Thermal Expansion — Question 1

Question 1

Two anchored rods close a shared clearance

Two rods point toward each other across a $2.10\,\mathrm{mm}$ gap at $20^\circ\mathrm C$. Their outer ends are fixed, while their facing ends can expand freely until contact. Rod A has $L_A=2.00\,\mathrm m$, $\alpha_A=12.0\times10^{-6}\,\mathrm{K^{-1}}$; B has $L_B=3.00\,\mathrm m$, $\alpha_B=20.0\times10^{-6}\,\mathrm{K^{-1}}$. Use $\Delta L=\alpha L\Delta T$.

1. Find the gap at $35^\circ\mathrm C$ and the temperature of first contact.

2. At $60^\circ\mathrm C$, calculate the free-expansion overlap. Explain why it is not a physical negative gap and what additional model is needed.

Question 1 - Solution

Two anchored rods close a shared clearance

(a) Method and calculation

$$g=g_0-(\alpha_AL_A+\alpha_BL_B)\Delta T,\quad\alpha_AL_A+\alpha_BL_B=84.0\,\mu\mathrm{m/K}.$$$$g(35)=2.10-.084(15)=.840\,\mathrm{mm},\quad\Delta T_c=2.10/.084=25.0\,\mathrm K.$$First contact occurs at $45.0^\circ$C.

(b) Interpretation and further calculation

$$\Delta L_{\mathrm{free}}=.084(40)=3.36\,\mathrm{mm},\quad\Delta L_{\mathrm{free}}-g_0=1.26\,\mathrm{mm}.$$The formal gap is $-1.26$ mm, signaling failure of the free-expansion assumption.$$g\ge0\quad\text{for noninterpenetrating rods}.$$Elastic compression, support motion or instability must accommodate the excess expansion.

Physical check

Both inward tip motions reduce the same gap, so their free expansions add.