12 a. You have two metal tubes of the same diameter. Explain briefly how you can use the principles from thermal expansion of matter to fit one of the tubes into the other. b. With a labelled diagram, briefly how you can find the centre of mass of a regular shaped lamina which is rectangular in shape using straight edge apparatus.

Tube P has known coefficient of linear expansion $\alpha_P$ (e.g., steel tube, $\alpha_P = 11 \times 10^{-6} \, \mathrm{K}^{-1}$). Tube Q has unknown coefficient $\alpha_Q$.

Attach the free ends to a rigid crossbar or use a micrometer screw gauge to measure relative displacement $\delta$.

The change in length for tube P: $\Delta L_P = L \alpha_P \Delta T$.

The change in length for tube Q: $\Delta L_Q = L \alpha_Q \Delta T$.

Relative displacement: $\delta = \Delta L_Q - \Delta L_P = L (\alpha_Q - \alpha_P) \Delta T$.

Units: $\delta$ in $\mathrm{m}$, $L$ in $\mathrm{m}$, $\Delta T$ in $\mathrm{K}$, $\alpha$ in $\mathrm{K}^{-1}$.

This uses the principle of linear thermal expansion $\Delta L = L \alpha \Delta T$.

12. (b)

For a uniform rectangular lamina ABCD, the centre of mass (COM) is at the geometric centre.

Method using straight edge:

Labelled diagram description:
Rectangle ABCD:

• A top-left, B top-right, C bottom-right, D bottom-left.
• Diagonal AC and BD cross at O (COM).

To verify with plumbline apparatus (using straight edge to mark lines):

COM at intersection of diagonals or plumb lines.