Tags: Universal Accounting Equation, UAE, mass balance, open system, closed system, extensive, intensive, damped oscillations, simple pendulum, SHM, professional ethics, overconfidence bias, finite difference, ENGR 216, Texas A&M
This covers the remaining topics on the ENGR 216 final: the Universal Accounting Equation (mass and population balances, steady-state conditions, moisture-content problems), oscillatory systems (pendulums, damped systems, spring-board SHM), thermodynamic system classifications, and the professional ethics content from the lab guidelines. It also includes the forward finite difference method for velocity.
Universal Accounting Equation (UAE)
Accumulation = Input - Output + Generation - Consumption. The master balance equation for any extensive quantity.
Steady state
A condition where nothing accumulates over time. Accumulation = 0. If the quantity is also conserved, then Net Generation = 0 as well.
Conserved quantity
A quantity (like total mass) that cannot be created or destroyed. Generation and Consumption are both zero.
Closed system
A system that does not allow mass to cross its boundaries. Energy can cross, but mass cannot.
Open system
A system that allows both mass and energy to cross its boundaries.
Extensive quantity
A property that scales with the size of the system (e.g. mass, volume, enthalpy, energy). Double the system, double the value.
Intensive quantity
A property that does not depend on system size (e.g. temperature, pressure, density).
Enthalpy
A thermodynamic quantity equal to internal energy plus pressure times volume. It is an extensive quantity.
Damping ratio
Compares actual damping to critical damping. Critical damping coefficient: c_c = 2 sqrt(m k). If c < c_c, the system is underdamped. If c = c_c, critically damped. If c > c_c, overdamped.
Simple harmonic motion (SHM)
Periodic oscillation where the restoring force (or torque) is proportional to displacement. Angular frequency omega = sqrt(restoring constant / inertia).
Overconfidence bias
Accepting information that supports a pre-existing belief while rejecting information that disproves it.
Suppression of open communication
Occurs when an organisation treats bad news or criticism as negatives, discouraging honest reporting.
Forward finite difference
A numerical method for estimating derivatives. The forward difference formula for velocity: v_i = (s_{i+1} - s_i) / delta-t.
The UAE in its general form:
Accumulation = Input - Output + Generation - Consumption
or equivalently:
Final - Initial = (In - Out) + (Gen - Con)
At steady state for a conserved quantity:
Accumulation = 0 (nothing changes over time)
Net Generation = 0 (conserved means no creation or destruction)
Therefore: Input = Output
The INPUT and OUTPUT terms in the UAE represent extensive quantities (amounts), not intensive ones. This is tested as true/false (the statement that they are intensive is false).
Example: Booneville, population 3428 in 1980, population 2783 in 1990. Births = 178, deaths = 87.
Accumulation = Final - Initial = 2783 - 3428 = -645
Generation - Consumption = Births - Deaths = 178 - 87 = 91
Accumulation = (In - Out) + (Gen - Con)
-645 = (In - Out) + 91
In - Out = -736
Net migration out = 736 people
For drying problems, the key insight is that solids are conserved (they do not evaporate).
Example: 1000 kg of wet casein at 80% moisture, dried to 10% moisture. Drying cost = $2 per kg water removed.
Step 1, find the mass of solids:
Solids = 1000 * (1 - 0.80) = 200 kg
Step 2, find the final mass:
Final product is 10% moisture, so 90% solids
0.90 * M_final = 200
M_final = 222.22 kg
Step 3, find water removed:
Water removed = 1000 - 222.22 = 777.78 kg
Step 4, find drying cost:
Cost = 777.78 * $2 = $1555.56, approximately $1556
A closed system does not allow mass to cross its boundaries. This is a core definition tested as true/false. The statement "mass can cross the boundaries of a closed system" is false.
Enthalpy is extensive (scales with system size). Tested as true/false: the statement is true.
Classification uses the critical damping coefficient:
c_c = 2 sqrt(m k)
If c < c_c: underdamped (oscillates, but amplitude decays)
If c = c_c: critically damped (returns to equilibrium fastest, no oscillation)
If c > c_c: overdamped (returns slowly, no oscillation)
Example: m = 0.5 kg, k = 60 N/m, c = 10 kg/s.
c_c = 2 sqrt(0.5 60) = 2 * sqrt(30) = 10.95 kg/s
Since c = 10 < 10.95, the system is underdamped
Period formula:
T = 2 pi sqrt(L / g)
Example on Mars (g = 3.721 m/s^2, L = 1 m):
T = 2 pi sqrt(1 / 3.721) = 2 pi 0.5184 = 3.26 s
Example on the Moon (g = 1.62 m/s^2, L = 1 m):
T = 2 pi sqrt(1 / 1.62) = 2 pi 0.7857 = 4.94 s
For a uniform board pinned at one end with a spring at the other end:
Moment of inertia about the pin: I = (1/3) m L^2
Restoring torque from the spring: tau = -k L^2 theta
Torque constant (kappa): kappa = k * L^2
Angular frequency: omega = sqrt(kappa / I)
Period: T = 2 * pi / omega
Example: m = 12 kg, L = 4 m, k = 1300 N/m.
I = (1/3)(12)(16) = 64 kg m^2
kappa = 1300 * 16 = 20800 Nm/rad
omega = sqrt(20800 / 64) = sqrt(325) = 18.03 rad/s
T = 2 * pi / 18.03 = 0.35 s
Overconfidence bias
Accepting information that confirms what you already believe, while rejecting information that contradicts it. The exam specifically tests the scenario of an engineer who accepts supporting information but rejects disproving information.
Suppression of open communication
When an organisation's culture treats bad news or criticism as negatives. People stop reporting problems, which leads to hidden risks. The exam tests this as a true/false: the statement is true.
These are not the same thing. Overconfidence bias is an individual cognitive error. Suppression of open communication is an organisational culture problem.
The forward finite difference formula for the first derivative (velocity) at time t_i:
v_i = (s_{i+1} - s_i) / delta-t
This uses the current position and the next position. Compare with:
Backward difference: v = (s_i - s_{i-1}) / delta-t
Central difference: v = (s_{i+1} - s_{i-1}) / (2 * delta-t)
The exam specifically tests the forward difference formula.
UAE: Accumulation = Input - Output + Generation - Consumption
Critical damping: c_c = 2 sqrt(m k)
Pendulum period: T = 2 pi sqrt(L / g)
Board-spring SHM: omega = sqrt(k L^2 / I_pin), T = 2 pi / omega
Forward finite difference: v_i = (s_{i+1} - s_i) / delta-t
⚠️ At steady state with a conserved quantity, both Accumulation and Net Generation are zero. Not just one of them.
⚠️ UAE input/output terms are extensive, not intensive. This false statement is tested directly.
⚠️ Mass cannot cross the boundary of a closed system. This definition is tested as true/false.
⚠️ Enthalpy is extensive. Tested as true/false.
⚠️ For damped systems, compute c_c = 2sqrt(mk) first, then compare c to c_c. Do not guess from the numbers.
⚠️ Pendulum period on different planets: just swap g. The exam tests both Mars (3.721) and Moon (1.62) values.
⚠️ Overconfidence bias is not the same as suppression of open communication. The exam tests both, and you need to distinguish them.
⚠️ The forward difference uses s_{i+1} and s_i, not s_{i-1}. The exam offers all three difference formulas as options.
Q: At steady state, what are the Accumulation and Net Generation terms for a conserved quantity in the UAE?
A: Both are zero.
Q: True or false: mass can cross the boundaries of a closed system.
A: False.
Q: True or false: enthalpy is an extensive quantity.
A: True.
Q: A system has m = 0.5 kg, k = 60 N/m, and damping coefficient c = 10 kg/s. Is it underdamped, critically damped, or overdamped?
A: Underdamped. c_c = 2*sqrt(0.5 * 60) = 10.95, and 10 < 10.95.
Q: What is the period of a 1 m pendulum on the Moon (g = 1.62 m/s^2)?
A: T = 2pisqrt(1/1.62) = 4.94 s.
Q: An engineer accepts supporting evidence but rejects contradicting evidence. What bias is this?
A: Overconfidence bias.
Q: In the forward finite difference method, how is velocity at time t_i calculated?
A: v_i = (s_{i+1} - s_i) / delta-t.
Q: 1000 kg of wet casein at 80% moisture is dried to 10% moisture. How much dried product is produced?
A: 222 kg (200 kg of solids at 90% solid fraction gives 200/0.9 = 222.22 kg).
Q: True or false: suppression of open communication occurs when bad news is treated as a negative in an organisation.
A: True.
Universal Accounting Equation, UAE, mass balance, population balance, steady state, conserved quantity, accumulation, open system, closed system, extensive property, intensive property, enthalpy, damped oscillation, underdamped, overdamped, critically damped, critical damping coefficient, simple pendulum, period, simple harmonic motion, SHM, spring-board oscillation, overconfidence bias, confirmation bias, suppression of open communication, forward finite difference, numerical differentiation, moisture content, mass balance on solids, casein drying, ENGR 216, Texas A&M