Question

A steel ball that is initially at a pressure of 1.0 × 10⁵ Pa is heated from 20°C to 120°C, keeping its volume constant.
Find the pressure inside the ball. Coefficient of linear expansion of steel = 12 × 10^–6 °C^–1 and bulk modulus of steel = 1.6 × 10¹¹ Nm^–2.

Answer 1

Given:
Initial pressure on the steel ball = 1.0 × 10⁵ Pa
The ball is heated from 20 °C to 120 °C.
So, change in temperature, $\Delta \theta$ = 100 °C.
Coefficient of linear expansion of steel, $\alpha$ = 12 × 10^–6 °C^–1
Bulk modulus of steel, B = 1.6 × 10¹¹ Nm^–2
Pressure is given as,
$$\begin{gathered} \Rightarrow P=B\times \gamma ∆\theta \\ \Rightarrow P=B\times 3\alpha ∆\theta \left(\because \gamma =3\alpha \right) \\ \Rightarrow P=1.6\times {10}^{11}\times 3\times 12\times {10}^{-6}\times \left(120-20\right) \\ =1.6\times 3\times 12\times {10}^{11}\times {10}^{-6}\times {10}^2 \\ =57.6\times {10}^7 \\ \Rightarrow P=5.8\times {10}^8\mathrm{Pa} \end{gathered}$$
Therefore, the pressure inside the ball is 5.8 × 10⁸ Pa.