Source: ENGR 216 Comprehensive Exam Practice Bank | Texas A&M University
Tags: dimensions, units, coherence, AES, American Engineering System, dimensional analysis, propagation of uncertainty, error propagation, RSS, root sum of squares, ENGR 216
Every physical equation must be dimensionally consistent, meaning the dimensions on the left side must match the right. The American Engineering System breaks "coherence" by defining force and mass independently, requiring a conversion factor g_c. When measurements carry uncertainty, that uncertainty propagates through calculations using partial derivatives and root-sum-of-squares (RSS).
Dimensional homogeneity
An equation is valid only if every term has the same net dimensions. You can solve for unknown constants by matching dimensions on both sides.
Coherent unit system
A system where the base units combine directly in equations without needing extra conversion factors. SI is coherent; AES is not.
American Engineering System (AES)
A non-coherent system that defines both the pound-mass (lbm) and pound-force (lbf) as independent base units. Because F = ma would not balance without help, the system introduces g_c.
g_c (gravitational conversion constant)
g_c = 32.174 lbm·ft / (lbf·s²). It patches the AES so that Newton's second law works: F = m·g / g_c.
Propagation of uncertainty (error propagation)
The method for calculating how measurement uncertainties feed through a formula to produce uncertainty in the result.
RSS (root sum of squares)
The standard combination rule for independent uncertainties: w_R = √( (∂R/∂x₁ · w₁)² + (∂R/∂x₂ · w₂)² + … )
When an equation contains an unknown constant, isolate it and solve for its dimensions.
Write out the dimensions of every known quantity.
Rearrange the equation to put the constant on one side.
Cancel and collect dimensions.
Worked example: power constant C
Given P = C · ω³ · D⁵
P has dimensions M·L² / T³
ω has dimensions 1/T, so ω³ = 1/T³
D has dimensions L, so D⁵ = L⁵
Solving for C:
C = P / (ω³ · D⁵) = (M·L² / T³) / (L⁵ / T³) = M / L³
C has dimensions of M / L³ (a density). It is not dimensionless.
In SI, 1 newton accelerates 1 kilogram at 1 m/s². The units slot together cleanly.
In AES, 1 lbf is defined as the weight of 1 lbm under standard gravity. That means F = ma does not balance on its own, because both sides would give "lbm·ft/s²" while force is measured in lbf.
The fix is g_c:
F = m · g / g_c
Worked example: weight of 50 lbm on Earth
F = (50 lbm × 32.174 ft/s²) / (32.174 lbm·ft/(lbf·s²)) = 50 lbf
The numerical value of g and g_c cancel here, but the units do not cancel without g_c. That dependence on a conversion factor is what makes the system non-coherent.
For a result R that depends on measured variables x₁, x₂, …, each with uncertainty w:
w_R = √( Σ (∂R/∂xᵢ · wᵢ)² )
Steps:
Find the partial derivative of R with respect to each measured variable.
Multiply each partial derivative by the corresponding uncertainty.
Square each product, sum them, take the square root.
F_max = μ_s · F_p, where μ_s = tan(θ)
So F_max = F_p · tan(θ)
F_p = 18 N (exact, so its uncertainty is zero). The only uncertain variable is θ.
∂F_max/∂θ = F_p · sec²(θ)
Convert uncertainty to radians: 0.5° × (π/180) = 0.008727 rad
Convert θ to radians: 36.3° = 0.6335 rad
sec²(0.6335) = 1 / cos²(0.6335) ≈ 1.539
∂F_max/∂θ = 18 × 1.539 ≈ 27.70 N/rad
w_F = 27.70 × 0.008727 ≈ 0.24 N
F_max = 18 × tan(36.3°) ≈ 13.22 N
Result: F_max = 13.2 ± 0.2 N
For resistors in series, R_total = R₁ + R₂ + R₃.
Every partial derivative ∂R_total/∂Rᵢ = 1, so:
R_total = 51 + 107 + 128 = 286 Ω
w_R = √(7² + 10² + 11²) = √(49 + 100 + 121) = √270 ≈ 16.4 Ω
Result: R_total = 286 ± 16 Ω
Dimensional analysis: isolate the unknown, cancel dimensions on both sides.
AES weight conversion: F = m · g / g_c, where g_c = 32.174 lbm·ft/(lbf·s²)
RSS uncertainty: w_R = √( Σ (∂R/∂xᵢ · wᵢ)² )
Derivative of tan(θ): sec²(θ). Always convert degrees to radians before differentiating.
⚠️ Dimensional analysis questions are common. If a constant "looks" dimensionless, prove it by working through the algebra, do not assume.
⚠️ AES problems almost always require you to show g_c explicitly. Stating "50 lbm = 50 lbf" without the conversion is incomplete.
⚠️ The single most common mistake in uncertainty propagation is forgetting to convert degrees to radians before taking derivatives or plugging into trig functions.
⚠️ When a quantity is stated as "exact," its uncertainty is zero, which eliminates its term from the RSS sum entirely.
Q: How do you determine the dimensions of an unknown constant in a physical equation?
A: Rearrange the equation to isolate the constant, substitute the known dimensions of every other quantity, then simplify. Whatever dimensions remain on the right-hand side are the dimensions of the constant.
Q: Why does the AES require g_c, and what is its value?
A: Because the AES defines lbm and lbf independently, Newton's second law does not balance without a conversion factor. g_c = 32.174 lbm·ft/(lbf·s²).
Q: In the welding failure problem, what would happen to the uncertainty if F_p also had an uncertainty of ±1 N?
A: You would add a second term to the RSS: (∂F_max/∂F_p · w_Fp)² = (tan(θ) · 1)². The total uncertainty would increase.
Q: For resistors in series, why does each partial derivative equal 1?
A: Because R_total = R₁ + R₂ + R₃ is a simple sum, and the derivative of a sum with respect to any one of its terms is 1.
dimensional analysis, unit consistency, homogeneity, coherent units, non-coherent units, AES, pound-mass, pound-force, lbm, lbf, g_c, gravitational constant, propagation of uncertainty, error propagation, RSS, root sum of squares, partial derivatives, series resistance, friction coefficient, welding static failure, ENGR 216, Texas A&M