32
-0.8T = 32
T = -40
-40°F is the same temperature as -40°C, which in Kelvins is:
T = -40°C + 273.15
T = 233.15 K
002
Kelvins scale equally with Celsius:
ΔT(K) = ΔT(°C)
Fahrenheit scales 1.8 times with Celsius, and therefore with Kelvins:
ΔT(°F) = 1.8 ΔT(°C) = 1.8 ΔT(K)
If ΔT(K) = 5 K, then:
ΔT(°C) = ΔT(K) = 5°C
ΔT(°F) = 1.8 ΔT(K) = 9°F
003
Linear thermal expansion is:
ΔL = L α ΔT
where ΔL is the change in length,
L is the original length,
α is the linear expansion coefficient,
and ΔT is the change in temperature.
Plugging in values:
ΔL = (31 m) (11×10⁻⁶ (°C)⁻¹) (40°C − (-5°C))
ΔL = 0.0153 m
004
Thermal stress is:
σ = E α ΔT
where E is Young's modulus.
Plugging in values:
σ = (20×10¹⁰ N/m²) (11×10⁻⁶ (°C)⁻¹) (40°C − (-5°C))
σ = 9.9×10⁷ N/m²
005
Volumetric thermal expansion is:
ΔV = V β ΔT
where ΔV is the change in volume,
V is the original volume,
β is the volumetric expansion coefficient,
and ΔT is the change in temperature.
The change in volume is:
ΔV = A Δh
ΔV = π r² Δh
ΔV = π (d/2)² Δh
ΔV = 1/4 π d² Δh
The volume of the bulb is:
V = 4/3 π r³
V = 4/3 π (D/2)³
V = 1/6 π D³
Substituting:
1/4 π d² Δh = 1/6 π D³ β ΔT
1/4 d² Δh = 1/6 D³ β ΔT
3 d² Δh = 2 D³ β ΔT
Assuming the diameter of the capillary tube is 0.0059 cm, and the diameter of the bulb is 0.12 cm, then:
3 (0.0059 cm)² Δh = 2 (0.12 cm)³ (0.000182 (°C)⁻¹) (33°C)
Δh = 0.199 cm
006
If each beam is 140 m, then the length they increase by is:
ΔL = L α ΔT
ΔL = (140 m) (1.2×10⁻⁵ (°C)⁻¹) (20°C)
ΔL = 0.0336 m
So the length of each beam increases to 140.0336 m. Using Pythagorean theorem to find the height:
(140.0336 m)² = (140 m)² + y²
y = 3.07 m
007
Use the ideal gas law:
PV = nRT
where P is the absolute pressure,
V is the volume,
n is the number of moles,
R is the gas constant,
and T is the absolute temperature.
Since V, n, and R are constant:
P₁ / T₁ = P₂ / T₂
Plugging in values:
(22 psi + 14.7 psi) / (0.17 + 273.15 K) = (P + 14.7 psi) / (21 + 273.15 K)
P = 24.8 psi
