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Question
An aluminium can of cylindrical shape contains 500 cm3 of water. The area of the inner cross section of the can is 125 cm2. All measurements refer to 10°C.
Find the rise in the water level if the temperature increases to 80°C. The coefficient of linear expansion of aluminium is 23 × 10–6 °C–1 and the average coefficient of the volume expansion of water is 3.2 × 10–4 °C–1.
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Solution
Given:
Volume of water contained in the aluminium can, V0 = 500 cm3
Area of inner cross-section of the can, A = 125 cm2
Coefficient of volume expansion of water, γ = 3.2 × 10–4 °C–1
Coefficient of linear expansion of aluminium,
\[\alpha_{AL}\] = 23 × 10–6 °C–1
If \[∆ \theta\] is the change in temperature, then final volume of water
\[\left( V \right)\] due to expansion,
V = V0(1 + γΔθ)
= 500 [1 + 3.2 × 10–4 × (80 – 10)]
= 500 [1 + 3.2 × 10–4 × 70]
= 511.2 cm3
The aluminium vessel expands in its length only.
So, area of expansion of the base can be neglected.
Increase in volume of water = 11.2 cm3
Consider a cylinder of volume 11.2 cm3
∴ Increase in height of the water
\[= \frac{11 . 2}{125}\] = 0.0896
= 0.089 cm
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