Principles and Practice of Engineering Exam (PE) Mechanical Engineering (PE Mechanical) Syllabus

State the hydrostatic pressure equation for a fluid at rest.

$$P = P_0 + \rho g h$$ Gauge pressure increases linearly with depth $h$; $\rho$ is fluid density and $g$ is gravitational acceleration. Pressure is independent of container shape.

State Archimedes' principle and the buoyant force expression.

A body immersed in a fluid experiences an upward buoyant force equal to the weight of displaced fluid: $$F_B = \rho_{fluid}\, g\, V_{disp}$$

Write the continuity equation for steady incompressible flow in a pipe.

$$A_1 V_1 = A_2 V_2 = Q$$ (volumetric flow rate constant). Generally, mass conservation gives $\dot{m} = \rho A V = \text{constant}$.

State Bernoulli's equation for steady, incompressible, inviscid flow along a streamline.

$$\frac{P}{\rho g} + \frac{V^{2}}{2g} + z = \text{constant}$$ The terms are pressure head, velocity head, and elevation head; total head is conserved with no losses.

Define the Reynolds number and give the typical laminar/turbulent transition value for internal pipe flow.

$$Re = \frac{\rho V D}{\mu} = \frac{V D}{\nu}$$ For pipe flow: laminar if $Re < 2300$, turbulent if $Re > 4000$, transitional in between.

Write the Darcy-Weisbach equation for head loss due to friction in a pipe.

$$h_f = f\frac{L}{D}\frac{V^{2}}{2g}$$ where $f$ is the Darcy friction factor, $L$ pipe length, $D$ diameter, and $V$ mean velocity.

Give the friction factor for fully developed laminar pipe flow.

$$f = \frac{64}{Re}$$ Valid for $Re < 2300$. For turbulent flow, $f$ is obtained from the Moody chart or the Colebrook equation.

List the three modes of heat transfer and the governing law for each.

Conduction (Fourier's law), convection (Newton's law of cooling), and radiation (Stefan-Boltzmann law). Conduction needs a medium; radiation requires none.

Write Fourier's law of heat conduction (1-D) and define each term.

$$q = -kA\frac{dT}{dx}$$ $q$ is heat rate, $k$ thermal conductivity, $A$ area, and $dT/dx$ the temperature gradient. The negative sign indicates heat flows down the gradient.

Write Newton's law of cooling for convection.

$$q = hA(T_s - T_\infty)$$ where $h$ is the convection coefficient, $A$ surface area, $T_s$ surface temperature, and $T_\infty$ the bulk fluid temperature.

Write the Stefan-Boltzmann law for radiation from a real (gray) surface.

$$q = \varepsilon \sigma A (T_s^{4} - T_{surr}^{4})$$ where $\varepsilon$ is emissivity, $\sigma = 5.67\times10^{-8}\ \text{W/m}^2\text{K}^4$, and temperatures are absolute.

Define thermal resistance for conduction through a plane wall and for convection.

Conduction: $$R_{cond} = \frac{L}{kA}$$ Convection: $$R_{conv} = \frac{1}{hA}$$ Resistances in series add: total $q = \Delta T / R_{total}$.

Define the Biot number and state what its magnitude tells you in transient conduction.

$$Bi = \frac{hL_c}{k}$$ If $Bi < 0.1$, internal conduction resistance is negligible and the lumped-capacitance method (uniform body temperature) is valid.

Define the Nusselt, Prandtl, and Grashof numbers used in convection.

Nusselt: $Nu = hL/k$ (dimensionless convection coefficient). Prandtl: $Pr = \nu/\alpha = c_p\mu/k$ (momentum vs thermal diffusivity). Grashof: $Gr = g\beta\Delta T L^3/\nu^2$ (buoyancy vs viscous forces, natural convection).

Compare parallel-flow and counter-flow heat exchangers in terms of effectiveness.

Counter-flow (fluids move in opposite directions) is more effective: it maintains a more uniform $\Delta T$, can raise the cold-fluid outlet above the hot-fluid outlet, and gives a larger LMTD than parallel-flow for the same terminal temperatures.

Write the heat exchanger rate equation using the log mean temperature difference (LMTD).

$$q = UA\,\Delta T_{lm}, \qquad \Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)}$$ where $U$ is the overall heat transfer coefficient and $A$ the area.

Define heat exchanger effectiveness $\varepsilon$ in the NTU method.

$$\varepsilon = \frac{q}{q_{max}} = \frac{q}{C_{min}(T_{h,in} - T_{c,in})}$$ where $C_{min}$ is the smaller of the two heat-capacity rates $\dot{m}c_p$. NTU $= UA/C_{min}$.

Define normal (axial) stress and shear stress.

Normal stress: $$\sigma = \frac{P}{A}$$ (force perpendicular to area). Shear stress: $$\tau = \frac{V}{A}$$ (force parallel to area). Units are typically Pa or psi.

State Hooke's law in 1-D and define Young's modulus.

$$\sigma = E\,\varepsilon$$ where $E$ is Young's (elastic) modulus and $\varepsilon = \delta/L$ is normal strain. Valid in the linear-elastic region below the proportional limit.

Write the flexure (bending stress) formula for a beam.

$$\sigma = \frac{Mc}{I}$$ where $M$ is bending moment, $c$ the distance from the neutral axis to the outer fiber, and $I$ the area moment of inertia of the cross section.

Write the torsion shear stress formula for a circular shaft.

$$\tau = \frac{Tr}{J}$$ where $T$ is torque, $r$ radial distance, and $J$ the polar moment of inertia. For a solid circular shaft, $J = \pi d^4/32$.

Give the principal stresses for a 2-D plane stress state.

$$\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}$$ These are the maximum/minimum normal stresses where shear is zero (Mohr's circle extremes).

State the maximum shear stress (Tresca) and distortion energy (von Mises) failure criteria.

Tresca: yielding when $\tau_{max} = \sigma_y/2$, i.e. $\sigma_1 - \sigma_3 = \sigma_y$. Von Mises: yielding when $$\sigma' = \sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2} = \sigma_y$$ Von Mises is less conservative and more accurate for ductile metals.

Write the Euler buckling load formula for a column.

$$P_{cr} = \frac{\pi^{2} E I}{(KL)^{2}}$$ where $K$ is the effective length factor (1 for pinned-pinned, 0.5 fixed-fixed, 2 fixed-free) and $L$ the column length.

What is a key-and-keyway used for, and how do bolted vs welded joints differ in load transfer?

A key transmits torque between a shaft and a hub, preventing relative rotation. Bolted joints transfer load by friction/shear and are removable; welded joints are permanent and transfer load through the fused metal (continuous load path).

For a helical compression spring, write the spring rate (stiffness) formula.

$$k = \frac{G d^{4}}{8 D^{3} N}$$ where $G$ is shear modulus, $d$ wire diameter, $D$ mean coil diameter, and $N$ the number of active coils. Force $F = k\delta$.

Define the AFBMA/bearing basic dynamic load relationship between life and load for rolling bearings.

$$L = \left(\frac{C}{P}\right)^{a}$$ where $L$ is rated life (millions of revolutions), $C$ the basic dynamic load rating, $P$ the equivalent load, and $a = 3$ for ball bearings, $10/3$ for roller bearings.

Differentiate ductile and brittle materials by their stress-strain behavior.

Ductile materials (e.g., mild steel) exhibit large plastic deformation, a distinct yield point, and necking before fracture (high $\%$ elongation). Brittle materials (e.g., cast iron, ceramics) fracture with little plastic deformation near the ultimate strength.

Define yield strength, ultimate tensile strength, and the modulus of resilience.

Yield strength: stress at onset of plastic deformation (often $0.2\%$ offset). Ultimate strength: maximum stress on the engineering stress-strain curve. Modulus of resilience: elastic strain energy per unit volume $= \sigma_y^2/(2E)$.

Define Poisson's ratio and give the relation between $E$, $G$, and $\nu$ for isotropic materials.

Poisson's ratio: $$\nu = -\frac{\varepsilon_{lateral}}{\varepsilon_{axial}}$$ Relation: $$G = \frac{E}{2(1+\nu)}$$ Typical metals have $\nu \approx 0.3$.

Define the endurance (fatigue) limit and the significance of fatigue in cyclic loading.

The endurance limit is the cyclic stress amplitude below which a material (notably steels) can endure infinite cycles without fatigue failure, seen as a horizontal asymptote on the S-N curve. Many nonferrous metals have no true endurance limit.

Write the equation of motion for an undamped single-DOF free vibration and its natural frequency.

$$m\ddot{x} + kx = 0, \qquad \omega_n = \sqrt{\frac{k}{m}}\ \text{(rad/s)}, \quad f_n = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$$

Define the damping ratio $\zeta$ and classify under-, critically, and over-damped systems.

$$\zeta = \frac{c}{c_c} = \frac{c}{2\sqrt{km}}$$ $\zeta < 1$ underdamped (oscillatory decay); $\zeta = 1$ critically damped (fastest non-oscillatory return); $\zeta > 1$ overdamped (slow, no oscillation).

What is resonance, and what is the resonance condition for a forced vibration?

Resonance is the large-amplitude response when the forcing frequency $\omega$ approaches the natural frequency $\omega_n$ ($\omega/\omega_n \to 1$). For an undamped system the amplitude grows without bound; damping limits the peak.

State Newton's second law in rotational form for a rigid body.

$$\sum M = I\alpha$$ The net torque (moment) about an axis equals the mass moment of inertia $I$ times angular acceleration $\alpha$. Linear analog: $\sum F = ma$.

Define accuracy vs precision in measurement.

Accuracy is closeness of a measurement to the true value (low systematic/bias error). Precision is repeatability—closeness of repeated measurements to each other (low random error). A device can be precise but inaccurate.

Differentiate systematic (bias) error from random error.

Systematic error is consistent and repeatable, shifting all readings in one direction (e.g., miscalibration); it can be corrected. Random error fluctuates unpredictably about the mean and is reduced by averaging multiple readings.

What is a thermocouple and what principle does it operate on?

A thermocouple measures temperature using the Seebeck effect: two dissimilar metals joined at a junction produce a temperature-dependent voltage (EMF) proportional to the temperature difference between the measuring and reference junctions.

How does a strain gauge measure strain, and what is the gauge factor?

A strain gauge changes electrical resistance as it deforms. Gauge factor: $$GF = \frac{\Delta R/R}{\varepsilon}$$ It is typically read with a Wheatstone bridge to detect small resistance changes.

Define the transfer function of a linear time-invariant control system.

The transfer function is the ratio of the Laplace transform of the output to that of the input at zero initial conditions: $$G(s) = \frac{Y(s)}{X(s)}$$ Its poles and zeros determine system stability and response.

Compare open-loop and closed-loop (feedback) control systems.

Open-loop: control action is independent of output—no feedback, simpler, but no error correction (e.g., a timer). Closed-loop: output is fed back and compared to the setpoint; the error drives the controller, improving accuracy and disturbance rejection.

Describe the three terms of a PID controller and what each contributes.

$$u(t) = K_p e + K_i\int e\,dt + K_d\frac{de}{dt}$$ Proportional ($K_p$): reacts to current error. Integral ($K_i$): eliminates steady-state offset. Derivative ($K_d$): anticipates error trend, improving damping/stability.