Mechanics formula solvers

Categories — 73 of them, 1787 solvers in all

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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.

ASME Required Wall Thickness (t = PR/(SE − 0.6P))

t = \frac{P R}{S E - 0.6 P}

Strength of MaterialsMechanicsCivil & SurveyingMinimum required wall thickness of a cylindrical shell under internal pressure, from ASME Section VIII Division 1 UG-27(c)(1): pressure times inside radius over allowable stress times joint efficiency, less the 0.6P correction.

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.

Bank Shot Rail Contact Point

x_c = \frac{x_1 y_2 + e\,x_2 y_1}{y_2 + e\,y_1}

Billiards & Cue SportsMechanicsWhere on the rail to send the ball for a one-cushion bank. The schoolbook mirror method reflects the target through the rail and aims at the image — which is this formula with e = 1. Real cushions rebound wider than they receive, so the true contact point sits shifted, and this is the mirror method corrected for restitution.

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.

Beam Reaction by Moment Equilibrium — Two Point Loads

R_A = \frac{P_1 (L - a_1) + P_2 (L - a_2)}{L}

Strength of MaterialsCivil & SurveyingMechanicsReaction at support A of a simply supported beam carrying two point loads, from ΣM = 0 taken about support B. Each load is weighted by its distance from the FAR support, which is the whole content of the moment equation.

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.

Bearing Stress on a Pin or Bolt (σ = P/dt)

\sigma_{b} = \frac{P}{d \, t}

Strength of MaterialsMechanicsCivil & SurveyingBearing stress where a pin or bolt presses on the side of its hole: the load divided by the PROJECTED area, bolt diameter times plate thickness. The third failure mode of a bolted lap joint, after net-section tension and bolt shear.

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 Mechanical Energy (½mv² + mgh)

\tfrac{1}{2} v_1^{2} + g h_1 = \tfrac{1}{2} v_2^{2} + g h_2

MechanicsPhysicsSpeed and height at two points on a frictionless path, from the fact that kinetic plus potential energy does not change along the way.

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.

Culmann Planar Wedge Factor of Safety

FS = \frac{2c'\sin\beta}{\gamma H\sin(\beta-\theta)\sin\theta} + \frac{\tan\phi'}{\tan\theta}

Soil MechanicsMechanicsFactor of safety of a rigid triangular wedge sliding on a single plane beneath a steep cut of height H and face angle β — Culmann's 1866 analysis, the oldest slope calculation still in use and the right one for a steep face where the infinite-slope assumption fails.

Cushion Rebound Angle

\tan\theta_{out} = \frac{\tan\theta_{in}}{e}

Billiards & Cue SportsMechanicsWhy a ball does not come off a cushion at the angle it went in. The cushion squashes and pushes back along its own normal, so only the perpendicular part of the velocity is affected — it comes back multiplied by e — while the part parallel to the rail sails through untouched. The result is always a wider angle out than in.

Cushion Rebound Speed

v_{out} = v_{in}\sqrt{e^{2}\cos^{2}\theta + \sin^{2}\theta}

Billiards & Cue SportsMechanicsHow much speed a cushion takes out of a ball. Only the perpendicular component is squeezed and returned at e times its size; the parallel component is untouched. Add the two back as vectors and the answer depends entirely on the angle — a ball hitting the rail square loses the most, and one grazing along it loses almost nothing.

Cut Angle from Ball Fraction

\sin\varphi = \frac{b}{2R}

Billiards & Cue SportsMechanicsThe whole of aiming geometry in one line. Two spheres of radius R touch when their centres are 2R apart, so if the cue ball's centre passes a perpendicular distance b to the side of the object ball's centre, the line of centres at contact — and therefore the object ball's departure direction — sits at sin⁻¹(b/2R) from the cue ball's path. A half-ball hit gives exactly 30°.

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.

Infinite Slope Factor of Safety with Cohesion and Pore Pressure

FS = \frac{c' + \left(\gamma z\cos^{2}\beta - u\right)\tan\phi'}{\gamma z\sin\beta\cos\beta}

Soil MechanicsMechanicsThe general infinite-slope factor of safety: Mohr-Coulomb strength on a slope-parallel plane at depth z, with cohesion in the numerator and the pore pressure eating the effective normal stress that friction acts on.

Infinite Slope Factor of Safety, Dry Cohesionless Soil

FS = \frac{\tan\phi'}{\tan\beta}

Soil MechanicsMechanicsFactor of safety of a long shallow slide in dry cohesionless soil, where the whole answer is the ratio of two tangents: the friction angle over the slope angle. Depth and unit weight cancel out completely.

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.

Lamé Hoop Stress in a Thick-Walled Cylinder

\sigma_{\theta} = \frac{p_i r_i^{2}}{r_o^{2} - r_i^{2}} \left(1 + \frac{r_o^{2}}{r^{2}}\right)

Strength of MaterialsMechanicsPhysicsCircumferential (hoop) stress at any radius r in a thick-walled cylinder with internal pressure only — the Lamé solution. Unlike the thin-wall formula it shows the stress varying through the wall, peaking at the bore.

Lamé Radial Stress in a Thick-Walled Cylinder

\sigma_{r} = \frac{p_i r_i^{2}}{r_o^{2} - r_i^{2}} \left(1 - \frac{r_o^{2}}{r^{2}}\right)

Strength of MaterialsMechanicsPhysicsRadial stress at any radius r in a thick-walled cylinder with internal pressure only. It is compressive throughout, equal to −p at the bore and exactly zero at the free outside surface.

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.

Membrane Stress in a Thin-Walled Sphere (σ = pr/2t)

\sigma = \frac{p r}{2 t}

Strength of MaterialsMechanicsPhysicsMembrane stress in the wall of a thin-walled sphere under internal pressure, σ = pr/2t. A sphere is stressed equally in every direction, which is why it is the most efficient pressure vessel shape there is.

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.

Net Force on One Body

F_{\text{net}} = F_1 - F_2

MechanicsPhysicsNet force on a single body along one axis: everything driving it forward minus everything resisting.

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.

Rockfall Kinetic Energy at a Barrier

E = \tfrac{1}{2} m v^{2}

Soil MechanicsMechanicsTranslational kinetic energy of a falling or bouncing block at the point it reaches a barrier — the number a rockfall net, fence or embankment is selected against, and the number that barrier ratings are published in.

Rolling Cue Ball Deflection (the real 30° rule)

\tan\alpha = \frac{5 \sin\varphi \cos\varphi}{2 + 5\sin^{2}\varphi}

Billiards & Cue SportsMechanicsWhere a ROLLING cue ball actually ends up after a cut. It leaves along the tangent line at 90° to the object ball, but it is still carrying its forward roll, so friction bends it forward again until it is rolling once more. The final path sits α off the original line of travel — and α hovers near 32° for every cut angle between 20° and 40°, which is the whole reason the 30° rule exists.

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.

Slide Distance Before Natural Roll

d = \frac{12 v_0^{2}}{49 \mu g}

Billiards & Cue SportsMechanicsHow far a cue ball struck with no spin slides before cloth friction has spun it up into a roll. It is the range over which a stun shot still behaves like a stun shot, and it is the reason the 90° rule works on a short shot and fails on a long one.

Slide Time Before Natural Roll

t = \frac{2 v_0}{7 \mu g}

Billiards & Cue SportsMechanicsHow long a cue ball struck with no spin slides before it is rolling. The ball decelerates at μg while friction spins it up at 5μg/2R, and the two meet when v = Rω — which happens after 2v₀/7μg, linearly in the starting speed rather than quadratically like the distance.

Speed at Natural Roll

v_f = \tfrac{5}{7} v_0 + \tfrac{2}{7} R\omega_0

Billiards & Cue SportsMechanicsThe speed a ball settles to once cloth friction has turned its slide into a roll. Friction acts at the contact point, so angular momentum about that point is conserved through the whole slide — and the answer comes out as a fixed weighted average of the starting speed and the starting spin. A ball struck with no spin loses exactly two-sevenths of its speed, always.

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.

Stun Shot — Cue Ball Speed

v_{CB} = u \sin\varphi

Billiards & Cue SportsMechanicsWhat the cue ball keeps after a stun shot: the component of its velocity ALONG THE TANGENT LINE, which the collision cannot touch. Together with the object ball's u cos φ this is the 90° rule — two perpendicular velocities whose squares add back to u², so no energy is lost and no momentum goes missing.

Stun Shot — Object Ball Speed

v_{OB} = u \cos\varphi

Billiards & Cue SportsMechanicsHow much of the cue ball's speed the object ball actually receives on a cut. The balls are smooth, so the impulse runs along the line of centres and only the component of the cue ball's velocity along that line is transferred — cos φ of it. A thin cut sends the object ball nowhere, which is why thin cuts have to be hit hard.

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.

Support Reaction — Simple Beam, Uniform Load (R = wL/2)

R = \frac{w L}{2}

Strength of MaterialsCivil & SurveyingMechanicsReaction at each support of a simply supported beam under a uniformly distributed load: half the total load wL. It is also the maximum shear in the beam, which is why it sizes the bearing and the end connection.

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.

Tip Offset to Spin Ratio

\frac{R\omega_0}{v_0} = \frac{5b}{2R}

Billiards & Cue SportsMechanicsHow much spin a given tip offset puts on the cue ball. Strike the ball a distance b off centre and the same impulse that gives it speed also gives it angular momentum about its centre — and because a solid sphere has I = ⅖mR², the ratio of surface spin speed to travel speed comes out as 5b/2R, independent of how hard you hit it.

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.

Von Mises Equivalent Stress (Plane Stress)

\sigma_v = \sqrt{\sigma_x^{2} - \sigma_x \sigma_y + \sigma_y^{2} + 3\tau_{xy}^{2}}

Strength of MaterialsMechanicsCivil & SurveyingVon Mises equivalent stress for a plane stress state: the single tensile stress that would do the same distortion damage as the combination of σx, σy and τxy. Compare it directly against the yield strength.

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.