Why measuring in multiples works
- Every measurement carries an uncertainty of about the smallest scale division of the instrument.
- A millimetre rule is trustworthy to about ±0.5 mm\pm 0.5\ \text{mm}±0.5 mm, and a hand-operated stopwatch carries a reaction-time error of about 0.3 s0.3\ \text{s}0.3 s.
- The problem comes when the quantity you want is smaller than that uncertainty, such as one sheet of paper or one swing of a pendulum.
- The fix is to measure a large, known number of identical items together and then divide.
Key Idea
- Measure a large, known number nnn of identical items together, then divide:
- The uncertainty in the total is no bigger than for a single reading, but it is now shared between nnn items, so the uncertainty in each item is nnn times smaller.
- The instrument has not improved; you have made the reading big enough to measure reliably.
Finding a small distance: measure many, then divide
- Stack a large, known number of identical sheets, 100 is sensible, and press them together so there are no air gaps.
- Check the rule for a zero error, then measure the total thickness with a millimetre rule, your line of sight perpendicular to the scale.
- Divide the total thickness by the number of sheets.
- Repeat with a fresh stack and take a mean.
- The same method finds any small length: wind 40 tight, touching turns of wire round a pencil and divide the length of the coil by 40 to get the wire's diameter.
Example
- A stack of 100 sheets of paper measures 9.8 mm9.8\ \text{mm}9.8 mm thick.
- In SI units that is 9.8×10−5 m9.8 \times 10^{-5}\ \text{m}9.8×10−5 m, quoted to 2 significant figures to match the reading, and with a unit.
Finding a short time: time many oscillations
Definition
Period of oscillation
The time taken for one complete oscillation, that is one full to-and-fro swing that returns the bob to its starting point moving in the same direction.
- Set the bob swinging through a small angle in one vertical plane.
- Place a fiducial mark, a pin or pointer, at the centre of the swing, directly below the pivot.
- Start the stopwatch as the bob passes the mark moving in one direction, counting that pass as zero, then count each later pass in the same direction as 1, 2, 3, up to 20.
- Stop on the count of 20 and record the total time ttt.
- Repeat three times, discard any anomaly, and take the mean of ttt.
- Calculate the period from T=t/nT = t/nT=t/n, with n=20n = 20n=20.
- The mark goes at the centre because the bob moves fastest there, so the instant it passes is easiest to judge; at the ends of the swing the bob is momentarily at rest and hard to time.
Example
- Timing 20 complete oscillations three times gives 28.5 s28.5\ \text{s}28.5 s, 28.7 s28.7\ \text{s}28.7 s and 28.6 s28.6\ \text{s}28.6 s.
- A 0.3 s0.3\ \text{s}0.3 s timing error is about 1%1\%1% of 28.6 s28.6\ \text{s}28.6 s, and dividing by 20 leaves only about 0.015 s0.015\ \text{s}0.015 s on the period.
Practical
Investigation: the period of a simple pendulum
- Aim: to measure the period of a simple pendulum and investigate how it depends on the length of the pendulum and the mass of the bob.
- Apparatus: retort stand and clamp, string, two bobs of different mass, stopwatch, metre ruler, and a fiducial marker (a pin or pointer).
- Method:
- Measure the length lll from the point of support to the centre of the bob.
- Set the fiducial mark at the rest position and pull the bob aside through a small angle, releasing it in one vertical plane.
- Start timing as the bob passes the mark, counting that pass as zero, and time ten complete oscillations.
- Record the time ttt and calculate T=t/10T = t/10T=t/10.
- Repeat and average, then repeat for a longer length, and again with a heavier bob at the same length.
- Measure and plot:
- Time ten oscillations to get the period TTT, and tabulate TTT against length.
- As an extension, plot T2T^2T2 against lll to get a straight line through the origin.
- Result: the period increases with length but does not depend on the mass of the bob.
- Watch out:
- Reaction-time error dominates a single swing, so time ten oscillations and divide by ten, then repeat and average.
- Keep the amplitude small and start the count at zero, not one.
- Safety: minimal risk; weight the base so the stand cannot topple.
Random and systematic errors behave differently
Definition
Random error
An error that varies unpredictably from reading to reading, scattering results on either side of the true value; it is reduced by repeating and averaging.
Definition
Systematic error
An error that shifts every reading by the same amount or in the same direction, such as a zero error or a worn scale; repeating cannot remove it, so it must be found and corrected.
- Repeating and averaging reduces random error, the scatter caused by reaction time varying from go to go.
- Spot and discard an obvious anomaly before averaging, not after.
- Measuring in multiples and repeating do not remove a systematic error.
- A worn rule end, a zero error, or a slow stopwatch shifts every reading the same way, so check for these at the start.
Common Mistake
- Do not count the first pass as "one": start the count on zero, or you time only 19 oscillations while dividing by 20 and make the period about 5%5\%5% too small.
- One oscillation is the complete to-and-fro journey; the swing from one extreme to the other is only half a period.
- Repeats do not cure a zero error; you simply record the same fault every time.
- Do not measure one sheet or one swing and then "check" it with multiples, because the multiple is the measurement.
Exam technique
- "Describe how you would determine..." is point-marked: state a sensible large number (100 sheets, 20 oscillations), because a mark is usually reserved for it.
- Say what you measure and with what: the total thickness with a millimetre rule, the total time with a stopwatch.
- Say that you divide by the number, the mark most often dropped.
- Add repeat and take a mean, then give the answer with its unit.
Self review
- Why does measuring a stack of 100 sheets and dividing give a better value than measuring one sheet?
- What is the period of a pendulum?
- When timing oscillations, why do you start the count at zero rather than one?
- Where do you place the fiducial mark, and why there?
- What is the difference between a random error and a systematic error?
