Answer:
a. 9.99625 cm b. 68 °C
Step-by-step explanation:
(a) What is the length of the rod at the freezing point of water (0 0C)?
Before we find the length of the rod, we need to find the coefficient of linear expansion, α = (L - L₀)/[L₀(T - T₀)] where L₀ = length of rod at temperature T₀ = 10.0 cm, T₀ = 20 °C, L = length of rod at temperature T = 10.015 cm and T = 100 °C
Substituting the values of the variables into the equation, we have
α = (L - L₀)/[L₀(T - T₀)]
α = (10.015 cm - 10.0 cm)/[10.0 cm(100 °C - 20 °C)]
α = 0.015 cm/[10.0 cm × 80 °C]
α = 0.015 cm/[800.0 cm °C]
α = 0.00001875 /°C
We now find the length L₁ at T₁ = 0 °C from
L₁ = L₀(1 + α(T₁ - T₀))
So, substituting the values of the variables into the equation, we have
L₁ = L₀(1 + α(T₁ - T₀))
L₁ = 10.0 cm[1 + 0.00001875 /°C(0° C - 20 °C)]
L₁ = 10.0 cm[1 + 0.00001875 /°C × -20° C]
L₁ = 10.0 cm[1 - 0.000375]
L₁ = 10.0 cm[0.999625]
L₁ = 9.99625 cm
(b) What is the temperature if the length of the rod is 10.009 cm?
With length L₃ = 10.009 cm at temperature T₃, using
L₃ = L₀(1 + α(T₃ - T₀))
making T₃ subject of the formula, we have
L₃/L₀ = 1 + α(T₃ - T₀)
L₃/L₀ - 1 = α(T₃ - T₀)
T₃ - T₀ = (L₃/L₀ - 1)/α
T₃ = T₀ + (L₃/L₀ - 1)/α
substituting the values of the variables into the equation, we have
T₃ = 20 °C + (10.009 cm/10.0 cm - 1)/0.00001875 /°C
T₃ = 20 °C + (1.0009 - 1)/0.00001875 /°C
T₃ = 20 °C + 0.0009/0.00001875 /°C
T₃ = 20 °C + 48 °C
T₃ = 68 °C