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BPhO R1 S1 · Section 1

Uncertainty and Estimation

5 questions — reveal each final answer.

2011 · nUncertainty and Estimation

A rectangular block has a mass of $1.5 \mathrm{~kg}$ with an uncertainty of magnitude $0.03 \mathrm{~kg}$, and a volume of $80 \mathrm{~mm} \times 50 \mathrm{~mm} \times 30 \mathrm{~mm}$, with uncertainties of magnitude $1 \mathrm{~mm}$ in each dimension. Determine the magnitude of the fractional uncertainty in the density of the block.

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Final result
  • The fractional uncertainty in the density is about $0.086$, i.e. $8.6\%$.

Worked solutions are in the book.

2012 · aUncertainty and Estimation

In an experiment to measure the Planck constant, $h$, the following results were obtained, in units of $10^{-34} \mathrm{Js}: 6.65,6.67,6.57,6.61,6.72,6.60,3.66,6.71,6.66,6.64$, 6.67.

What would you give as:

(i) The best estimate of $h$ ?

(ii) The uncertainly of $h$ from your reading? [3]

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Final result
  • Excluding the clear outlier $3.66$, the best estimate is the mean, $h = (6.65 \pm 0.01) \times 10^{-34}\,\mathrm{Js}$, where the uncertainty is the standard error of the mean $\sigma/\sqrt{n} \approx 0.015 \times 10^{-34}\,\mathrm{Js}$.

Worked solutions are in the book.

2015 · aUncertainty and Estimation

A measurement is carried out to check the speed of a camera shutter of $1 / 15 \mathrm{~s}$. The camera is focused symmetrically on a rotating turntable which revolves at $33.3 \pm 0.1$ revolutions per minute and has a spot at its centre and at its circumference. A photograph shows the arc produced by the spot on the circumference subtends an angle of $12.4 \pm 0.1^{\circ}$ at the centre of rotation. What is the correct exposure time? [3]

Interactive walkthrough →

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Final result
  • $\tau\approx0.062062\,\mathrm{s}$. Worst-case uncertainty: $1.11\%$, $\Delta\tau\approx6.9\times10^{-4}\,\mathrm{s}$. Independent standard uncertainties in quadrature: $0.86\%$, $u_\tau\approx5.3\times10^{-4}\,\mathrm{s}$; either convention is accepted. Clarified — the printed ±0.0005 s assumes quadrature; the uncertainty convention was not stated; see the interactive walkthrough.

Worked solutions are in the book.

2017 · aUncertainty and Estimation

Physicists sometimes use the approximation that light travels in a vacuum at a speed of 1 foot in $1 \mathrm{~ns}$. What is the percentage error in using this value? [3]

$$ (1.000 \mathrm{~m}=1.094 \text { yards and } 1.000 \text { yard }=3.000 \text { feet }) $$

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Final result
  • The approximation “1 foot per nanosecond” overstates the speed of light by about $1.6\%$, so it is a convenient rule of thumb for rough timing estimates.

Worked solutions are in the book.

2019 · cUncertainty and Estimation

Estimate the mass of a piece of paper the size of a pinhead (the blunt end of a sewing pin). Show your calculation. [2]

Interactive walkthrough →

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Final result
  • The mass of a pinhead-sized piece of paper is about $1 \times 10^{-7} \, \text{kg}$ ($\sim 0.1 \, \text{mg}$).

Worked solutions are in the book.

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