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.

12. (a)

Step 1: Select two metal tubes P and Q of identical diameter and same initial length $L$. Tube P has known coefficient of linear expansion $\alpha_P$ (e.g., steel tube, $\alpha_P = 11 \times 10^{-6} \, K^{-1}$). Tube Q has unknown coefficient $\alpha_Q$.

Step 2: Fix one end of both tubes rigidly to a common support. Attach the free ends to a rigid crossbar or use a micrometer screw gauge to measure relative displacement $\delta$.

Step 3: Heat both tubes uniformly (e.g., with steam jacket), raising temperature by $\Delta T$ (measured by thermometer in $^\circ\mathrm{C}$ or $\mathrm{K}$).

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

Step 4: Solve for $\alpha_Q$:

$\alpha_Q = \alpha_P + \frac{\delta}{L \Delta T}.$

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:

Step 1: Place lamina on white paper.

Step 2: Use straight edge to draw diagonal from corner A to C.

Step 3: Use straight edge to draw diagonal from corner B to D.

Step 4: Mark intersection point O — this is the COM.

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):
Step 1: Punch three holes near edges (e.g., close to A, B, midway on CD).
Step 2: Suspend lamina from hole 1 via pin on stand; hang plumb bob, mark plumb line on paper using straight edge.
Step 3: Repeat for holes 2 and 3.
Step 4: Lines intersect at COM (same as O).

COM at intersection of diagonals or plumb lines.