Measuring Things — Question 2

Question 2

A repeatable result with a moving standard

A rigid bench has not changed length. In January its length is reported as $80.0$ local units. One local unit is defined by the physical length of a particular reference strip. Repeated placements agree closely.

By July the reference strip has stretched: its length is now $1.010$ times its January length. The same strip still defines one local unit. Assume the placement method can resolve fractional units without an additional bias.

1. Express the bench length in July local units, and calculate the percentage change in its numerical reading relative to January.

2. Has the bench physically shrunk? Explain the distinction between a quantity and its numerical value in a chosen unit.

3. Would making 100 repeated July measurements expose this problem by itself? Propose a reference-check procedure that could detect it, and state what independence that procedure requires.

Question 2 - Solution

A repeatable result with a moving standard

(a) Relate the reference lengths.

$$L=n_{\mathrm{Jan}}u_{\mathrm{Jan}}=n_{\mathrm{Jul}}u_{\mathrm{Jul}},\qquad u_{\mathrm{Jul}}=1.010u_{\mathrm{Jan}}.$$

Let $u_J$ denote the January reference length and $u_L$ the July reference length. The fixed bench length is $L=80.0u_J$, while $u_L=1.010u_J$. Hence $$\frac{L}{u_L}=\frac{80.0u_J}{1.010u_J}=79.2079\ldots, \qquad \boxed{L\approx79.2\ \text{July local units}}.$$ Using the unrounded ratio, the numerical change is $$\left(\frac{1}{1.010}-1\right)100\%\approx\boxed{-0.990\%}.$$

(b) Separate object and reference.

The bench has not shrunk. The same physical length is divided by a larger unit, so its numerical value becomes smaller. A 1% increase in unit size produces a decrease of approximately, but not exactly, 1% in the reading.

(c) Check stability independently.

Repeating the July procedure can reveal variation in placement, but every repeat uses the same stretched reference. Agreement cannot establish the reference’s stability. One suitable procedure is to compare the strip periodically with an independently maintained length reference under controlled conditions, recording the comparison and its uncertainty. The second reference must not share the strip’s stretching mechanism or be redefined from that same strip. A second strip that stretches by the same factor would conceal the problem.

Check: As the reference length increases, fewer reference lengths fit along the unchanged bench, consistent with the sign above.