1.3.2Development of Practical Skills in Physics

Analogue Apparatus and Interpolation

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Lesson summary

1.3.2 Analogue Apparatus and Interpolation — OCR A Level Physics A notes: Read analogue scales to the correct precision by interpolating between divisions, and state a sensible reading uncertainty.An analogue instrument shows a quantity by the position of a pointer, a mark or a l…

Measuring cylinder and burette
Thermometer
Moving-coil meters
Protractor
Spring balance
Worked example 1

Section 1 of 8What analogue means

An analogue instrument shows a quantity by the position of a pointer, a mark or a liquid level against a scale.

Examples include rulers, protractors, liquid-in-glass thermometers, measuring cylinders, burettes, spring balances and moving-coil ammeters and voltmeters.

Estimate

Unlike a digital display, the position may fall between two marked divisions, so the person taking the reading must estimate the extra fraction.

Because this estimate is a judgement, analogue readings always carry a reading uncertainty that depends on how finely the scale is marked and how clearly it can be seen.

Section 2 of 8Interpolation

Interpolation means estimating a value that lies between two marked divisions. Look at where the pointer sits between the lines and decide whether it is about a quarter, a half or three quarters of the way across. Then add that fraction of one division to the lower marking.

Suppose a voltmeter scale is marked every 0.2 V and the needle is roughly one third of the way from 1.4 V to 1.6 V.

Interpolation

The reading is 1.4 + 0.07 = 1.47 V, which is sensibly written as 1.47 V or, being cautious, 1.5 V.

As a guide, interpolate to the nearest half or fifth of a division, but do not claim more digits than your eyes can justify.

The same idea appears when reading values from a graph: the value between two gridlines is found by proportion, and on a data table, a value between two tabulated points is estimated by assuming a straight line between them (linear interpolation).

Section 3 of 8Reading uncertainty

The reading uncertainty for an analogue instrument is normally taken as half of the smallest division. On a ruler marked in millimetres this is ± 0.5 mm. If a scale is hard to read, for example a thick pointer or widely spaced marks, you may quote one full division as the uncertainty.

A length measurement involves two positions, one at each end of the object.

Reading uncertainty

If each end is read separately the uncertainties add, so with a millimetre ruler the length has an uncertainty of up to ± 1 mm.

If the object is placed with one end exactly at the zero mark, the uncertainty is in principle smaller, but the ends of rulers are often worn, so it is safer to start from a clearly marked line (for instance 10.0 cm) and subtract.

Percentage uncertainty = (reading uncertainty / reading) × 100%. Large readings have small percentage uncertainties, which is why you should use as much of the scale as you can, for example choosing a meter range so the needle deflects more than halfway.

Section 4 of 8Technique with common analogue instruments

Technique with common analogue instruments

Measuring cylinder and burette

Measuring cylinder and burette read the bottom of the meniscus at eye level. Water curves downwards where it touches glass, so the reading is at the lowest point of the curve.

Thermometer

Thermometer allow time to reach thermal equilibrium, keep the bulb fully immersed, and avoid touching the glass. Typical resolution is 1 °C, so a reading is ± 0.5 °C.

Moving-coil meters

Moving-coil meters check the needle rests on zero when disconnected; use the mirror strip so that the needle and its reflection coincide, which removes parallax.

Protractor

Protractor place the centre exactly on the pivot point and line the zero with one arm. Typical uncertainty is ± 0.5° or ± 1°.

Spring balance

Spring balance check for zero error by reading with no load, and hold it vertical so friction against the casing does not alter the reading.

Section 5 of 8Recording with suitable precision

The final recorded value should be consistent with the resolution.

If a ruler has millimetre divisions then a length is written to the nearest 0.5 mm, so 3.65 cm is acceptable but 3.654 cm is not.

In a table of results every value of the same quantity should have the same number of decimal places, and the column heading should show the quantity and its unit, for example l / cm.

Calculated quantities are then given to a number of significant figures no greater than the least precise data used.

This links with 1.3.1, where percentage uncertainty is calculated, and with later work on error bars on graphs, where the reading uncertainty gives the length of each bar.

Section 6 of 8Worked examples

An ammeter scale has divisions every 0.05 A. The pointer lies about halfway between 0.30 A and 0.35 A. Give the reading and a sensible uncertainty.
A measuring cylinder has divisions every 2 cm³. The water level starts at 46 cm³. After a stone is added the bottom of the meniscus is half a division above the 58 cm³ mark. Find the volume of the stone and its uncertainty.
A student measures a pendulum length of 85.3 cm with a metre rule marked in mm and reads each end with an uncertainty of ± 0.5 mm. Find the percentage uncertainty.

Section 7 of 8Key terms

Term	Meaning
Analogue	Instrument displaying a quantity by pointer position or liquid level on a scale.
Interpolation	Estimating a value between two marked divisions.
Meniscus	Curved surface of a liquid in a narrow tube.
Reading uncertainty	Uncertainty due to the limits of reading a scale, often half the smallest division.
Parallax	Apparent shift of a reading when viewed off-axis.
Smallest division	Spacing between adjacent scale marks, which sets the resolution.

Section 8 of 8Exam technique

Reading the top of a water meniscus instead of the bottom.

Quoting a reading with more precision than the scale justifies, e.g. 3.654 cm from a ruler marked in mm.

Using the end of a worn ruler as zero without checking it.

Forgetting that a difference of two readings has the uncertainties added, not subtracted.

Viewing the pointer from an angle so that parallax shifts the reading.

Try it in the virtual labCircular motion: centripetal force \n

Section 1 of 8What analogue means

Section 2 of 8Interpolation

Section 3 of 8Reading uncertainty

Section 4 of 8Technique with common analogue instruments

Section 5 of 8Recording with suitable precision

Section 6 of 8Worked examples

An ammeter scale has divisions every 0.05 A. The pointer lies about halfway between 0.30 A and 0.35 A. Give the reading and a sensible uncertainty.

A measuring cylinder has divisions every 2 cm3. The water level starts at 46 cm3. After a stone is added the bottom of the meniscus is half a division above the 58 cm3 mark. Find the volume of the stone and its uncertainty.

A student measures a pendulum length of 85.3 cm with a metre rule marked in mm and reads each end with an uncertainty of ± 0.5 mm. Find the percentage uncertainty.

Section 7 of 8Key terms

Section 8 of 8Exam technique

Marked practiceQuizzes for this lesson

A measuring cylinder has divisions every 2 cm³. The water level starts at 46 cm³. After a stone is added the bottom of the meniscus is half a division above the 58 cm³ mark. Find the volume of the stone and its uncertainty.