Thermal Expansion — Question 2

Question 2

Heating a ring enlarges its empty hole

At the reference temperature, a ring has inner diameter $D_h=50.00\,\mathrm{mm}$ and a shaft diameter $D_s=50.04\,\mathrm{mm}$. Their linear expansion coefficients are $\alpha_h=20.0\times10^{-6}\,\mathrm{K^{-1}}$ and $\alpha_s=10.0\times10^{-6}\,\mathrm{K^{-1}}$. Treat linear diameter expansion as exact in the exercise; assume uniform free heating.

1. Find the minimum ring temperature rise for just fitting over a shaft held at its original temperature.

2. If both ring and shaft receive the same temperature rise, find the new fitting threshold. Explain why the hole does not contract when the ring expands.

Question 2 - Solution

Heating a ring enlarges its empty hole

(a) Method and calculation

Every unconstrained linear dimension scales by the same factor, including the hole:$$D_h(T)=D_h(1+\alpha_h\Delta T).$$$$\Delta T_{\mathrm{ring\ only}}=\frac{50.04-50.00}{50.00(20\times10^{-6})}=40.0\,\mathrm K.$$

(b) Interpretation and further calculation

$$D_h(1+\alpha_h\Delta T)=D_s(1+\alpha_s\Delta T).$$$$\Delta T=\frac{.04}{50(.000020)-50.04(.000010)}\,\mathrm K\approx80.064\,\mathrm K.$$Imagine the hole filled with identical material: the fictitious plug expands and the surrounding boundary moves with it. Removing the plug does not reverse that displacement.

Physical check

Heating both objects takes a larger rise because shaft expansion offsets part of the increased hole diameter.