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Measuring Things — Question 7

Section 1-1 · Physics pages 24–25

Question 7 Calibration that changes during an experiment A balance has an exact scale factor of one, but its additive zero offset may drift. A reference mass has assigned mass 50.00 g. During one measurement sequence, the balance displays: t (min) 0 2 4 Reference 50.20 g Sample 63.70 g Reference 50.60 g Indicated masses in time order. The reference’s assigned mass is 50.00 g. Assume the sample mass is constant, readings and the reference are exact for this exercise, and the additive offset varies linearly in time from minute 0 to minute 4. (a) Determine the offset at each reference measurement and its interpolated value when the sample is weighed. (b) Calculate the corrected sample mass. What would you obtain by applying only the initial calibration, and in which direction would that result be wrong? (c) Does agreement with the two endpoint reference readings prove that the drift was linear? Explain what additional check near the sample measurement would help, and why one check still does not prove an arbitrary drift history. The relation is displayed mass = true mass + offset.
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Question 7 - Solution Calibration that changes during an experiment t (min) 0 2 4 Reference 50.20 g Sample 63.70 g Reference 50.60 g (a) Find the offset at the reference times. b(0) = 50.20 −50.00 = 0.20 g, b(4) = 50.60 −50.00 = 0.60 g. Linear drift gives, with t in minutes, b(t) = 0.20 g + 0.40 g 4 min t, b(2) = 0.40 g. (b) Correct at the relevant time. minitial only −mcorrected = b(2) −b(0) = 0.20 g. m = 63.70 −0.40 = 63.30 g . Using only the initial offset would give 63.70−0.20 = 63.50 g, an overestimate of 0.20 g. Subtracting too small a positive offset leaves a result that is too large. (c) Distinguish a model from its endpoint data. Many nonlinear offset histories pass through b(0) = 0.20 g and b(4) = 0.60 g. The endpoints alone do not establish b(2) = 0.40 g. A reference check immediately before and after the sample weighing, with a justified bound on change during the short interval, would test local drift more effectively. A further intermediate reference reading could also test the linear model. However, finitely many checks cannot exclude arbitrary fluctuations between them without an additional smoothness or rate-of-change assumption. Check: The sample time is halfway between the reference times, so linear interpolation uses the mean of the two offsets.

Question 7

Calibration that changes during an experiment

A balance has an exact scale factor of one, but its additive zero offset may drift. A reference mass has assigned mass 50.0050.00 g. During one measurement sequence, the balance displays:

Diagram shown in the original SVG worksheet.

Assume the sample mass is constant, readings and the reference are exact for this exercise, and the additive offset varies linearly in time from minute 0 to minute 4.

  1. Determine the offset at each reference measurement and its interpolated value when the sample is weighed.

  2. Calculate the corrected sample mass. What would you obtain by applying only the initial calibration, and in which direction would that result be wrong?

  3. Does agreement with the two endpoint reference readings prove that the drift was linear? Explain what additional check near the sample measurement would help, and why one check still does not prove an arbitrary drift history.

The relation is displayed mass=true mass+offset\text{displayed mass}=\text{true mass}+\text{offset}.

Question 7 - Solution

Calibration that changes during an experiment

Diagram shown in the original SVG worksheet.

(a) Find the offset at the reference times.

b(0)=50.20−50.00=0.20g,b(4)=50.60−50.00=0.60g.b(0)=50.20-50.00=0.20\ \mathrm{g},\qquad b(4)=50.60-50.00=0.60\ \mathrm{g}. Linear drift gives, with tt in minutes, b(t)=0.20g+0.40g4mint,b(2)=0.40g.b(t)=0.20\ \mathrm{g}+\frac{0.40\ \mathrm{g}}{4\ \mathrm{min}}t, \qquad b(2)=0.40\ \mathrm{g}.

(b) Correct at the relevant time.

minitialonly−mcorrected=b(2)−b(0)=0.20g.m_{\mathrm{initial\ only}}-m_{\mathrm{corrected}}=b(2)-b(0)=0.20\ \mathrm g.

m=63.70−0.40=63.30g.\boxed{m=63.70-0.40=63.30\ \mathrm{g}}. Using only the initial offset would give 63.70−0.20=63.5063.70-0.20=63.50 g, an overestimate of 0.200.20 g. Subtracting too small a positive offset leaves a result that is too large.

(c) Distinguish a model from its endpoint data.

Many nonlinear offset histories pass through b(0)=0.20b(0)=0.20 g and b(4)=0.60b(4)=0.60 g. The endpoints alone do not establish b(2)=0.40b(2)=0.40 g. A reference check immediately before and after the sample weighing, with a justified bound on change during the short interval, would test local drift more effectively. A further intermediate reference reading could also test the linear model. However, finitely many checks cannot exclude arbitrary fluctuations between them without an additional smoothness or rate-of-change assumption.

Check: The sample time is halfway between the reference times, so linear interpolation uses the mean of the two offsets.

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