Source: Midterm Make Up Assignment S26-4-1-1
Tags: error propagation, uncertainty, quadrature, partial derivatives, static friction, series resistors, root sum of squares, RSS, propagated error, experimental physics
When a measured quantity feeds into a formula, its uncertainty carries through to the result. For independent, random errors you propagate using partial derivatives and add uncertainties in quadrature (root sum of squares). The two cases below cover a single-variable chain rule propagation and a multi-variable linear sum.
Uncertainty (δx)
The estimated range within which the true value of a measurement is expected to fall. Often reported as ± after a measured value.
Error propagation
The process of determining how measurement uncertainties affect the uncertainty of a calculated result. Uses partial derivatives of the formula with respect to each measured variable.
Quadrature (root sum of squares, RSS)
The method of combining independent, random uncertainties: square each, sum them, take the square root. Written as δf = √(δf₁² + δf₂² + …).
Partial derivative
The derivative of a multivariable function with respect to one variable while holding all others constant. Central to the general propagation formula.
General propagation formula
For a result f(x₁, x₂, …, xₙ) with independent uncertainties δxᵢ:
δf = √[ Σ (∂f/∂xᵢ · δxᵢ)² ]
Coefficient of static friction (μs)
The ratio of maximum static friction force to normal force. Here defined as μs = tan(θ), where θ is the angle at which sliding begins.
When a formula depends on one uncertain variable, the propagation simplifies to:
δf = |df/dx| · δx
The key steps are:
Differentiate the formula with respect to the uncertain variable.
Evaluate the derivative at the measured value.
Multiply by the uncertainty of that variable, making sure the units are consistent (convert degrees to radians if differentiating trig functions).
Given: Fmax = Fp · tan(θ), where Fp = 18 N (exact) and θ = 36.3° ± 0.5°.
Step 1: Differentiate with respect to θ.
dFmax/dθ = Fp · sec²(θ)
Step 2: Convert the angle uncertainty to radians.
δθ = 0.5° × (π / 180) = 0.008727 rad
Step 3: Evaluate the derivative.
sec²(36.3°) = 1 / cos²(36.3°) ≈ 1.5395
dFmax/dθ = 18 × 1.5395 ≈ 27.71
Step 4: Multiply.
δFmax = 27.71 × 0.008727 ≈ 0.24
⚠️ The most common mistake here is forgetting to convert δθ from degrees to radians before multiplying.
For a result that is a simple sum of measured quantities, each partial derivative equals 1, so the formula becomes:
δR_T = √(δR₁² + δR₂² + δR₃²)
This only applies when the errors are independent and random. If errors were systematic (all biased the same way), you would add them directly instead.
Given: R1 = 51 ± 7 Ω, R2 = 107 ± 10 Ω, R3 = 128 ± 11 Ω, connected in series (R_T = R1 + R2 + R3).
δR_T = √(7² + 10² + 11²) = √(49 + 100 + 121) = √270 ≈ 16.43
Note: the total resistance itself is 286 Ω, but the question only asks for the propagated error.
One uncertain variable: single derivative, multiply by δx.
Multiple uncertain variables, linear sum: RSS of individual uncertainties (all partials = 1).
Multiple uncertain variables, nonlinear formula: full general formula with each partial derivative computed separately.
General propagation formula:
δf = √[ Σᵢ (∂f/∂xᵢ · δxᵢ)² ]
Single-variable case:
δf = |df/dx| · δx
Linear sum case (f = x₁ + x₂ + … + xₙ):
δf = √(δx₁² + δx₂² + … + δxₙ²)
Derivative of tan:
d/dθ [tan(θ)] = sec²(θ) = 1 / cos²(θ)
Degree-to-radian conversion:
θ_rad = θ_deg × π / 180
⚠️ Always convert angular uncertainties to radians before propagating through trig functions. This is the single most common error on these problems.
⚠️ "Independent and random" is your cue to use quadrature (RSS). If the problem does not state this, consider whether simple addition is required instead.
⚠️ When Fp (or any parameter) is stated as "exactly" some value, its uncertainty is zero, so it drops out of the propagation and acts as a constant multiplier.
⚠️ Read carefully whether the question asks for the uncertainty of the result or the result itself. These midterm questions ask only for the uncertainty.
Q: A force is given by F = mg sin(θ) where m = 5.0 kg (exact), g = 9.81 m/s² (exact), and θ = 28.0° ± 0.3°. What is the uncertainty in F?
A: dF/dθ = mg cos(θ). Convert δθ = 0.3° to radians = 0.005236 rad. dF/dθ = 5.0 × 9.81 × cos(28°) = 49.05 × 0.8829 = 43.31. δF = 43.31 × 0.005236 ≈ 0.23 N.
Q: Three measurements with uncertainties ±3, ±4, and ±5 are summed. What is the propagated uncertainty, assuming independent random errors?
A: δ = √(9 + 16 + 25) = √50 ≈ 7.07.
Q: Why do we add uncertainties in quadrature rather than directly for independent errors?
A: Because independent random errors are unlikely to all be at their maximum simultaneously. Quadrature reflects the statistical reality that some errors partially cancel. Direct addition gives the worst-case (systematic) bound.
Q: If f = A · tan(θ) and A has zero uncertainty, how many uncertain variables contribute to δf?
A: Only one (θ). The propagation reduces to |df/dθ| · δθ.
error propagation, uncertainty propagation, quadrature, root sum of squares, RSS, partial derivatives, propagated error, series resistors, total resistance uncertainty, coefficient of static friction, tan uncertainty, sec squared, degree to radian conversion, independent random errors, ENGR lab midterm, experimental physics, Texas A&M