2018 A/L Physics Past Paper · Question 19 (English Medium)

Mercury-in-glass thermometers P and Q

Two mercury-in-glass thermometers $P$ and $Q$ are to be constructed with $P$ having a larger bulb of mercury than that of $Q$ , and both calibrated in the range $0\ ^\circ\text{C} - 100\ ^\circ\text{C}$ . Assume that walls of both bulbs have the same thickness. Consider the following statements.

Using capillary tubes with appropriate uniform bore radii, the two thermometers can be constructed to have

(A) the same capillary length between $0\ ^\circ\text{C}$ and $100\ ^\circ\text{C}$ markings.

(B) the same response time for rapid changes in the measuring temperature.

Of the above statements,

(1) only A is true.
(2) only B is true.
(3) only B and C are true.
(4) only A and C are true.
(5) all A, B and C are true.

Answer and explanation

Answer: (4) only A and C are true.

What the question asks

We know: P has a bigger bulb than Q.

Both bulbs have walls of the same thickness.

We want: what a suitable bore radius can change, and what it cannot.

Relevant theory

The length of a column of mercury in a glass capillary tube which has a uniform cross section, is the thermometric property of this thermometer.

Thermal capacity of the junction is too small. Therefore, it can be used to measure varying temperatures because of its quick response.

The key idea.

The extra volume of the mercury spreads along the bore. That is how far the column grows.

$\Delta l = \frac{V\gamma\,\Delta\theta}{\pi r^{2}}$

Here $V$ is the volume of the bulb. $r$ is the bore radius.

The response time depends on the heat capacity of the bulb and the thickness of its wall. The bore does not change it.

Reading the options

The options are made of A, B and C.

Test each statement on its own. Then find the matching option.

Reasoning and calculation

The same capillary length between 0 °C and 100 °C

P has the bigger bulb. So its mercury expands more.

Give P a wider bore. Then its column rises the same length.

So both can have the same length from 0 °C to 100 °C. A is true.

The same response time

P has more mercury. So more heat must flow in.

The walls have the same thickness. So the heat does not flow in faster in the same ratio.

P warms more slowly. The bore cannot change this. B is false.

Higher sensitivity of P

Sensitivity is how far the column moves for each $1\ ^\circ\text{C}$ .

It goes with $V/r^{2}$ .

P has the larger $V$ . Give it the same or a thinner bore. Then P is more sensitive. C is true.

Only B is false.

A bigger bulb needs more heat. With the same walls, it responds more slowly.

Why the other options fail

This option says: only A is true.

A is true. But C is true too, and this option leaves it out.

So it is wrong.

This option says: only B is true.

But B is false. A bigger bulb always responds more slowly.

This option says: only B and C are true.

C is true. But B is false.

This option says: A, B and C are all true.

A and C are true. But B is false.

Conclusion

A is true. B is false. C is true.

Whatever the bore, a bigger bulb responds more slowly.

The answer is option (4): only A and C are true.

Explore

P has a bigger bulb than Q.

With a wider bore, P can rise the same length from 0 to 100. So A is true.

With a thin bore, P's thread rises further for each $1\ ^\circ\text{C}$ . So C is true.

Dipped into hot water, the small bulb warms first. The big bulb lags behind. So B is false.

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