Source: ENGR 216 Comprehensive Exam Practice Bank
Tags: propagation of error, uncertainty, error analysis, partial derivatives, RSS, root sum of squares, finite differences, forward difference, Fourier's law, thermal conduction, numerical derivatives, ENGR 216, Texas A&M
Error propagation tells you how measurement uncertainties combine when you calculate a derived quantity. For independent, random errors, the total uncertainty is found by root-sum-of-squares (RSS) of the individual contributions. Finite differences approximate derivatives from discrete data points, which feeds directly into applied formulas like Fourier's law of thermal conduction.
Propagation of error (uncertainty propagation)
The method for determining the uncertainty in a calculated result based on the uncertainties of the measured inputs.
Root sum of squares (RSS)
For independent, random errors, the total uncertainty is the square root of the sum of the squared individual uncertainties. This is the standard combination rule.
Partial derivative (sensitivity coefficient)
The rate of change of the output quantity with respect to one input variable, holding all others constant. Each partial derivative scales how much that input's uncertainty contributes to the total.
Forward finite difference
A numerical approximation of a derivative using the data point at the current position and the next point forward:
$$\frac{dT}{dx}\bigg|i \approx \frac{T{i+1} - T_i}{x_{i+1} - x_i}$$
Backward finite difference
Uses the current point and the previous point:
$$\frac{dT}{dx}\bigg|i \approx \frac{T_i - T{i-1}}{x_i - x_{i-1}}$$
Central finite difference
Uses the points on either side of the current position:
$$\frac{dT}{dx}\bigg|i \approx \frac{T{i+1} - T_{i-1}}{x_{i+1} - x_{i-1}}$$
Fourier's law of thermal conduction
$$q = kA\left(\frac{dT}{dx}\right)$$
where q is the energy conducted, k is thermal conductivity, A is cross-sectional area, and dT/dx is the temperature gradient.
For a derived quantity $f(x_1, x_2, \ldots, x_n)$ with independent random errors:
$$\delta f = \sqrt{\left(\frac{\partial f}{\partial x_1}\delta x_1\right)^2 + \left(\frac{\partial f}{\partial x_2}\delta x_2\right)^2 + \cdots}$$
Steps:
Take the partial derivative of your formula with respect to each measured variable.
Multiply each partial derivative by the corresponding uncertainty.
Square each term, sum them, and take the square root.
The critical force is $F_{\max} = \mu_s F_p$ where $\mu_s = \tan(\theta)$, so:
$$F_{\max} = F_p \tan(\theta)$$
Given:
$F_p = 18$ N (exact, so $\delta F_p = 0$)
$\theta = 36.3° \pm 0.5°$
Since only $\theta$ carries uncertainty:
$$\delta F_{\max} = F_p \cdot \sec^2(\theta) \cdot \delta\theta$$
Convert the angle uncertainty to radians: $\delta\theta = 0.5° \times \frac{\pi}{180} = 0.008727$ rad.
$$\delta F_{\max} = 18 \times \sec^2(36.3°) \times 0.008727$$
$\cos(36.3°) = 0.80674$, so $\sec^2(36.3°) = 1/0.80674^2 = 1.5362$.
$$\delta F_{\max} = 18 \times 1.5362 \times 0.008727 = 0.2413 \approx 0.24$$
For $R_T = R_1 + R_2 + R_3$, each partial derivative is 1, so:
$$\delta R_T = \sqrt{\delta R_1^2 + \delta R_2^2 + \delta R_3^2}$$
$$\delta R_T = \sqrt{7^2 + 10^2 + 11^2} = \sqrt{49 + 100 + 121} = \sqrt{270} = 16.43$$
Key insight: when quantities are simply added (or subtracted), the RSS formula simplifies to the square root of the sum of squared uncertainties, with no partial-derivative scaling needed.
The choice of method depends on the position in the dataset:
At the first point, you can only go forward.
At the last point, you can only go backward.
At interior points, all three methods are available, but the question will specify which to use.
Pay close attention to which method the exam question asks for. Using the wrong one is a common error.
Data:
Point | Position (m) | Temperature (°C) |
|---|---|---|
4 | 0.04 | 82.6 |
5 | 0.05 | 94.5 |
Forward difference at Point 4:
$$\frac{dT}{dx}\bigg|_4 = \frac{T_5 - T_4}{x_5 - x_4} = \frac{94.5 - 82.6}{0.05 - 0.04} = \frac{11.9}{0.01} = 1190 \text{ °C/m}$$
Apply Fourier's law with $k = 0.053$ kW/(m·°C) and $A = 0.73$ m²:
$$q = 0.053 \times 0.73 \times 1190 = 46.04 \text{ kW}$$
Formula | Use |
|---|---|
$\delta f = \sqrt{\sum\left(\frac{\partial f}{\partial x_i}\delta x_i\right)^2}$ | General RSS error propagation |
$\delta R_T = \sqrt{\sum \delta R_i^2}$ | Error for a simple sum |
$\frac{dT}{dx}\bigg|i \approx \frac{T{i+1} - T_i}{x_{i+1} - x_i}$ | Forward finite difference |
$q = kA\left(\frac{dT}{dx}\right)$ | Fourier's law of thermal conduction |
Remember: angle uncertainties must be in radians when derivatives involve trig functions.
⚠️ Always convert angle uncertainties from degrees to radians before plugging into derivative-based error propagation.
⚠️ For a simple sum ($f = x_1 + x_2 + \ldots$), the partial derivatives are all 1, so the formula reduces to RSS of the raw uncertainties.
⚠️ "Exact" values (like $F_p = 18$ N exactly) have zero uncertainty. Do not invent an uncertainty for them.
⚠️ Forward difference uses points $i$ and $i+1$. Backward uses $i$ and $i-1$. Central uses $i+1$ and $i-1$. Read carefully which the question specifies.
⚠️ The derivative of $\tan(\theta)$ is $\sec^2(\theta)$, not $1/\cos(\theta)$. This is a frequent slip.
Q: Given $F_{\max} = F_p \tan(\theta)$, $F_p = 18$ N (exact), and $\theta = 36.3° \pm 0.5°$, what is the uncertainty of $F_{\max}$?
A: 0.24. Convert 0.5° to radians (0.008727), then compute $18 \times \sec^2(36.3°) \times 0.008727$.
Q: Three resistors in series have uncertainties of $\pm 7$, $\pm 10$, and $\pm 11$ ohms. What is the total propagated error?
A: $\sqrt{49 + 100 + 121} = \sqrt{270} = 16.43$ ohms.
Q: Using forward finite difference with $T_4 = 82.6°C$ at $x = 0.04$ m and $T_5 = 94.5°C$ at $x = 0.05$ m, plus $k = 0.053$ kW/(m·°C) and $A = 0.73$ m², what is $q$ at Point 4?
A: 46.04 kW. The temperature gradient is $(94.5 - 82.6) / 0.01 = 1190$ °C/m, then $q = 0.053 \times 0.73 \times 1190$.
Q: Why must angle uncertainties be converted to radians in error propagation?
A: Because the derivative of trigonometric functions (e.g. $d/d\theta[\tan\theta] = \sec^2\theta$) assumes the angle is in radians. If you leave the uncertainty in degrees, the propagated error will be off by a factor of $\pi/180$.
error propagation, uncertainty, propagation of uncertainty, RSS, root sum of squares, partial derivatives, sensitivity coefficients, independent errors, random errors, forward difference, backward difference, central difference, finite differences, numerical derivative, Fourier's law, thermal conduction, heat transfer, series resistors, static friction, ENGR 216, Texas A&M