Measuring and recording practical data

Every conclusion in an experiment rests on its readings, so a result is only as good as the instruments chosen and the care taken reading them. Choosing a suitable instrument, reading it without zero or parallax errors, timing many swings of a pendulum and recording the readings in a clear table are skills every other investigation uses. They build on the units and instruments in measuring instruments: range and precision.

Choose and read the instrument

Pick an instrument whose range covers the value you expect and whose resolution (smallest step it can show) is small compared with the changes you want to see. A 30 cm rule cannot measure a 1.5 m bench in one go; a rule marked in millimetres is unsuitable for a wire whose diameter is less than 1 mm.

The resolutions below are typical. Check the scale of the instrument you are using: a finer display does not make a measurement more accurate if, for example, the instrument has a zero error.

Check the instrument and your viewing position:

• An uncorrected zero error adds an offset to the readings. Calipers that read 0.02 mm with the jaws closed make every length 0.02 mm too large. Subtract the zero reading from each measurement.
• A parallax error occurs when a pointer, liquid level or object edge is separated from the scale and you view it at an angle. Put your eye level with the reading and look straight at the scale.

Time a pendulum

The period T is the time for one complete oscillation. The bob returns to the same position moving in the same direction. At the central marker it passes twice per oscillation; counting every pass would time half-oscillations.

Your reaction time when starting and stopping a stopwatch adds an error of a few tenths of a second, whatever you are timing. For one swing of about 1 s, an error of 0.2 s is about 20%. Time 20 swings instead and the same 0.2 s is spread over all of them, adding only about 0.01 s to each period. Secure the stand, check the bob is firmly attached and keep the swing clear of people.

1. Measure the length l from the point of suspension to the centre of the bob.
2. Put a marker (a pin or a line on card) level with the lowest point of the swing. This is where the bob moves fastest, so you can judge the moment it passes most precisely.
3. Pull the bob aside through a small angle (for example, less than 10°) and release it without pushing. Start the stopwatch as it passes the marker in your chosen direction and count that pass as “zero”.
4. Count only returns through the marker in that same direction. Stop at count 20 and record the elapsed time t.
5. Repeat, find the mean t and calculate T = t ÷ 20.

Worked example 1

From stopwatch readings to a period

Problem

A pendulum of length 0.500 m is timed twice over 20 oscillations: 28.4 s and 28.2 s. Find the period to a suitable precision.

Show full solution

1. Check the repeats agree

   Method

   Compare the two readings before averaging.

   Reason

   Check that both runs used the same length, release angle and count. A 0.2 s difference is small compared with the roughly 28 s total, so these two readings give no obvious reason to reject either.

   Working

   Answer

   28.4 - 28.2 = 0.2 s, so both readings are kept.

2. Average, then divide by 20

   Method

   Find the mean time for 20 oscillations, then the time for one.

   Reason

   The period is the time for one oscillation.

   Working

   Answer

   t bar = (28.4 + 28.2)/2 = 28.3 s, T = 28.3/20 = 1.415 s

3. Round to match the data

   Method

   Present T to three significant figures for this example.

   Reason

   The timings in this example are recorded to 0.1 s. Three significant figures are a sensible way to present the period here; follow any precision instructions in your task.

   Working

   Answer

   T = 1.42 s. Keep 1.415 in your calculator for any later step such as T².

Quick check

One period of about 1.4 s is sensible for a 0.5 m pendulum; a value near 28 s would mean you forgot to divide by 20.

Record readings in a table

A good table lets someone else check every step:

• Each column heading gives the quantity and its unit, such as l / m or t / s. Units go in the heading, not in the cells.
• Raw readings come before processed values such as the mean, T and T².
• Use consistent recording precision for the same instrument and range, following the task instructions. Here lengths are recorded to 0.001 m and the example times to 0.1 s. Check the instrument’s display step and follow the task’s recording instructions.
• Keep extra digits during calculations, then round the presented results sensibly. Assess uncertainty from the method and the spread of repeats, as well as the instrument’s resolution.

The following constructed example data represent two timings of 20 oscillations at each of six lengths. They are provided to practise checking and analysis, rather than reported measurements.

Guided practice 2

Which row needs another reading?

About 4 min

Problem

Look at the table above. Which row should the student not average yet? Give a likely cause and say what the student should do.

Decide before opening the hints

The row, the likely cause and the next step

Hints

Hint 1

compare the spreads

Compare the size of t₁ - t₂ for every row. Which difference is much bigger than the others?

Hint 2

think about counting

By how much would the time change if one oscillation were missed? One period here is about 1.4 s.

Show solution step by step

1. Find the odd spread

   Method

   The 0.500 m row: 28.4 s and 26.9 s differ by 1.5 s.

   Reason

   Every other pair differs by 0.2 s. The 1.5 s difference stands out against that pattern and needs investigation.

   Working

   Answer

   28.4 - 26.9 = 1.5 s; about one period (1.4 s).

2. Suggest a cause from the evidence

   Method

   One possible explanation is that the shorter timing missed one complete oscillation.

   Reason

   If the observer counted the starting pass as “one” and stopped at count 20, only 19 complete oscillations were timed. Other timing or counting mistakes are also possible.

   Working

   Answer

   19 × 1.415 ≈ 26.9 s, which matches.

3. Repeat, then decide

   Method

   Take a third reading before choosing which to keep.

   Reason

   More readings and a check of the counting method help investigate the discrepancy. Check the count and method before excluding a reading.

   Working

   Answer

   For this example, the observer confirms that the 26.9 s run timed only 19 oscillations. A correctly counted repeat gives 28.2 s. Keep all three raw readings, note the confirmed miscount, and calculate the mean from the two valid 20-oscillation timings.

Quick check

Checking the spread of repeats is the reason for repeating. An odd reading is evidence to investigate, not a number to hide.

Check your understanding 1 Can close repeats still share a zero error?

A digital caliper reads +0.04 mm when closed. It reads 0.62, 0.63 and 0.62 mm for a wire, with 0.01 mm display steps. Find the corrected mean to 0.01 mm. Does averaging remove the zero error? What should you check before measuring again?

Show the steps Optional working

Step 1: Uncorrected mean diameter

mmUnit: mm

Corrected mean, to 0.01 mm

mmUnit: mm

Does averaging remove the zero error?

Yes: close repeats cancel all errorsCorrectIncorrectNo: every reading has the same offsetCorrectIncorrect

Before further readings

Keep extra digits and ignore the offsetCorrectIncorrectRemove low readings and keep high readingsCorrectIncorrectCheck closed jaws and correct their offsetCorrectIncorrect

Check answer

Show answer

Subtract the positive zero reading: corrected values are 0.58, 0.59 and 0.58 mm; their mean is about 0.583 mm, sensibly reported as 0.58 mm for this instrument and example. Close repeats show little scatter, but all uncorrected readings share the +0.04 mm bias. Averaging does not remove it. Calculator digits do not improve the caliper’s 0.01 mm resolution. Close the jaws gently, check the zero and zero the instrument or record the offset before further measurements.

Plot and analyse the completed table in Graphs and data analysis. Explain its limitations and propose a modification in Evaluating investigations.

Common mistakes

• Choosing an instrument whose range cannot cover the measurement, or whose scale is too coarse for the changes you want to see.
• Forgetting to check and subtract a zero error.
• Reading a scale or a meniscus from above or below instead of at eye level.
• Counting every pass of the bob through the marker, which times half-oscillations.
• Putting units in the table cells instead of the column headings.
• Averaging repeats that disagree without first finding out why.