University Physics I · topic UPI-01.03
Significant figures and measurement uncertainty
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Significant Figures & Uncertainty: predict a finite change
The uncertainty of a measurement is a number in the same unit as the value, saying how far the true value could reasonably sit from it — by convention one standard deviation of the spread, unless a coverage factor is stated. Use the supplied topic model Y_out = (s/b)^2, where Y_out is BREAK-EVEN N (readings). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. b is Systematic uncertainty b, measured in ms. Keep s (Scatter of one reading s) = 80 ms; N (Readings averaged N) = 25 fixed. For this exercise use only 5.14375 ms ≤ b ≤ 35.8562 ms on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at b = 11.9005 ms and 16.2002 ms. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δb across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
BREAK-EVEN N against b
Predictions from the supplied Significant Figures & Uncertainty model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): BREAK-EVEN N / readings
Horizontal axis (x): b: Systematic uncertainty b / ms
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| b: Systematic uncertainty b / ms | BREAK-EVEN N / readings |
|---|---|
| 5.14375 | 241.8913113125232 |
| 12.821874999999999 | 38.92932707852021 |
| 20.499999999999996 | 15.229030339083884 |
| 28.178124999999998 | 8.060385095180033 |
| 35.856249999999996 | 4.977946669310391 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of b separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At b = 11.9005 ms, substitute into Y_out = (s/b)^2: Y₀ = 45.1908 readings.
- At b = 16.2002 ms, the same fixed-parameter model gives Y₁ = 24.3858 readings.
- Signed change: ΔY_out = Y₁ − Y₀ = -20.805 readings; the input change is Δb = 4.29975 ms.
- Average sensitivity is ΔY_out/Δb = -4.83865 readings per ms. Its negative sign means that this response decreases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 45.1908 readings, Y₁ = 24.3858 readings; ΔY_out = -20.805 readings; average sensitivity = -4.83865 readings per ms.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Significant Figures & Uncertainty: infer an input with uncertainty
The uncertainty of a measurement is a number in the same unit as the value, saying how far the true value could reasonably sit from it — by convention one standard deviation of the spread, unless a coverage factor is stated. Use the supplied topic model Y_out = (s/b)^2, where Y_out is BREAK-EVEN N (readings). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. b is Systematic uncertainty b, measured in ms. Keep s (Scatter of one reading s) = 80 ms; N (Readings averaged N) = 25 fixed. For this exercise use only 5.14375 ms ≤ b ≤ 35.8562 ms on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 11.224589 readings, with a bounded reading uncertainty ±0.937 readings. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown b by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00307125 ms, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
BREAK-EVEN N against b
Predictions from the supplied Significant Figures & Uncertainty model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): BREAK-EVEN N / readings
Horizontal axis (x): b: Systematic uncertainty b / ms
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| b: Systematic uncertainty b / ms | BREAK-EVEN N / readings |
|---|---|
| 5.14375 | 241.8913113125232 |
| 12.821874999999999 | 38.92932707852021 |
| 20.499999999999996 | 15.229030339083884 |
| 28.178124999999998 | 8.060385095180033 |
| 35.856249999999996 | 4.977946669310391 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(b) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(5.14375) = 241.891 readings and Y_out(35.8562) = 4.97795 readings. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives b_line = 5.14375 + (11.2246 − 241.891)(35.8562 − 5.14375)/(4.97795 − 241.891) = 35.0465 ms. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives b = 23.8784 ms; substituting back gives Y_out = 11.224589 readings.
- The interpolation error in the input is |b_line − b| = 11.1681 ms. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 10.287589 readings and 12.161589 readings. Sort the roots even if the response decreases: 22.9401 ms ≤ b ≤ 24.9421 ms.
- The lower and upper input deviations are 0.9383 ms and 1.06373 ms. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer b ≈ 23.8784 ms; compatible interval [22.9401, 24.9421] ms. Endpoint interpolation gives 35.0465 ms.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.