Newtonian Mechanics — Question 6

Question 6

An idealization is testable against a tolerance

A simplified model predicts 100 mm of displacement. Including a measured weak air effect changes the prediction to 99.6 mm. Required agreement is within 0.2 mm for experiment A and within 1 mm for experiment B.

1. Find the absolute and fractional discrepancy relative to the simple prediction.

2. For which experiment is omitting this effect within the stated tolerance?

3. Does agreement at one condition establish that the effect is always negligible?

Question 6 - Solution

An idealization is testable against a tolerance

(a) Quantify the omission.

$$\delta=|100-99.6|=0.4\,\mathrm{mm},\quad \boxed{\epsilon=\frac{0.4}{100}=0.004=0.4\%}$$ The discrepancy is $0.4$ mm, with fraction $0.4/100=0.004=0.4\%$. This is a model comparison, not automatically a statistical uncertainty.

(b) Use the actual acceptance scale.

$$\delta>\tau_A=0.2\,\mathrm{mm},\qquad\delta<\tau_B=1\,\mathrm{mm}$$ It exceeds A’s 0.2 mm tolerance but lies within B’s 1 mm tolerance. The same approximation can be inadequate for one purpose and adequate for another.

(c) Restrict the conclusion to tested conditions.

$$\text{Required over domain }D:\quad\max_{u\in D}|x_{\mathrm{simple}}(u)-x_{\mathrm{refined}}(u)|\le\tau$$ Changing speed, duration or geometry may change the omitted effect. Adequacy at this condition does not establish universal validity. Check the size of the omitted contribution over the intended operating range.