Concluding and evaluating practice questions.

An experiment gives g = 9.2 ± 0.1 m s⁻², while the accepted value is 9.81 m s⁻². What is the most valid conclusion?

• The result is inconsistent with theory, suggesting a systematic error
• The result agrees with the accepted value within its uncertainty range
• The measurement has a large random error but no systematic error
• The accepted value of g must be wrong for this particular location

Answer: The result is inconsistent with theory, suggesting a systematic error

The accepted value (9.81) lies well outside the range 9.1 to 9.3, so the difference is too large to be random and points to a systematic error in the method.

Which change would best reduce the random error when timing the period of a pendulum with a stopwatch?

• Time 20 oscillations and divide the total time by 20
• Use a more expensive stopwatch that displays more digits
• Round each measured time to the nearest whole second
• Release the pendulum from a much larger starting angle

Answer: Time 20 oscillations and divide the total time by 20

Timing many oscillations and dividing spreads the human reaction-time error over each period, reducing its effect. More digits or a larger angle do not reduce random timing error.

While plotting data, a student finds one point lying far from the straight-line trend of all the others. What is the best practice?

• Check or repeat it, and exclude it if still anomalous
• Draw the best-fit line so that it passes through it
• Adjust all of the other readings until they agree with it
• Report it as the most reliable reading that was taken

Answer: Check or repeat it, and exclude it if still anomalous

An outlier should be checked and, if it cannot be explained or reproduced, left out of the best-fit line. Forcing the line through it or altering good data would corrupt the result.

A result has a large percentage uncertainty because the measured length was very small. Which change would most improve the experiment?

• Use a longer length so the fractional error is smaller
• Repeat the very same short measurement a great many more times
• Record the short length using many more decimal places than before
• Swap to a ruler that has a much longer overall measuring scale

Answer: Use a longer length so the fractional error is smaller

The absolute uncertainty of a ruler is roughly fixed, so measuring a larger length makes that uncertainty a smaller fraction of the reading. Repeating or writing more digits does not shrink the fractional error.

An experiment gives a spring constant of (2.45 ± 0.15) N m⁻¹ and the accepted value is 2.50 N m⁻¹. Which conclusion is valid?

• The result disagrees with the accepted value, since 2.45 is not 2.50
• The result agrees with the accepted value, which lies inside the range
• The result is invalid because the uncertainty was not made smaller first
• The result shows a systematic error of exactly 0.05 N m⁻¹ in the method

Answer: The result agrees with the accepted value, which lies inside the range

The measured range runs from 2.30 to 2.60 N m⁻¹ and contains 2.50, so the two are consistent. Demanding exact equality ignores the uncertainty the experiment reports.

A graph of resistance against length is a straight line with a large positive intercept on the resistance axis. A student concludes that resistance is proportional to length. What is the flaw in this conclusion?

• A straight line can never be used to support a proportional relationship
• The gradient of the line has to be exactly 1 for proportionality to hold
• Resistance and length are quantities of different kinds, so they cannot be compared
• Proportionality requires the line to pass through the origin, and it does not

Answer: Proportionality requires the line to pass through the origin, and it does not

y = mx + c with c ≠ 0 is linear but not proportional. The intercept here is physical — the resistance of the leads and contacts — and it should be reported, not ignored.