2020 A/L Physics Past Paper · Question 27 (English Medium)

A lagged rod that narrows — the steady-state temperature distribution

A solid metal rod, uniform in cross section, has the cross-sectional area of its section BCBC gradually reduced as shown in the figure. This body is well lagged, and its two ends are maintained at temperatures 100∘C100^\circ\text{C} and 20∘C20^\circ\text{C}. Which graph best describes the variation of temperature (θ)(\theta) along the axis (l)(l) of the body in the steady state?

  1. (1) Graph (1)
  2. (2) Graph (2)
  3. (3) Graph (3)
  4. (4) Graph (4)
  5. (5) Graph (5)
Answer and explanation

Answer: (4) Graph (4)

What the question asks

We must find: the shape of the θ\theta–ll graph along the axis in the steady state — the slope and curvature in each section.

Relevant theory

When heat is supplied to one end of the rod, lagging ensures that the heat flow is axial along the rod. This is because no heat can flow through the curved surface.

In the steady state, the rate of flow of heat through any cross section like P or R should be the same because the flow of heat is axial.

In the steady state of flow of heat, the temperature gradient along the axis of a rod which is lagged is constant.

Required equations

ΔQΔt=KAΔθΔl=constant\frac{\Delta Q}{\Delta t} = K A \frac{\Delta\theta}{\Delta l} = \text{constant}

∣ΔθΔl∣∝1A\left|\frac{\Delta\theta}{\Delta l}\right| \propto \frac{1}{A}

Reasoning and calculation

Axes: horizontal axis the distance ll from A, vertical axis the temperature θ\theta. The slope of the graph is the temperature gradient.

Starting conditions: at l=0l = 0 (A), θ=100∘C\theta = 100^\circ\text{C}; at D, 20∘C20^\circ\text{C}; in between the temperature keeps falling (heat flows from A towards D).

Sections: the area changes at B and C — three pieces AB (constant large area), BC (shrinking area), CD (constant small area).

AB

Area is large and constant⇒gradient is small and constantArea\,is\,large\,and\,constant \Rightarrow gradient\,is\,small\,and\,constant: a gentle straight line.

BC

Area shrinks gradually⇒gradient grows graduallyArea\,shrinks\,gradually \Rightarrow gradient\,grows\,gradually: a curve bending downward, steepening.

CD

Area is very small and constant⇒gradient is very large and constantArea\,is\,very\,small\,and\,constant \Rightarrow gradient\,is\,very\,large\,and\,constant: a steep straight line.

Boundary values: the ends are 100∘C100^\circ\text{C}, 20∘C20^\circ\text{C}; the graph is continuous at B and C (temperature does not jump), the slope changes continuously from AB's slope at B to CD's slope at C.

Slope, curvature, sign: the slope is negative throughout; its magnitude is AB<BC<CDits\,magnitude\,is\,AB < BC < CD; since the magnitude keeps growing through BC, the curve bends downward there.

Reading the options

Comparing now only against the options: of the five, only graph (4) matches the three-section behaviour above.

Why the other options fail

Steep in AB, gentle in CD — the reverse.

Nearly one slope throughout — only fits a uniform rod.

Becomes gentle again in CD — but that is exactly where the area is smallest.

Nearly flat in BC — that would mean heat flow stops there.

Conclusion

In the steady state, gradient ∝1/A\propto 1/A: a gentle straight line in AB, a steepening curve in BC, a steep straight line in CD — the answer is (4).

In a lagged rod, the key fact is "the same heat flows through every cross section" — wherever the area is smaller, the temperature falls faster.

Exam tip

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