An aluminium can of cylindrical shape contains 500 cm^{3} of water. The area of the inner cross section of the can is 125 cm^{2}. 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}.

#### Solution

Given:

Volume of water contained in the aluminium can, V_{0} = 500 cm^{3}

Area of inner cross-section of the can, A = 125 cm^{2}^{}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 = V_{0}(1 + γΔθ)

= 500 [1 + 3.2 × 10^{–4} × (80 – 10)]

= 500 [1 + 3.2 × 10^{–4 }× 70]

= 511.2 cm^{3}

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 cm^{3}^{ }

Consider a cylinder of volume 11.2 cm^{3}

∴ Increase in height of the water

\[= \frac{11 . 2}{125}\] = 0.0896

= 0.089 cm