A hollow cylinder of aluminium 1 m long with a 40 mm outside diameter is at a temperature of 20 °C. What is the change in length and circumference of the cylinder when it is cooled to −25 °C?
(Note: the thermal co-efficient of expansion of aluminium is 24 × 10−6 K−1.)
To find the change in length and circumference of the aluminum cylinder when it is cooled from 20 °C to -25 °C, we can use the formula:
ΔL = L * α * ΔT
where:
ΔL is the change in length,
L is the original length,
α is the coefficient of linear expansion, and
ΔT is the change in temperature.
Given:
Original length, L = 1 m
Outside diameter, D = 40 mm
Outside radius, r = D/2 = 40/2 = 20 mm = 0.02 m
Original temperature, T₁ = 20 °C
Final temperature, T₂ = -25 °C
Coefficient of linear expansion, α = 24 × 10^(-6) K^(-1)
To find the change in length, ΔL:
First, we need to find the change in temperature, ΔT = T₂ - T₁ = -25 - 20 = -45 °C.
ΔL = L * α * ΔT
= 1 * 24 × 10^(-6) * -45
= -0.00108 m
Therefore, the change in length of the cylinder when cooled from 20 °C to -25 °C is -0.00108 meters. Since the length decreases, the negative sign indicates contraction.
To find the change in circumference, ΔC:
The original circumference, C₁, can be calculated using the formula:
C₁ = 2πr, where r is the original radius.
C₁ = 2π * 0.02
= 0.04π m
The change in circumference, ΔC, is given by the formula:
ΔC = ΔL / (2πr) * (C₁)
ΔC = -0.00108 / (2π * 0.02) * (0.04π)
= -0.00108 / 0.04
= -0.027 m
Therefore, the change in circumference of the cylinder when cooled from 20 °C to -25 °C is -0.027 meters. Again, the negative sign indicates a decrease in circumference.