Inconsistent Pendulum Results in a Class VIII Practical
Five groups of Class VIII students measure the time period of a simple pendulum of length 80 cm using stopwatches with a resolution of 0.01 s. Each group times 10 oscillations. The results are 18.02 s, 17.65 s, 18.10 s, 17.30 s and 18.55 s. The teacher notices that one group starts the watch when the bob is at the extreme of its swing while another starts at the mean position; one group counts the first extreme as 'one', another counts it as 'zero'. The students' reaction time is about 0.2 s. Using T = 2π√(L/g) with L = 0.80 m and g = 9.8 m/s², the expected time for 10 oscillations is about 17.95 s. One group argues that its value of 17.30 s must be right because the stopwatch reads to 0.01 s and is therefore 'very accurate'.
1. Distinguish the sources of error here, and say which one dominates the spread of results.
Read the worked answer
Three different error sources are being confused.
Instrument resolution: 0.01 s. Over 10 oscillations this contributes at most about 0.005 s of reading uncertainty - negligible compared with the observed 1.25 s spread. So the student's claim that a 0.01 s display makes the result accurate is wrong: resolution is not accuracy.
Human reaction time: about 0.2 s at start and 0.2 s at stop, so up to ~0.4 s in 10 oscillations, i.e. ~4% in T. This is a systematic error if the same student always starts late/early, random if it varies.
Reference-point inconsistency: timing from an extreme is unreliable because the bob moves slowest there and it is hard to judge the exact turning point; timing from the mean position is better because the bob moves fastest and crosses a fixed mark sharply. Counting 'one' at the first extreme versus 'zero' is a counting convention error, and mixing conventions across groups shifts results by a whole oscillation.
Dominant cause of the 1.25 s spread: inconsistent reference points and counting convention, not the stopwatch.
2. State how the procedure should be redesigned, and estimate how much the uncertainty in T would fall.
Read the worked answer
Redesign for precision:
- Choose a fixed fiducial mark at the mean (lowest) position, viewed against a vertical reference, and time from the bob passing it in one direction.
- Time 50 oscillations instead of 10. Reaction-time error is roughly fixed at ~0.4 s, so it is now 0.4/50 = 0.008 s per oscillation, or about 0.4% of T, compared with about 4% with 10 oscillations - roughly a tenfold reduction.
- Let one student start and stop throughout, and repeat three times, taking the mean.
- Fix the counting convention in writing: release from the extreme, start timing at the first mean-position crossing, count that as zero.
- Increase length to reduce percentage error if timing precision is limiting (since T ∝ √L, a longer pendulum gives a longer time).
After this, the spread between groups should drop to well under 0.1 s in 10 oscillations.
3. The group insists its 17.30 s result is valid because the stopwatch is accurate to 0.01 s. How do you correct this reasoning, and what would you do with the anomalous value?
Read the worked answer
Correct the concept first: instrument accuracy describes the smallest readable change, not how close the measurement is to the true value. A precise-looking number produced by a faulty method is still wrong - this is the difference between precision and accuracy.
Evidence that 17.30 s is an outlier rather than a discovery: the expected value is 17.95 s, and the deviation of 0.65 s is far more than could arise from the stated resolution. A likely cause is one oscillation being lost or gained through a counting-convention mistake (T ≈ 1.795 s, so one oscillation ≈ 1.8 s - much larger than 0.65 s, but a miscount at the half-swing level fits).
Handling: do not simply delete the value. Re-run that group's measurement under the standardised procedure with the teacher watching, check the count, then either retain the corrected value or report the mean and note the outlier with its cause. In senior practical work this is the same discipline as rejecting a reading only with a documented physical reason, never because it looks inconvenient.