Mechanics formulas — 135 solvers

Acceleration Down a Frictionless Incline

a = g \sin\theta

MechanicsPhysicsAcceleration of an object sliding freely down a frictionless incline, set only by gravity and the slope angle.

Acceleration Down an Incline with Friction

a = g\left(\sin\theta - \mu_k \cos\theta\right)

MechanicsPhysicsAcceleration of an object sliding down an incline once kinetic friction opposes the motion, from the slope angle and μₖ.

Active Thrust on a Retaining Wall

P_a = \tfrac{1}{2}\,K_a\,\gamma\,H^{2}

Soil MechanicsMechanicsTotal Rankine active thrust per unit length of wall from a dry cohesionless backfill, acting at one third of the wall height above the base.

Angle of Repose (μ = tan θ)

\mu_s = \tan\theta

MechanicsPhysicsSteepest angle a surface can be tilted before an object slides, where the coefficient of static friction equals tan θ.

Angle of Twist (φ = TL/JG)

\varphi = \frac{T L}{J G}

Strength of MaterialsMechanicsPhysicsAngle of twist of a round shaft under torque, φ = TL/JG, the stiffness check that governs long drive and torque shafts.

Angular Acceleration

\alpha = \frac{\omega - \omega_0}{t}

MechanicsPhysicsAverage angular acceleration: the change in angular velocity divided by the time taken.

Angular Displacement (θ = ω₀t + ½αt²)

\theta = \omega_0 t + \tfrac{1}{2} \alpha t^{2}

MechanicsPhysicsAngle turned under constant angular acceleration, the rotational twin of x = v₀t + ½at².

Angular Momentum (L = Iω)

L = I \omega

MechanicsPhysicsAngular momentum of a rotating body: moment of inertia times angular velocity.

Angular Velocity (ω = θ/t)

\omega = \frac{\theta}{t}

MechanicsPhysicsAverage angular velocity: the angle swept divided by the time taken.

Angular Velocity from Period

\omega = \frac{2\pi}{T}

MechanicsWaves & OscillationsPhysicsOne full revolution is 2π radians, so angular velocity is 2π divided by the period.

Area Moment of Inertia — Rectangle

I = \frac{b h^{3}}{12}

Strength of MaterialsMechanicsGeometryArea moment of inertia of a rectangle about its centroidal axis, I = bh³/12, returned in m⁴ and entered as a plain number.

Area Moment of Inertia — Solid Round Bar

I = \frac{\pi d^{4}}{64}

Strength of MaterialsMechanicsGeometryArea moment of inertia of a solid round bar about a diameter, I = πd⁴/64, returned in m⁴ and entered as a plain number.

At-Rest Earth Pressure Coefficient (Jaky)

K_0 = 1 - \sin\phi

Soil MechanicsMechanicsJaky's 1944 empirical coefficient of earth pressure at rest for a normally consolidated soil that is not permitted to strain laterally.

Atwood Machine Acceleration

a = \frac{\left(m_1 - m_2\right) g}{m_1 + m_2}

MechanicsPhysicsAcceleration of an Atwood machine — two masses joined by a rope over a frictionless pulley, driven by their difference.

Average Shear Stress (τ = V/A)

\tau = \frac{V}{A}

Strength of MaterialsMechanicsPhysicsAverage shear stress on a bolt, pin or weld throat: the transverse force divided by the area resisting it, in Pa or psi.

Axial Deformation (δ = PL/AE)

\delta = \frac{P L}{A E}

Strength of MaterialsMechanicsPhysicsElongation of an axially loaded bar from load, length, area and Young's modulus — the workhorse δ = PL/AE of hanger design.

Banked Curve Angle

\theta = \arctan\!\left(\frac{v^{2}}{r g}\right)

MechanicsPhysicsBank angle that lets a vehicle round a curve of a given radius at a given speed with no reliance on sideways friction.

Barlow's Formula (Pipe Pressure Rating)

P = \frac{2 S t}{D}

HVAC & HydronicsFluid MechanicsMechanicsInternal pressure a pipe can hold from wall stress, wall thickness and outside diameter — the thin-wall hoop-stress relation used by pipeline codes.

Basquin S-N Relation

\sigma_a = \sigma_f' \, (2N_f)^{b}

Strength of MaterialsMechanicsThe straight line an S-N curve becomes on log-log paper: alternating stress against reversals to failure, with the fatigue strength coefficient and exponent from a materials database.

Beam Deflection — Simply Supported, Centre Load

\delta = \frac{P L^{3}}{48 E I}

Strength of MaterialsMechanicsPhysicsMaximum midspan deflection of a simply supported beam with a central point load, δ = PL³/48EI, with I entered in m⁴.

Beam Deflection — Simply Supported, Uniform Load

\delta = \frac{5 w L^{4}}{384 E I}

Strength of MaterialsMechanicsPhysicsMaximum midspan deflection of a simply supported beam under a uniform load, δ = 5wL⁴/384EI, with I entered in m⁴.

Bearing Capacity Factor Nc

N_c = (N_q - 1)\cot\phi

Soil MechanicsMechanicsPrandtl's cohesion bearing capacity factor Nc derived from Nq and the friction angle, tending to 5.14 as the friction angle goes to zero.

Bearing Capacity Factor Nq

N_q = e^{\pi\tan\phi}\,\tan^{2}\!\left(45^\circ + \frac{\phi}{2}\right)

Soil MechanicsMechanicsPrandtl–Reissner surcharge bearing capacity factor Nq from the friction angle, the value tabulated by Meyerhof, Hansen and Vesic.

Bending Stress (σ = Mc/I)

\sigma = \frac{M c}{I}

Strength of MaterialsMechanicsPhysicsBending stress at a distance c from the neutral axis of a beam, with the area moment of inertia I entered in m⁴.

Bending Stress from Section Modulus (σ = M/S)

\sigma = \frac{M}{S}

Strength of MaterialsMechanicsPhysicsBending stress straight from the moment and a tabulated section modulus S in m³, the everyday form used with steel tables.

Bolt Preload from Torque (T = KDF)

T = K D F

Strength of MaterialsMechanicsPhysicsBolt preload from tightening torque using the nut factor K, T = KDF, the field method behind every published torque spec.

Bounce Height from Coefficient of Restitution

h_2 = e^{2} h_1

MechanicsPhysicsHeight a dropped ball rebounds to, from the drop height and the coefficient of restitution of the bounce.

Bulk Modulus (K = ΔP·V₀/ΔV)

K = \frac{\Delta P \, V_0}{\Delta V}

Strength of MaterialsMechanicsPhysicsBulk modulus from the pressure rise and the volume change it produces; water sits near 2.2 GPa and hydraulic oil near 1.5 GPa.

Cantilever Deflection — End Load

\delta = \frac{P L^{3}}{3 E I}

Strength of MaterialsMechanicsPhysicsTip deflection of a cantilever carrying a point load at its free end, δ = PL³/3EI, with I entered in m⁴ as a plain number.

Cantilever Deflection — Uniform Load

\delta = \frac{w L^{4}}{8 E I}

Strength of MaterialsCivil & SurveyingMechanicsTip deflection of a cantilever carrying a uniformly distributed load along its whole length, δ = wL⁴/8EI — three-eighths of the sag the same total load would cause at the tip.

Centripetal Acceleration (a = v²/r)

a_c = \frac{v^2}{r}

MechanicsPhysicsInward acceleration of an object moving in a circle at constant speed.

Centripetal Acceleration (a = ω²r)

a_c = \omega^{2} r

MechanicsPhysicsCentripetal acceleration written in terms of angular velocity rather than linear speed.

Centripetal Force (F = mv²/r)

F_c = \frac{m v^2}{r}

MechanicsPhysicsNet inward force required to keep a mass moving in a circle at constant speed.

Coefficient of Restitution

e = \frac{v_2 - v_1}{u_1 - u_2}

MechanicsPhysicsRatio of separation speed to approach speed in a collision, measuring how much of the relative motion survives impact.

Combined Axial and Bending Stress

\sigma = \frac{P}{A} + \frac{M c}{I}

Strength of MaterialsCivil & SurveyingMechanicsExtreme-fibre stress where an axial force and a bending moment act together, as in an eccentrically loaded column or a beam-column. Enter a negative c for the relieved face.

Conservation of Momentum (Two Bodies)

m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2

MechanicsPhysicsConservation of linear momentum in a two-body collision, solving any one mass or velocity from the other five.

Displacement (Uniform Acceleration)

d = v_0 t + \tfrac{1}{2} a t^2

MechanicsPhysicsDistance travelled under constant acceleration, starting from an initial velocity, over a time t.

Displacement from Average Velocity

d = \frac{v_0 + v}{2} \, t

MechanicsPhysicsDisplacement as the average of initial and final velocities multiplied by the elapsed time, valid for uniform acceleration.

Displacement from Final Velocity (d = vt − ½at²)

d = v t - \tfrac{1}{2} a t^2

MechanicsPhysicsThe fifth kinematic equation: displacement from the FINAL velocity and the time, for when the starting speed is the unknown.

Drag Force (F = ½CdρAv²)

F_D = \tfrac{1}{2} C_d \rho A v^{2}

MechanicsPhysicsAerodynamic drag on a body moving through a fluid, growing with the square of speed and with frontal area and density.

Drop Height of a Horizontally Launched Projectile

y = \tfrac{1}{2} g t^{2}

MechanicsPhysicsDistance a horizontally launched projectile falls in a given time, independent of how fast it was thrown sideways.

Effective Stress (Terzaghi, σ′ = σ − u)

\sigma' = \sigma - u

Soil MechanicsMechanicsTerzaghi's effective stress principle: the grain-to-grain stress that controls soil strength equals total stress minus pore water pressure.

Elastic Collision — Final Velocity of Body 1

v_1 = \frac{\left(m_1 - m_2\right) u_1 + 2 m_2 u_2}{m_1 + m_2}

MechanicsPhysicsFinal velocity of the first body in a one-dimensional elastic collision, where both momentum and kinetic energy survive.

Elastic Potential Energy

U = \tfrac{1}{2} k x^{2}

MechanicsPhysicsEnergy stored in an ideal spring displaced x from its rest length.

Elastic Section Modulus (S = I/c)

S = \frac{I}{c}

Strength of MaterialsMechanicsGeometryElastic section modulus S = I/c, in m³, the single number that turns a bending moment straight into a bending stress.

Escape Velocity

v = \sqrt{\frac{2GM}{r}}

Astronomy & GravitationMechanicsPhysicsMinimum launch speed needed to escape the gravity of a mass M starting from distance r, with no further propulsion.

Euler Critical Buckling Load

P_{cr} = \frac{\pi^{2} E I}{(K L)^{2}}

Strength of MaterialsMechanicsPhysicsEuler's critical buckling load for a slender column, using the end-condition factor K and the area moment of inertia in m⁴.

Expansion Loop Leg Length (Guided Cantilever)

L = \sqrt{\frac{3 E D \, \Delta}{S_a}}

HVAC & HydronicsFluid MechanicsMechanicsLeg length an expansion loop or offset needs to absorb a given thermal movement without exceeding the pipe's allowable stress.

Factor of Safety

FS = \frac{\sigma_{u}}{\sigma_{allow}}

Strength of MaterialsMechanicsPhysicsFactor of safety as ultimate or yield strength divided by the allowable working stress, the engineer's declared margin of ignorance.

Factor of Safety Against Sliding

FS = \frac{W\tan\delta}{P_h}

Soil MechanicsMechanicsFactor of safety of a retaining structure against base sliding, comparing frictional resistance under its weight with the driving horizontal thrust.

Final Velocity (Uniform Acceleration)

v = v_0 + a t

MechanicsPhysicsFinal velocity after accelerating uniformly from an initial velocity for a given time.

Fixed-End Moment — Fixed-Fixed Beam, Uniform Load

M_F = \frac{w L^{2}}{12}

Strength of MaterialsCivil & SurveyingMechanicsMoment at each built-in end of a fixed-fixed beam under a uniformly distributed load, wL²/12. Midspan carries only wL²/24, half as much again the other way.

Gear Ratio

GR = \frac{N_{out}}{N_{in}}

MechanicsPhysicsGear ratio of a meshing pair as the driven gear's tooth count divided by the driver's, setting the torque and speed trade.

Goodman Fatigue Criterion

\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_u} = \frac{1}{n}

Strength of MaterialsMechanicsThe modified Goodman line for infinite life under a fluctuating stress: alternating stress over endurance limit plus mean stress over ultimate strength equals one over the factor of safety.

Gravitational Field Strength

g = \frac{GM}{r^{2}}

Astronomy & GravitationMechanicsPhysicsGravitational acceleration produced by a mass M at distance r from its center.

Gravitational Potential Energy (Orbital)

U = -\frac{GMm}{r}

Astronomy & GravitationMechanicsPhysicsGravitational potential energy of a mass m at distance r from a central mass M, taking zero at infinite separation.

Gravitational Potential Energy (U = mgh)

U = m g h

MechanicsPhysicsEnergy stored by raising a mass to height h near Earth's surface, with g = 9.80665 m/s².

Hooke's Law

F = k x

MechanicsPhysicsRestoring force of an ideal spring, proportional to its displacement from rest.

Hoop Stress in a Thin-Walled Cylinder

\sigma_{h} = \frac{p d}{2 t}

Strength of MaterialsMechanicsPhysicsHoop (circumferential) stress in a thin-walled pipe or pressure vessel, σ = pd/2t — exactly twice the longitudinal stress.

Horizontal Velocity Component

v_x = v \cos\theta

MechanicsPhysicsHorizontal component of a projectile's launch velocity — the part of the speed that carries it downrange at a constant rate.

Hydrostatic Pressure (P = ρgh)

P = \rho g h

MechanicsPhysicsWater TreatmentGauge pressure at depth h in a fluid of density ρ, using g = 9.80665 m/s².

Impulse (J = FΔt)

J = F \, \Delta t

MechanicsPhysicsImpulse delivered by an average force acting over a contact time, equal to the change in momentum.

Kepler's Third Law (Ratio Form)

\frac{T_1^{2}}{T_2^{2}} = \frac{a_1^{3}}{a_2^{3}}

Astronomy & GravitationMechanicsPhysicsFor two bodies orbiting the same central mass, the squares of their periods are in the same ratio as the cubes of their orbital sizes.

Kinetic Friction Force (f = μₖN)

f_k = \mu_k N

MechanicsPhysicsFriction force resisting a sliding surface, equal to the coefficient of kinetic friction times the normal force.

Linear Momentum (p = mv)

p = m v

MechanicsPhysicsMomentum as the product of an object's mass and velocity.

Linear Speed from Rotation (v = ωr)

v = \omega r

MechanicsPhysicsA point at radius r on a rotating body moves with linear speed ωr.

Longitudinal Stress in a Thin-Walled Cylinder

\sigma_{l} = \frac{p d}{4 t}

Strength of MaterialsMechanicsPhysicsLongitudinal (axial) stress in a thin-walled cylinder under internal pressure, σ = pd/4t — exactly half the hoop stress.

Machine Efficiency

\eta = \frac{W_{out}}{W_{in}}

MechanicsPhysicsEfficiency of a machine as useful work out divided by work in, with the shortfall lost to friction, heat, and noise.

Max Bending Moment — Centre Point Load

M = \frac{P L}{4}

Strength of MaterialsMechanicsPhysicsMaximum bending moment in a simply supported beam carrying one point load at midspan, M = PL/4, occurring under the load.

Max Bending Moment — Uniform Load

M = \frac{w L^{2}}{8}

Strength of MaterialsMechanicsPhysicsMaximum bending moment at midspan of a simply supported beam under a uniformly distributed load, the classic M = wL²/8.

Max Moment — Simple Beam, Off-Centre Point Load

M = \frac{P a (L - a)}{L}

Strength of MaterialsCivil & SurveyingMechanicsMaximum bending moment in a simply supported beam with a single point load at distance a from one support, M = Pab/L, occurring directly under the load.

Maximum In-Plane Shear Stress

\tau_{max} = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}

Strength of MaterialsCivil & SurveyingMechanicsRadius of Mohr's circle: the largest shear stress on any plane through a plane-stress element, equal to half the difference of the two principal stresses.

Maximum Principal Stress (Mohr's Circle)

\sigma_1 = \frac{\sigma_x + \sigma_y}{2} + \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}

Strength of MaterialsCivil & SurveyingMechanicsLarger of the two principal stresses for a plane-stress element, from the normal stresses σx and σy and the shear τxy — the circle's centre plus its radius.

Maximum Speed on a Flat Curve

v_{\max} = \sqrt{\mu_s g r}

MechanicsPhysicsFastest a vehicle can round a flat, unbanked curve before friction can no longer supply the centripetal force.

Maximum Static Friction (f = μₛN)

f_{s,\max} = \mu_s N

MechanicsPhysicsLargest static friction force available before an object breaks loose and slides, from the static coefficient and normal force.

Mechanical Advantage of a Lever

MA = \frac{d_e}{d_l}

MechanicsPhysicsMechanical advantage of a lever as the ratio of effort arm to load arm, showing how much the lever multiplies force.

Miner's Cumulative Damage Rule (Three Blocks)

D = \frac{n_1}{N_1} + \frac{n_2}{N_2} + \frac{n_3}{N_3}

Strength of MaterialsMechanicsLinear damage summation over three blocks of a variable-amplitude load history. Failure is predicted when the damage fraction D reaches 1.

Minimum Principal Stress (Mohr's Circle)

\sigma_2 = \frac{\sigma_x + \sigma_y}{2} - \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}

Strength of MaterialsCivil & SurveyingMechanicsSmaller of the two principal stresses for a plane-stress element — the circle's centre minus its radius. Often the compressive one, and the value brittle materials care about least.

Mohr–Coulomb Shear Strength

\tau_f = c' + \sigma'\tan\phi'

Soil MechanicsMechanicsMohr–Coulomb failure criterion giving the shear strength of soil from effective cohesion and the friction mobilised by effective normal stress.

Moment of Inertia: Point Mass

I = m r^{2}

MechanicsPhysicsRotational inertia of a compact mass circling at radius r from the axis.

Moment of Inertia: Solid Disk

I = \tfrac{1}{2} m r^{2}

MechanicsPhysicsRotational inertia of a uniform solid disk or cylinder about its central axis.

Moment of Inertia: Solid Sphere

I = \tfrac{2}{5} m r^{2}

MechanicsPhysicsRotational inertia of a uniform solid sphere about an axis through its center.

Net Allowable Bearing Pressure

q_{all} = \frac{q_u - q}{FS}

Soil MechanicsMechanicsNet allowable bearing pressure for a footing, the ultimate capacity less the existing overburden, divided by the chosen factor of safety.

Newton's Law of Universal Gravitation

F = \frac{G m_1 m_2}{r^2}

MechanicsPhysicsAttractive force between two masses, with G = 6.6743 × 10⁻¹¹ m³ kg⁻¹ s⁻² (CODATA 2018).

Newton's Second Law for Rotation (τ = Iα)

\tau = I \alpha

MechanicsPhysicsNet torque equals moment of inertia times angular acceleration — F = ma for spinning things.

Normal (Axial) Stress

\sigma = \frac{P}{A}

Strength of MaterialsMechanicsPhysicsAxial stress in a bar or hanger rod — the internal force divided by the cross-sectional area that carries it, in Pa or psi.

Normal Force on an Incline (N = mg cos θ)

N = m g \cos\theta

MechanicsPhysicsNormal force pressing a resting mass into an incline, equal to the component of its weight perpendicular to the slope.

Normal Strain (ε = δ/L)

\varepsilon = \frac{\delta}{L}

Strength of MaterialsMechanicsPhysicsNormal strain as the change in length divided by the original length, a dimensionless ratio usually quoted in microstrain.

Orbital Period

T = 2\pi \sqrt{\frac{r^{3}}{GM}}

Astronomy & GravitationMechanicsPhysicsTime for one circular orbit of radius r around a central mass M — Kepler's third law in Newtonian form.

Orbital Velocity

v = \sqrt{\frac{GM}{r}}

Astronomy & GravitationMechanicsPhysicsSpeed of a body in a circular orbit of radius r around a central mass M.

Perfectly Inelastic Collision

v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}

MechanicsPhysicsCommon velocity of two bodies that stick together after a perfectly inelastic collision, from conservation of momentum.

Period of a Spring-Mass Oscillator

T = 2\pi \sqrt{\tfrac{m}{k}}

Waves & OscillationsMechanicsPhysicsPeriod of a mass bouncing on a spring, set only by the mass and the spring stiffness.

Poisson's Ratio

\nu = \frac{\varepsilon_{lat}}{\varepsilon_{ax}}

Strength of MaterialsMechanicsPhysicsPoisson's ratio, the lateral contraction per unit of axial extension — close to 0.30 for steel and 0.33 for aluminium.

Polar Moment of Inertia — Solid Shaft

J = \frac{\pi d^{4}}{32}

Strength of MaterialsMechanicsGeometryPolar moment of inertia of a solid round shaft, J = πd⁴/32, in m⁴ — exactly twice the diametral moment of inertia.

Power (P = W/t)

P = \frac{W}{t}

MechanicsPhysicsAverage power as work or energy delivered per unit time.

Power from Force and Velocity (P = Fv)

P = F v

MechanicsPhysicsInstantaneous power delivered by a force parallel to the velocity.

Pressure (P = F/A)

P = \frac{F}{A}

MechanicsPhysicsPressure as perpendicular force spread over an area.

Projectile Maximum Height

H = \frac{v_0^{2} \sin^{2}\theta}{2g}

MechanicsPhysicsPeak height reached by a projectile launched at a given speed and angle above level ground, ignoring air resistance.

Projectile Range on Level Ground

R = \frac{v_0^{2} \sin 2\theta}{g}

MechanicsPhysicsHorizontal distance a projectile covers over level ground, from its launch speed and angle, ignoring air resistance.

Projectile Time of Flight

T = \frac{2 v_0 \sin\theta}{g}

MechanicsPhysicsTotal time a projectile stays airborne before returning to its launch height, set by launch speed and angle with g = 9.80665 m/s².

Pulley System Effort Force

F = \frac{W}{n}

MechanicsPhysicsEffort force needed to lift a load with a pulley system, divided down by the number of rope sections supporting the load.

Radius of Gyration (r = √(I/A))

r = \sqrt{\frac{I}{A}}

Strength of MaterialsMechanicsGeometryRadius of gyration of a cross-section, r = √(I/A), the shape property that decides how slender a column really is.

Rankine Active Earth Pressure Coefficient

K_a = \tan^{2}\!\left(45^\circ - \frac{\phi}{2}\right)

Soil MechanicsMechanicsRankine coefficient of active earth pressure for a smooth vertical wall retaining level cohesionless backfill that has yielded away from the soil.

Rankine Passive Earth Pressure Coefficient

K_p = \tan^{2}\!\left(45^\circ + \frac{\phi}{2}\right)

Soil MechanicsMechanicsRankine coefficient of passive earth pressure, the resistance mobilised when a wall or footing is pushed into level cohesionless soil.

Relation Between E, G and ν

E = 2G(1 + \nu)

Strength of MaterialsMechanicsPhysicsThe isotropic elastic identity E = 2G(1 + ν), linking Young's modulus, the shear modulus and Poisson's ratio in one step.

Rope Tension When Lifting a Mass

T = m\left(g + a\right)

MechanicsPhysicsTension in a rope lifting a mass with an upward acceleration, exceeding the static weight by the factor (g + a).

Rotational Kinetic Energy

KE_{rot} = \tfrac{1}{2} I \omega^{2}

MechanicsPhysicsKinetic energy stored in rotation: half the moment of inertia times angular velocity squared.

Rotational Power (P = τω)

P = \tau \omega

MechanicsPhysicsMechanical power delivered by a torque turning at angular velocity ω.

Shear Flow (q = VQ/I)

q = \frac{V Q}{I}

Strength of MaterialsCivil & SurveyingMechanicsLongitudinal shear force per unit length that must cross a joint in a built-up beam — the number that sets nail spacing, bolt pitch and weld size.

Shear Modulus (G = τ/γ)

G = \frac{\tau}{\gamma}

Strength of MaterialsMechanicsPhysicsShear (rigidity) modulus as shear stress divided by shear strain, roughly 0.38 of Young's modulus for common metals.

SHM Displacement at Time t

x = A \cos\left(\omega t\right)

MechanicsPhysicsDisplacement of a simple harmonic oscillator at time t, a cosine of amplitude A and angular frequency ω released from full stretch.

SHM Maximum Acceleration

a_{\max} = A \omega^{2}

MechanicsPhysicsMaximum acceleration of a simple harmonic oscillator, reached at the turning points where the restoring force is largest.

SHM Maximum Velocity

v_{\max} = A \omega

MechanicsPhysicsMaximum speed of a simple harmonic oscillator, reached at the equilibrium point, equal to amplitude times angular frequency.

Simple Pendulum Period

T = 2\pi \sqrt{\frac{L}{g}}

Waves & OscillationsMechanicsPhysicsPeriod of a simple pendulum swinging through small angles, with g = 9.80665 m/s² (standard gravity).

Slenderness Ratio (KL/r)

\lambda = \frac{K L}{r}

Strength of MaterialsMechanicsPhysicsSlenderness ratio KL/r of a compression member, the single number that decides whether a column crushes or buckles.

Speed in Circular Motion (v = 2πr/T)

v = \frac{2\pi r}{T}

MechanicsPhysicsSpeed of an object in uniform circular motion: one circumference (2πr, with π ≈ 3.14159265) per period.

SPT Overburden Correction (Liao–Whitman)

(N_1)_{60} = N_{60}\sqrt{\frac{p_a}{\sigma'_v}}

Soil MechanicsMechanicsCorrects a field SPT blow count to a reference overburden of one atmosphere using the Liao and Whitman square-root factor CN.

Stress Concentration (σmax = Kt σnom)

\sigma_{max} = K_t \, \sigma_{nom}

Strength of MaterialsMechanicsPeak elastic stress at a hole, notch, groove or fillet: the nominal stress on the net section multiplied by a geometric factor read from a Peterson chart.

Support Reaction — Simple Beam, Off-Centre Point Load

R_A = \frac{P (L - a)}{L}

Strength of MaterialsCivil & SurveyingMechanicsReaction at the near support of a simply supported beam with one point load at distance a from it. The far reaction is the remainder, P − R_A = Pa/L.

Terminal Velocity

v_t = \sqrt{\frac{2 m g}{\rho A C_d}}

MechanicsPhysicsSteady falling speed at which drag balances weight, from mass, air density, frontal area, and the drag coefficient.

Terzaghi Ultimate Bearing Capacity (Strip Footing)

q_u = c\,N_c + q\,N_q + \tfrac{1}{2}\,\gamma\,B\,N_\gamma

Soil MechanicsMechanicsTerzaghi's three-term ultimate bearing capacity of a shallow strip footing, summing the cohesion, surcharge and footing-width contributions.

Thermal Stress in a Restrained Member

\sigma = E \alpha \Delta T

Strength of MaterialsMechanicsPhysicsStress raised in a fully restrained member that is heated or cooled — the cause of rail sun kinks and cracked pipe anchors.

Torque

\tau = r F \sin\theta

MechanicsPhysicsTurning effect of a force applied at distance r from a pivot, at angle θ to the lever arm.

Torque with a Lever Arm (τ = rF sin θ)

\tau = r F \sin\theta

MechanicsPhysicsTorque produced by a force applied at distance r from the pivot, at angle θ to the lever.

Torsional Shear Stress (τ = Tr/J)

\tau = \frac{T r}{J}

Strength of MaterialsMechanicsPhysicsTorsional shear stress at radius r in a round shaft, τ = Tr/J, peaking at the surface, with J entered in m⁴ as a plain number.

Total Vertical Stress (σ = γz)

\sigma_v = \gamma z

Soil MechanicsMechanicsTotal vertical stress at depth in a uniform soil layer, the weight of the overburden column standing on one unit of area.

Transverse Shear Stress (τ = VQ/Ib)

\tau = \frac{V Q}{I b}

Strength of MaterialsCivil & SurveyingMechanicsShear stress at any height in a beam cross-section, peaking at the neutral axis. For a rectangle it works out to exactly 1.5 times the average V/A.

Velocity-Displacement Relation (v² = v₀² + 2ad)

v^2 = v_0^2 + 2 a d

MechanicsPhysicsLinks initial and final speeds to acceleration and displacement without involving time.

Vertical Velocity Component

v_y = v \sin\theta

MechanicsPhysicsVertical component of a projectile's launch velocity — the part of the speed that fights gravity and sets the time aloft.

Weight (W = mg)

W = m g

MechanicsPhysicsWeight of a mass at Earth's surface, using standard gravity g = 9.80665 m/s² (exact by definition).

Weight Component Along an Incline (mg sin θ)

F_{\parallel} = m g \sin\theta

MechanicsPhysicsComponent of an object's weight acting down the slope of an incline — the force that drives it toward the bottom.

Work (W = Fd cos θ)

W = F d \cos\theta

MechanicsPhysicsWork done by a constant force acting at an angle to the displacement.

Work from Force and Displacement Components

W = F_x d_x + F_y d_y

Vectors & MatricesPhysicsMechanicsComputes the work done by a force from the components of the force and the displacement, without needing the angle between them.

Work–Energy Theorem

W = \tfrac{1}{2} m \left(v^{2} - v_0^{2}\right)

MechanicsPhysicsNet work done on an object equals its change in kinetic energy, linking force and distance to a change in speed.

Young's Modulus (E = σ/ε)

E = \frac{\sigma}{\varepsilon}

Strength of MaterialsMechanicsPhysicsYoung's modulus as the ratio of normal stress to normal strain, the stiffness constant of a material in its elastic range.