Mechanical Engineering Basics

This deck gathers together the core concepts that every mechanical engineer encounters early in their studies. The cards move through the fundamentals of statics — equilibrium, free body diagrams, supports, and moments — before stepping into dynamics topics like Newton's laws, momentum, and the work-energy theorem. Together, these questions form the foundation you'll lean on in nearly every later course, from machine design to fluid mechanics.

It's a great fit if you're a first- or second-year engineering student working through introductory mechanics, or if you're returning to these topics after a break and want a quick refresher before a more advanced class. The questions stay at a definitional and conceptual level, so they're also useful for exam prep, interview warm-ups, or anyone curious about how engineers reason about forces and motion.

Because many of these ideas build on each other — for example, understanding supports leads into equilibrium, which leads into moments — it's worth reviewing the cards regularly rather than trying to push through them in one sitting. Spacing your sessions across a few days helps the definitions settle into long-term memory, and pairing each card with a quick sketch or example problem on paper will make the concepts stick far better than memorization alone.

Statics is the branch of mechanics that analyzes bodies at rest or in equilibrium. The first condition of equilibrium states that the sum of all external forces acting on a body must equal zero, written as \(\Sigma F = 0\). The second condition requires that the sum of all moments about any chosen point also equals zero, \(\Sigma M = 0\). Together, these two conditions provide the three scalar equations (in two dimensions) or six (in three dimensions) needed to solve for unknown forces and reactions in a structure.

To apply these equations, engineers draw a free body diagram (FBD), which is a sketch of the body isolated from its surroundings with all external forces and moments clearly indicated. When the reactions of a structure can be found using the equations of equilibrium alone, the structure is called statically determinate. The type of support influences the number and direction of reaction forces available. A roller support provides only one reaction force normal to the surface on which it rests, while a pin support provides two reaction forces, both horizontal and vertical.

A key concept in statics is the moment of a force about a point, defined as \(M = F \times d\), where \(F\) is the force magnitude and \(d\) is the perpendicular distance from the point to the line of action of the force. Two equal, opposite, and non-collinear forces form a couple, which produces a pure rotational effect with no net translational force. The centroid of a body is the geometric center where the body would balance perfectly if supported there. The moment of inertia, expressed as \(I = \int y^2\,dA\), measures a cross-section's resistance to bending, and the parallel axis theorem \(I = I_c + A d^2\) allows the moment of inertia about any parallel axis to be found from the value about the centroidal axis.

Dynamics extends statics by considering bodies in motion and the forces that cause that motion. The foundation is built on Newton's three laws of motion. The first law states that a body remains at rest or in uniform motion unless acted upon by a net external force. The second law relates the net force to mass and acceleration through \(F = ma\). The third law asserts that for every action there is an equal and opposite reaction, \(F_{12} = -F_{21}\).

Two important theorems follow from Newton's laws. The work-energy theorem states that the net work done on a body equals its change in kinetic energy, \(W_{net} = \Delta KE = \tfrac{1}{2}mv_2^2 - \tfrac{1}{2}mv_1^2\). The impulse-momentum theorem states that the impulse (force multiplied by time) equals the change in momentum, \(F\Delta t = \Delta(mv)\), where linear momentum itself is defined as \(p = mv\). It is important to distinguish kinematics, which describes motion without regard to its causes, from kinetics, which relates forces to the resulting motion.

For rotational motion, angular velocity \(\omega = d\theta/dt\) is the rate of change of angular displacement, measured in rad/s. A body moving along a circular path experiences centripetal acceleration directed toward the center, given by \(a_c = v^2/r = \omega^2 r\). D'Alembert's principle provides a useful bridge between dynamics and statics: by introducing an inertial force equal to \(-ma\) into a dynamic system, the problem can be analyzed as if it were in static equilibrium.

Thermodynamics governs the relationships among heat, work, temperature, and energy. The Zeroth Law establishes the concept of temperature by stating that if two systems are each in thermal equilibrium with a third, they are in thermal equilibrium with each other. The First Law expresses conservation of energy: for a closed system, the change in internal energy equals the heat added minus the work done, \(\Delta U = Q - W\). The Second Law has two commonly cited statements. The Clausius statement says that heat cannot spontaneously flow from a colder body to a hotter body without external work, while the Kelvin-Planck statement says that no heat engine can convert all absorbed heat into work; some heat must always be rejected to a cold reservoir. The Third Law states that the entropy of a perfect crystal approaches zero as the temperature approaches absolute zero.

Two derived properties are particularly useful. Enthalpy, defined as \(H = U + PV\), combines internal energy with flow work and is convenient for analyzing open systems such as turbines and heat exchangers. Entropy is a measure of molecular disorder or energy dispersal, defined for a reversible process as \(dS = \delta Q_{rev}/T\). The Carnot efficiency, \(\eta = 1 - T_{cold}/T_{hot}\) (with temperatures in Kelvin), gives the maximum possible efficiency of any heat engine operating between two thermal reservoirs.

Thermodynamic processes are often idealized. An adiabatic process involves no heat transfer with the surroundings (\(Q = 0\)), while an isothermal process occurs at constant temperature. Specific heat capacity, \(c = Q/(m\Delta T)\), specifies the heat required to raise the temperature of a unit mass by one degree. For ideal gases, the behavior is captured by the ideal gas law, \(PV = nRT\), where \(P\) is pressure, \(V\) is volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is absolute temperature.

Fluid mechanics is concerned with the behavior of liquids and gases. Density \(\rho = m/V\), measured in kg/m³, is the most fundamental property. Pascal's law states that pressure applied to a confined fluid is transmitted equally in all directions throughout the fluid, a principle that underlies hydraulic systems. Archimedes' principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid displaced.

Two cornerstone equations govern the flow of fluids. The continuity equation for incompressible flow, \(A_1 v_1 = A_2 v_2\), shows that the product of cross-sectional area and velocity is constant along a streamline. Bernoulli's equation expresses energy conservation along a streamline for steady, incompressible, inviscid flow: \(P + \tfrac{1}{2}\rho v^2 + \rho g h =\) constant. Together, these equations explain many practical phenomena, from the lift on an aircraft wing to the flow through a Venturi meter.

The Reynolds number \(Re = \rho v D / \mu\) is a dimensionless quantity that predicts whether flow is laminar or turbulent. For pipe flow, the transition typically occurs at a critical Reynolds number of about 2300. Laminar flow is characterized by smooth, orderly layers, while turbulent flow involves chaotic fluctuations and vigorous mixing. Viscosity \(\mu\) is the fluid property that quantifies internal resistance to flow, with the shear stress related to the velocity gradient by \(\tau = \mu (dv/dy)\).

Mechanics of materials studies how solids deform and fail under load. Stress is force per unit area, \(\sigma = F/A\), measured in Pascals (Pa). Strain is the ratio of deformation to original length, \(\varepsilon = \Delta L / L_0\), and is dimensionless. Within the elastic limit, stress is proportional to strain according to Hooke's law, \(\sigma = E \varepsilon\), where \(E\) is Young's modulus, a measure of a material's stiffness.

Other elastic constants describe different deformation modes. Shear stress \(\tau = V/A\) acts parallel to a surface, and the shear modulus \(G = \tau / \gamma\) is the ratio of shear stress to shear strain \(\gamma\). Poisson's ratio \(\nu = -\varepsilon_{lateral}/\varepsilon_{axial}\) captures the lateral contraction that accompanies axial stretching; for most metals it lies between 0.25 and 0.35. These three constants are not independent but are related by \(G = E / [2(1+\nu)]\).

The strength of a material is described by several characteristic stresses. The yield strength is the stress at which the material begins to deform plastically (permanently). The ultimate tensile strength (UTS) is the maximum stress a material can withstand before necking begins. Engineers apply a factor of safety, \(FoS = \sigma_{failure}/\sigma_{actual}\), to ensure designs remain well below these limits. Real components can fail in other ways as well: fatigue failure occurs under repeated cyclic loading at stresses below the static UTS, while creep is the time-dependent permanent deformation of a material under constant stress at elevated temperature. Thermal expansion causes dimensions to change with temperature according to \(\Delta L = \alpha L_0 \Delta T\), where \(\alpha\) is the coefficient of thermal expansion.

Heat transfer is the movement of thermal energy driven by temperature differences. There are three modes of heat transfer. Conduction transfers heat through a solid by molecular and electronic interactions. Convection transfers heat through the bulk motion of a fluid, either naturally or forced. Radiation transfers energy as electromagnetic waves and does not require a medium.

Each mode is governed by its own law. Fourier's law of conduction states that heat flux is proportional to the negative temperature gradient, \(q = -k(dT/dx)\), where \(k\) is the thermal conductivity of the material. Metals have high \(k\) and are good conductors, while insulators such as foams and ceramics have low \(k\). Newton's law of cooling expresses the convective heat transfer rate as \(Q = hA(T_s - T_{\infty})\), where \(h\) is the convection coefficient that depends on the fluid and the flow conditions. The Stefan-Boltzmann law describes radiation from a black body as \(Q = \sigma \varepsilon A T^4\), where \(\sigma = 5.67 \times 10^{-8}\) W/m²K⁴ and \(\varepsilon\) is the emissivity of the surface.

In practical engineering, multiple resistances often act in series. The overall heat transfer coefficient \(U\) combines conduction, convection, and fouling resistances into a single value, and the heat transfer rate becomes \(Q = UA\Delta T\). For heat exchangers, the log mean temperature difference, \(LMTD = (\Delta T_1 - \Delta T_2)/\ln(\Delta T_1/\Delta T_2)\), provides the appropriate driving temperature difference between the two fluid streams, accounting for the fact that this difference usually varies from one end of the exchanger to the other.

Machine design involves selecting and sizing components to transmit motion and force safely. A keyway is a slot machined into a shaft and hub to receive a key that transmits torque between them. A flywheel stores rotational kinetic energy to smooth out fluctuations in angular velocity during a machine cycle. The gear ratio \(GR = N_2/N_1 = \omega_1/\omega_2\) determines how rotational speed and torque are traded between meshing gears. Bearings reduce friction and support loads between moving parts; journal bearings rely on a fluid film of sliding contact, while rolling-element bearings use balls or rollers to minimize friction.

Springs are characterized by a stiffness \(k = F/\delta\). When springs combine in series, the equivalent stiffness obeys \(1/k_{eq} = 1/k_1 + 1/k_2 + \ldots\) and is softer than any individual spring, while in parallel the equivalent stiffness is \(k_{eq} = k_1 + k_2 + \ldots\) and is stiffer than any individual spring. The critical speed of a shaft is the rotational speed at which its natural frequency of vibration matches the operating speed, leading to dangerous resonance. Power screws convert rotary motion into linear motion and are used in jacks and presses. The mechanical advantage of a lever, \(MA = F_{out}/F_{in} = d_{in}/d_{out}\), relates output force to the ratio of lever arm distances.

Rotational quantities play a central role: torque \(T = F \times r\) is the rotational equivalent of force, and power is related to torque and angular velocity by \(P = T\omega\). Two key formulas govern the stress state in common machine elements. The torsion formula \(\tau = Tr/J\) gives the shear stress in a circular shaft, where \(J\) is the polar moment of inertia. The flexure formula \(\sigma = My/I\) gives the bending stress in a beam, where \(M\) is the bending moment, \(y\) is the distance from the neutral axis, and \(I\) is the moment of inertia of the cross-section.

Manufacturing processes shape raw materials into finished parts. Casting pours molten metal into a mold to solidify into the desired shape. Forging shapes metal using compressive forces, refining its grain structure. Hot working is performed above the recrystallization temperature, while cold working is done below it and increases hardness through strain hardening. Turning on a lathe rotates a workpiece against a stationary cutting tool to produce cylindrical shapes, while milling uses a rotating multi-point cutter to remove material from a stationary workpiece. Welding joins materials by applying heat, pressure, or both. MIG (GMAW) uses a consumable wire electrode and is faster, while TIG (GTAW) uses a non-consumable tungsten electrode for higher precision. Powder metallurgy compacts metal powder in a die and sinters it below the melting point, while injection molding forces molten plastic into a mold cavity under high pressure.

Material selection depends on the required properties. Hardness measures resistance to localized plastic deformation (indentation or scratching) and is commonly measured by Brinell, Rockwell, or Vickers tests. Brittle fracture occurs suddenly with little plastic deformation, whereas ductile fracture involves substantial plastic deformation before breaking. An alloy is a mixture of two or more elements, at least one a metal, designed to achieve improved mechanical or chemical properties, and ferrous metals are those in which iron is the primary element, such as steel and cast iron.

Finally, all mechanical engineering calculations rely on a consistent set of units. The SI system defines the Newton (N) as the unit of force (1 kg·m/s²), the Pascal (Pa) as the unit of pressure (1 N/m²), the Joule (J) as the unit of energy and work (1 N·m), and the Watt (W) as the unit of power (1 J/s). Common conversions include 1 bar = 100,000 Pa, 1 N ≈ 0.2248 lbf, 1 hp ≈ 745.7 W, 1 atm = 101,325 Pa, 1 BTU ≈ 1055.06 J, and the temperature shift \(K = °C + 273.15\).