Question

A circular hole of diameter 2.00 cm is made in an aluminium plate at 0°C. What will be the diameter at 100°C? α for aluminium = 2.3 × 10^–5 °C^–1.

Answer 1

Given:
Diameter of a circular hole in an aluminium plate at 0°C, d₁ = 2 cm = 2 × 10^–2 m
Initial temperature, t₁ = 0 °C
Final temperature, t₂ = 100 °C
So, the change in temperature, ($\Delta$t) = 100°C - 0°C = 100°C
The linear expansion coefficient of aluminium, α_al​ = 2.3 × 10^–5 °C^–1
Let the diameter of the circular hole in the plate at 100^oC be d₂ , which can be written as:
$d_2=d_1\left(1+\alpha \Delta t\right)$
$\Rightarrow d_2$= 2 × 10^–2 (1 + 2.3 × 10^–5 × 10²)
$\Rightarrow d_2$= 2 × 10^–2 (1 + 2.3 × 10^–3)
$\Rightarrow d_2$= 2 × 10^–2 + 2.3 × 2 × 10^–5
$\Rightarrow d_2$ = 0.02 + 0.000046
$\Rightarrow d_2$= 0.020046 m
$\Rightarrow d_2$≈ 2.0046 cm
Therefore, the diameter of the circular hole in the aluminium plate at 100^oC is ​2.0046 cm.