study of forces {dynamics}.
In a constant-force field, particles take shortest time and shortest distance, to minimize force F times distance change ds times time change dt {action, physics}|: F * ds * dt. Action tends to minimize {principle of least action, physics} {least-action principle}. Particles take shortest space-time path. Particle trajectories follow geodesics. Particles take least-resistance path.
In quantum mechanics, action has quanta, which have size Planck constant h. Action values are multiples of h.
Energy is force times distance change, so F * ds * dt = dE * dt. For energy, action is energy change times time change: dE * dt. By the least-action principle, particle trajectories take the shortest time with least energy change. Total action is sum of kinetic-energy KE to potential-energy PE difference over time dt: integral of (KE - PE) * dt. Least action over time makes conservation of energy.
Energy change dE is force F times distance change ds: dE = F * ds. Force F is momentum change dp divided by time change dt: F = dp / dt. Therefore, energy change dE times time change dt equals momentum change dp times distance change ds: dE * dt = F * ds * dt = dp * ds. For momentum, action is momentum change times distance change. By the least-action principle, particle trajectories take the shortest path length with least momentum change. Least action over translation makes conservation of momentum.
Tangential momentum p is angular momentum L divided by radius r: p = L / r, so dp = dL / r. Distance s is radius times angle A in radians, so ds = r * dA. Therefore, momentum change dp times distance change ds equals angular-momentum change dL times angle change dA in radians: dp * ds = (dL / r) * (r * dA) = dL * dA. For angular momentum, action is angular-momentum change times angle change. By the least-action principle, particle trajectories take the shortest rotation with least angular momentum change. Least action over rotation makes conservation of angular momentum.
If force varies, action minimizes dF * ds * dt. For force, action is force change times space-time change. By the least-action principle, particle trajectories take the geodesic with least force change. Least action over space-time makes force zero or flat space-time.
Over an instant or at a point or through an infinitesimal angle, action is zero or a limiting value.
Two colliding objects {collision}| have compressive force. Initial positions and velocities cannot be precise, but deviations are small, so future behavior mostly predictable. For collisions between three objects simultaneously, future behavior can deviate far from predicted behavior. If first two objects collide first, followed by third object, resulting motions can be much different than if last two objects collide first, followed by first object.
If two dense hard objects collide, all energy stays in motion, no heat is made, and objects bounce off each other {elastic collision}|. In elastic collisions, forces are equal and opposite, and momentum (m*v) before and after collision is constant: m1 * v1 = m2 * v2. In elastic collisions, with no heat, total energy E is kinetic energy KE plus potential energy PE and stays constant: E = KE1 + PE1 = 0.5 * m1 * v^2 + m1 * g * h1 = 0.5 * m2 * v^2 + m2 * g * h2 = KE2 + PE2.
examples
Superballs are denser and harder than regular balls. Karate experts try to make elastic collisions, rather than inelastic collisions, to break objects. Atomic-particle collisions are elastic, because they do not deform.
If either colliding object is soft or has low density, collision {inelastic collision}| permanently deforms surface, some collision energy becomes heat, and objects can stick together.
If physical-system coordinates transform, some physical properties remain unchanged {conservation laws}|.
fermions
all same-type fermions are identical. For example, all electrons are identical. Physical laws are symmetric for fermion replacement with same-type fermion.
mass
For non-relativistic conditions, mass stays constant. For example, mass does not change in chemical reactions. However, physical laws are not symmetric with respect to matter-antimatter for weak force.
baryon number
Baryon number stays constant
lepton number
Lepton number stays constant.
parity
Parity conserves, except for weak force. Physical laws are not symmetric with respect to reflection in space for weak force.
strangeness
Strangeness conserves, except for weak force.
no conservation
Physical laws are not symmetric with respect to scale. Physical laws are not symmetric with respect to uniform angular velocity.
symmetries
Conservation laws are about minimizations and symmetries. Symmetries require reference point, feature, and reference frame. Symmetry types depend on feature types. For example, rotating spheres with no features have no detectable spin. Particles with dipoles have detectable spin, which can be right or left. Particles must have mass, spin, or other feature to be detectable. Featureless objects or spaces have no symmetries. Symmetries can cancel large physical quantities. Physical theories have one symmetry for each conserved quantity (Noether) [1915].
energy
Energy conservation requires time symmetry: forward and backward in time are usually the same physically. By observing a physical process, observers cannot tell if time flows backwards or forwards.
Total closed-system energy is constant. However, energy can exchange between potential and kinetic energy. Kinetic energy minus potential energy {Lagrangian} measures energy exchange. The path integral of Lagrangian over time is the physical action. For cyclic processes, the system periodically returns to the same Lagrangian value, Lagrangian change is zero, and action is zero. For cyclic processes, the wave equations of motion are path integrals of Lagrangians over time set equal to zero.
momentum
Momentum conservation requires special-relativity constant-velocity reference-frame equivalence. When observing a physical process, observers have no preferred reference frame. The distance metric is the same for all constant-velocity observers (Lorentz invariance).
angular momentum
Angular momentum conservation requires right-left (parity) symmetry. When observing a physical process, observers cannot tell if it is right-handed or left-handed. Clockwise and counterclockwise rotations have same physics.
electric charge
Electric charge stays constant. Electric-charge conservation requires electromagnetism gauge invariance.
Basic space-time symmetries keep physical laws the same under various conditions {invariance, physics}. Baryon number, spatial rotation, and space-time translation are always invariant.
Charge conjugation, parity, and time reversal combined are invariant for all physical laws. Charge conjugation and parity together are invariant, except for strange-particle decays in weak nuclear forces. Mass-strength and strong-force-strength differences in up and down and other quarks cause charge-conjugation symmetry breaking. Parity breaks down in weak nuclear forces. Time reversal breaks down in weak nuclear forces.
Heat and work are kinetic energy. Force fields cause potential energy. Total energy is sum of kinetic and potential energies, which can interconvert. Isolated-system total energy is constant {energy conservation, dynamics}| {conservation of energy, dynamics}. Energy is invariant through time-coordinate translations. Physical laws are symmetric with respect to time dimension, so physics does not change if time reverses direction. Physical laws remain true at all times. All physical interactions are the same if time reverses, charges reverse, and positions reverse. However, weak-force physical laws are not symmetric with respect to time.
cause
Isolated systems have no added forces and so no added potential energy. Isolated systems have no volume changes and so no added distances or potential energy. Object movements interchange potential energies and kinetic energies, no matter which space-time path objects take.
vacuum energy
Kinetic energy and potential energy exert pressure on background vacuum energy. Kinetic energy has particle motions that make internal pressure. Potential energy has fields that make pressure by causing particle self-energy. Motions and fields pressure space-time points through which they pass. Space-time points have energy flux. Kinetic energy and potential energy both contribute to vacuum energy in the same way. Only energy amount counts. As masses move, vacuum adjusts to keep potential constant. Potential, flux, or pressure is constant at all vacuum points, making a new conservation law.
relativity
Mass and energy can interchange in space-time. By equipartition, all partition kinetic energies must be equivalent. Energy conservation remains true under relativistic conditions.
In general relativity, accelerations are equivalent to forces, which cause accelerations. Accelerations are velocity changes. Velocity changes change kinetic energy. Objects change velocity as they change field position and potential energy. Kinetic and potential energies are equivalent in general relativity.
quantum mechanics
Quantum mechanically, mass and energy states are the same. Energy conservation remains true under quantum mechanics.
dark energy
Energy conservation remains true for dark energy, which is symmetric in time.
Momentum conservation {conservation of momentum} means that total momentum is constant, no matter which direction objects take through space. Momentum is invariant under spatial-coordinate translations. All directions are equivalent. Physical laws are symmetric with respect to space dimensions. Physics does not change if space directions reverse, rotate, or translate. Physical laws remain true at all space points.
System total momentum stays constant. For interacting objects, one object's momentum change balances other object's momentum change, because both objects interact over same time. For non-interacting objects, motion states and object masses do not change, so momenta do not change. Fields and their bosons carry or contain momentum and inertia.
Angular momentum conservation {conservation of angular momentum} means that total angular momentum is constant, no matter what rotations (spins or orbits) objects take, at any orientation. Angular momentum is invariant under rotations. Angular momentum is invariant under handedness change, right-handed or left-handed. All rotations are equivalent. Physical laws are symmetric with respect to rotation. Physics typically does not change if spin directions reverse or change orientation. Physical laws remain true for right-handed or left-handed arrangements.
System total angular momentum, from spins, orbits, and curved trajectories, stays constant. Using infinite radius, angular-momentum conservation is equivalent to momentum conservation. Angular-momentum conservation implies that system mass center stands still, so universe either does not rotate or does not move. Angular-momentum conservation implies that forces have equal and opposite reaction-forces and that masses have inertia.
Force can exert over distance {energy}|. Energy is scalar, because motion component does not matter since force vector and distance vector have same direction. Net force can act over distance in direction {work, energy}. Forces can act over distances in all random directions {heat}. Heat has no net force. Forces must act for some time over some distance, so energy can exchange and motion can change.
Energy {kinetic energy}| can involve motion. Kinetic energy equals one-half times mass m times velocity v squared: KE = 0.5 * m * v^2. If object does work and loses speed, object loses kinetic energy. If object receives work and increases speed, object gains kinetic energy.
force
Collision, pushing, pulling, or other contact force F acting over distance ds makes mass m accelerate a to velocity v: dKE = F * ds = m * a * ds = m * (dv / dt) * ds = m * dv * ds / dt = m * v * dv, where t is time. Integral of dv = v/2, so KE = 0.5 * m * v^2.
comparison
Energy is either kinetic expressed energy or potential stored energy, because force times distance makes kinetic energy, and kinetic energy can act against force over distance to make potential energy.
Energy {potential energy}| can depend on force exerted over distance against field. Potential energy PE equals m times acceleration a times distance moved in field h: PE = m * a * h. If object moves to position with less force, object gains potential energy. If object moves to position with more force, object loses potential energy.
field
Field is gravitational, electric, or nuclear force field.
position
Potential energy depends on field force and object field position. If object moves farther away from attraction center, object gains potential energy. If object moves farther away from repulsion center, object loses potential energy. Small movements in strong fields can equal large movements in weak fields.
Exerted force can work against field force, and object gains potential energy. Field force can move object to do work, and object loses potential energy.
action
Going from one point to another point in potential field has only one path, with least average difference between kinetic and potential energy over time.
potential
Fields have measures {electric potential} {potential, electricity} of potential energy that depend on only source mass or charge, not on test-object mass or charge. Gravitational potential V is gravitational constant G times mass m at field center, divided by distance r from center: V = G * m / r. Electrical potential V is electric constant k times charge q at field center, divided by distance r from center: V = k * q / r.
Energy can flow per unit time {power, physics}|. Time t divides into energy E: P = (Ef - Ei) / (tf - ti). Power is constant force F times constant velocity v: P = F*v = F * (ds / dt) = (F * ds) / dt = dE / dt. Power is scalar, because energy is scalar.
Spinning or orbiting objects have rotation energy {rotational kinetic energy}. Because tangential velocity v equals angular velocity w times radius r, rotational kinetic energy KE equals half moment of inertia I times angular velocity w squared: KE = 0.5 * m * v^2 = 0.5 * m * (w*r)^2 = 0.5 * m * r^2 * w^2 = 0.5 * (m * r^2) * w^2 = 0.5 * I * w^2, where moment of inertia I = m * r^2.
work
Energy can be force over distance around axis {rotational work}. Rotational work W equals torque T times angle A in radians: W = F*s = (T/r) * (r*A) = T*A.
power
Energy can be force over time around axis {rotational power}. Rotational power P equals torque T times angular velocity w: P = E/t = (F*s) / t = ((T/r) * (r*A)) / t = T * (A/t)= T*w.
Force can act over distance in direction to transfer energy {work, physics}|. Net force F acts over distance change (sf - si) to perform work E: E = F * (sf - si). Small force Fs exerted over large distance sl can do same work as large force Fl exerted over short distance ss: Fs * sl = Fl * ss.
Objects in fluids have forces and motions {fluid dynamics}|.
thrust
Forward force {thrust, fluid} pushes objects through fluid.
drag
Friction {drag} retards moving objects in fluid. Drag rises as velocity increases.
velocity
If thrust stays constant, velocity rises and drag increases, until force balance makes no more acceleration, at terminal velocity. Example is feather falling through air under gravity.
energies
At pipe points, energies are kinetic energy from fluid flow, potential energy from liquid standing in open pipes, and/or energy from outside forces and pressures. Pipe fluids have energy conservation, by Bernoulli's theorem. For streamline flow, sum of pressure P and kinetic energy KE per volume V is constant: P + (KE / V) = constant. P * V = PE, so (PE / V) + (KE / V) = KE + PE = constant total energy.
pressure
Outside force can exert pressure on fluid. Force moves fluid small distance, and kinetic energy distributes throughout fluid, increasing fluid pressure. Outside pressure P is kinetic energy per volume and is force F times distance s divided by volume V: P = KE / V = (F * s) / V.
depth
At fluid depth, gravity causes pressure. Stationary pressure P is potential energy per volume and is density d times gravity acceleration g times depth h: P = PE / V = (m*g*h) / V = (m/V) * g * h = d * g * h.
flow
Fluid flow causes kinetic energy, which exerts pressure in flow direction. Directed pressure P is kinetic energy divided by volume and is half density d times velocity v squared: P = KE / V = (0.5 * m * v^2) / V = 0.5 * (m/V) * v^2 = 0.5 * d * v^2.
Rotating one cylinder inside another causes intervening liquid to flow {Couette-Taylor flow}. First, flow streamlines. At faster speed, fluid cylinder separates into separate layers along cylinder axis, so fluid goes up and down in cylinder. At higher frequency, flow is chaotic, with no defined frequencies.
At fluid boundaries {no-slip condition}, fluid does not slip.
Mixtures of large organic positive ions and inorganic negative ions {ionic liquid}| can be liquid at room temperature, because large charge is spread over large space, so crystal is loose. Liquid has polar and non-polar parts, so it can dissolve organic materials.
non-turbulent flow {laminar flow}|.
Material can become fluid {liquefaction}|.
After torpedo goes 50 meters per second in water, water pressure is low enough to allow water vapor to make vapor cavity around object {supercavitation, fluid}|, allowing high speed.
Wax surfaces repel water very well {superhydrophobicity}|.
City water pressure {water pressure}| is 30 to 50 pounds per square inch, which can lift water 25 to 30 meters.
Friction {drag, fluid}| slows objects moving through fluid. Drag increases if velocity increases. Pipe walls retard fluid flow by friction one millimeter into fluid.
Wing induces drag as it lifts {lifting line theory}.
Drag rises as velocity increases, while forward force stays constant, until forces balance with no more acceleration {terminal velocity}|. Feathers fall through air under gravity with terminal velocity.
Material density and water density have ratio {specific gravity}|. Specific gravity is one for water. Metals have higher specific gravities and sink in water. Wood has lower specific gravity and floats in water. Density D multiplied by gravity acceleration g is weight m*g per volume V {specific weight}: d*g = (m / V) * g = m*g / V.
Instruments {hydrometer}| can measure specific gravity.
Forces between molecules make fluid stick together {viscosity}|.
causes
In liquids, van der Waals forces cause viscosity. In gases, non-ideal molecular collisions cause viscosity.
pressure
Gas viscosity increases if pressure increases.
temperature
Temperature increase increases gas viscosity and decreases liquid viscosity.
factors
Fluid viscosity depends on fluid density, pressure, temperature, and velocity. In pipe, pipe-opening size affects viscosity. Intermolecular forces tend to pull fluid sideways in pipes and contribute to turbulence. Fluid sideways pressure P equals viscosity V times velocity change dv divided by length change dl: P = v * dV/dl.
Pipe flow with incompressible fluid has two regions. A thin layer {boundary layer} touches tube or obstruction and has viscous effects, because surface interacts thermally and mechanically with fluid. Center has flow with no turbulence.
Temperature, viscosity, and fluid depth relate {Rayleigh number}. Reynolds number and Rayleigh number together account for flow effects, viscosity, thermal conductivity, linear-expansion or volume-expansion coefficient, fluid depth, and temperature gradient.
Fluids have ratios {Reynolds number} of internal force to viscous force. Reynolds number measures fluid momentum change. If Reynolds number is small, smooth pipe decreases drag, because flow is laminar. If Reynolds number is high, vortices in smooth pipe increase drag.
Fluids have attractive electric forces among molecules {cohesion}|. Surface tension has cohesion.
Inside fluid, cohesive forces are symmetric and cancel each other. At fluid surfaces, cohesion pulls molecules closer together {surface tension}|, to make surface density more than inside density. Increased-surface-density layer width is 20 molecules. Air above fluid has small density and has little attraction for fluid. Surface chemical potential is greater than inside fluid, because net force is more, and fluid is denser and so more organized.
floating
Surface tension can make density great enough to float objects, such as steel pins.
examples
Glue, waterproofing, detergents, wicks in candles, blotters, towels, bubbles, milk drops, camphor dance, soap film, salts, and needles on water illustrate surface tension.
drops
Droplets have more surface area and high surface tension.
factors
Solute can lower solvent-molecule cohesion by disrupting cohesive forces and lowering chemical potential. Soaps and detergents lessen water surface tension by blocking water-molecule attractions. However, ions in water increase surface tension by increasing electric forces.
Gravity causes fluid molecules to press on molecules below {pressure, fluid} {fluid pressure}|. Deeper molecules have more pressure, because more molecules are above them. Pressure P, force F per area A, at point below fluid surface is density d times depth h times gravity acceleration g: P = F / A = (m * g) * (h / V) = (m / V) * g * h = d * g * h. Pressure is directly proportional to gravity acceleration, because acceleration times mass is force. Pressure is directly proportional to density, because density relates to molecule mass. Pressure is directly proportional to depth, because depth relates to molecule number. Pressure does not depend on total surface area, because pressure is force per unit area.
fluid level
Liquids rise to equal heights at all openings to atmosphere, because pressures and potential energies are equal at liquid surfaces.
cause
Random molecule motions cause fluid pressure. Random motion exerts force and pressure equally in all directions, even upward or at angle. Container wall slope has no effect. Pressure is the same at all points at same depth. The net effect of random motions is that pressure is perpendicular to fluid surface, because random motions are symmetric around perpendicular.
temperature
Temperature increase increases fluid pressure, because molecules move faster.
density
More and/or heavier molecules have higher density and exert more pressure.
gas
Gas has random translational kinetic energy per unit volume, making force per unit area on container walls. Random translational kinetic energy depends on mass and molecule average velocity. If volume decreases, pressure goes up. If temperature decreases, pressure goes down. If pressure decreases, volume goes up. If pressure increases, temperature goes up.
Because pressure transmits equally throughout fluids, pressure sum, and energy sum, is constant {Bernoulli's theorem} {Bernoulli theorem} {Bernoulli's principle}. At pipe points, energy conserves. Pressure stays constant throughout fluid.
If fluid goes through narrower pipe area, fluid speeds up, because mass cannot build up. Higher speed makes more kinetic energy and directed pressure. Sideways pressure decreases, to conserve energy. If fluid slows, kinetic energy goes down, and potential energy and sideways pressure increase.
examples
Fountains, tanks with holes, aspirators, sprayers, Venturi meters, wings, pipes, rubber tubes, and curve balls demonstrate Bernoulli's principle.
In confined fluids, force in one direction can transmit pressure to all directions {hydraulics}|. Increased pressure changes into increased random molecule motions. For example, pushing a piston in a long thin cylinder can do work on fluid and increase molecule random kinetic energy, which can do work on a piston in a short wide cylinder.
area
At depth, force per area pressure is the same throughout confined fluid. Small force acting over long distance on small area can make big force acting over short distance on large area, because energy in and energy out are equal. Large area can apply large total force.
examples
Dams, breathing using lung double cavity, manometer, Bourdon gauge, barometer, hydraulic brakes, hydraulic lifts, syringe, and fluid tank show hydraulic effects.
Simple-fluid hydrodynamics {hydrodynamics}| has no viscosity or heat exchange and follows Euler equation: mass m per volume V times acceleration g equals negative of pressure P gradient perpendicular to velocity vector: (m / V) * g = - dP / ds.
Equal areas have same pressure in confined fluid {Pascal's principle} {Pascal principle}.
Fluid discharge velocity from small hole at depth below open surface is square root of two times gravity acceleration g times depth h {Torricelli's theorem} {Torricelli theorem}: (2*g*h)^0.5.
Removing gas molecules {vacuum, gas}| reduces pressure. Vacuum pumps remove gas molecules.
Oil with iron filings {magnetorheological fluid} can turn solid in magnetic field.
Fluids {electrorheological fluid} can become solid, or have lower viscosity, in high electric fields.
Fluid mildly heated from bottom at first has temperature gradient with no net flow. More heat creates alternating hexagonal cells that allow hot fluid to rise and cold fluid to fall. Even more heat makes turbulent motion, with no net flow {Bénard problem}.
Heated fluids can have convection {Rayleigh-Bénard convection} with circular motions.
Two surfaces can stick to each other {adhesion, surface}|.
Fluids have electric forces between molecules and container surfaces. Fluids that physically adhere to surface can rise in small-diameter tubes {capillary rise}|. Clinging force pulls fluid up tube sides. Fluid rises until potential-energy increase balances air pressure.
When fluids leave holes, fluid tends to flow around hole edge {Coanda effect}.
If fluid is adhesive, fluid curves up container walls {meniscus}|. If fluid is not adhesive, fluid curves down container walls.
Van der Waals forces can cause molecules to bind to surfaces {physisorption}|. Vibrations then cause molecules to leave surface, within 10^-8 seconds, heating surface. Surface chemical bonds do not form or break.
Chemical bonds between surface molecules and fluid molecules can bind molecules to surfaces chemically {chemisorption}|. Molecule stays at surface from 1 to 1000 seconds and then has desorption. Chemisorption has activation energy. At low pressure and low absorption, chemisorption fraction depends on pressure. At high pressure or for strong electrical forces, chemisorption fraction depends on pressure inverse.
Chemisorbed molecules stay at surface from 1 second to 1000 seconds and then leave {desorption, surface}|, heating surface. Surface chemical bonds break. Desorption has activation energy.
Objects in fluids have more pressure on bottom surface than on top surface {buoyancy}|, because bottom surface is deeper in fluid. The greater force on bottom pushes object up. Buoyancy equals difference between object-bottom pressure and object-top pressure. Buoyant force is in opposite direction from gravity. Objects that sink have more force than fluid weight pushed up. Objects that sink are denser than fluid.
If objects float in fluid, fluid weight pushed up around object equals upward buoyant force on object {Archimedes principle, buoyancy}|.
Object in fluid pushes fluid out {displacement, fluid}|. Fluid tries to return to original position by gravity. Displaced fluid and object both want to occupy same place. Equilibrium happens when both forces push down equally.
Objects can sink until buoyant force balances gravity {floating}|. Displaced fluid and object both want to occupy same place, so object is at equilibrium when fluid force pushing down equals object force pushing down. Object that floats is less dense, including air spaces, than fluid. Submarines, fish, boats, balloons, and ice cubes demonstrate buoyancy. If floating-object mass center is not along buoyant-force line, object rotates around mass center.
Fluid mass can go past point or through area over time {fluid flow}|.
pipe
Fluid velocity at different pipe radii differs. Highest velocity is in center. Velocity is zero at pipe walls.
conservation
Same fluid amount at one point must be at another point. Otherwise, fluid builds. Same fluid volume passes any point, during time. At pipe points, inflow equals outflow.
pressure
Around pipe loops, pressures add to zero.
rate
Flow rate increases with increase in molecule velocity, temperature, pressure, and/or mean free path. Flow rate decreases with increase in cross-sectional area, molecule mass, and/or molecule collision frequency.
Flow in pipes can have constant velocity at each radius, with no sideways motion {streamline flow}|.
Flow in pipes can have sideways motion or different velocities at same pipe radius {turbulent flow}|. Trapped gases in fluid can cause turbulence.
High-speed flow and/or pipe edges can pull fluid apart, making vacuum spaces {cavitation, fluid}|.
Fluids have flow rate through area {flux, fluid}|. Flux is energy, mass, momentum, or charge change D divided by cross-sectional area A times time t: D / (A * t).
pipe
In pipes, masses entering and leaving cross-sectional areas are equal. Otherwise, fluid builds, or vacuum happens. Mass m flowing through pipe equals fluid density d times fluid velocity v times cross-sectional area A: m = d*v*A. For liquid, fluid density is constant, and fluid velocity going in vi times cross-sectional area at entrance Ai equals fluid velocity going out vo times cross-sectional area at exit Ao: vi * Ai = vo * Ao. For gas, fluid density varies, and fluid density at entrance di times fluid velocity going in vi times cross-sectional area at entrance Ai equals fluid density at exit do times fluid velocity going out vo times cross-sectional area at exit Ao: di * vi * Ai = do * vo * Ao.
Flux equals constant times gradient {Fick's first law of diffusion} {Fick first law of diffusion}: dm / (A * dt) = dC / ds, where m is mass, A is cross-sectional area, t is time, C is concentration difference, and s is distance.
Pressure, temperature, concentration, or force change over time relates to quantity change over distance {Fick's second law of diffusion} {Fick second law of diffusion}: dP / dt = dm / ds, where P is pressure, t is time, m is mass, and s is distance.
Other objects can cause object to tend to change motion {force, physics}|. Force requires interaction between two objects. All forces are pushes or pulls, such as when two objects collide. Gravity and electric forces are interactions of objects with second-object fields. Force F that object receives equals object mass m times object acceleration a: F = m*a. Force has direction and amount and so is vector. Mechanical force can be interaction between two colliding masses, but repulsions between electrons around molecules mediate contact between objects.
Gravity and electric force can act between two objects over distance {force field}.
exchange
All forces transmit bosons. In all field interactions, two objects exchange bosons. Gravity exchanges graviton bosons. Electromagnetism exchanges photon bosons. Strong nuclear force exchanges gluon bosons. Weak nuclear force exchanges W-particle and Z-particle bosons.
field
All forces are metric gauge fields. Bosons form field around object. Field changes space curvature, from flat to curved. When two objects interact, they go through curved space and change motion accordingly, just as cars turn on banked curves. Acceleration involves boson exchange. Boson exchange and curved spaced have identical effects, because bosons curve space, and space-curvature acceleration releases bosons to interact.
mass and distance
Bosons with no mass, such as gravitons and photons, can exchange to infinite distances. Bosons with mass, such as gluons, W particles, and Z particles, have short action distances.
time
Particles exchanged by forces take time to act, based on force strength. Strong force takes 10^-23 seconds. Electric force takes 10^-21 seconds. Weak force takes 10^-18 seconds to 1000 seconds. Gravitational force takes much longer.
electromagnetism
Electrical force is interaction between two charges. Magnetic force is interaction between two relativistic charges.
weak force
Electric force is 1000 times stronger than weak force.
gravity
Gravitation and electromagnetism are similar, because interactions cause both forces and both forces radiate in all directions. Because masses are positive, gravitational force is attractive. Because charges are positive and negative, electromagnetic force is attractive or repulsive. Electric-force to gravitational-force ratio is 10^36. If gravity is stronger, universe is smaller, and stars are smaller and exist shorter. If gravity is weaker, universe is larger, and stars are larger and exist longer.
Electromagnetic and weak forces unify {electroweak force}| {electric-weak unification theory} under SU(2) x U(1) Lie symmetry group, with gauge symmetry. An adjoint representation uses U(1) subgroup of SU(2), making electroweak Higgs field. W+, W-, and Z bosons and photons are equivalent at proton-diameter distances and high energies. Spontaneous symmetry breaking in current universe causes W+, W-, and Z bosons to have mass and photons to have zero rest mass.
All forces are metric fields {gauge invariant field} {gauge field}| that transmit bosons.
Gravitational force {gravity, mass} is interaction between two masses. Gravitons mediate gravity, by exchange at light speed. Gravity curves space. If only gravity shapes space-time, why does light speed, which depends on electromagnetic force, determine space-time boundaries?
Mass is always positive, and gravity is always attractive. Negative energy is repulsive {antigravity} {anti-gravity}. Negative internal pressure is also repulsive.
Atomic particles that have quarks and gluons interact {strong nuclear force}|. Strong nuclear force is positive and constant over distances more than 10^-14 centimeters, is repulsive over shorter distances, depends on quark number, and is 100 to 1000 times stronger than electric force.
Perhaps, nuclear forces {superweak force}| can mediate between strong and weak nuclear forces.
Leptons and quarks interact {weak nuclear force}|. Quarks and leptons have left-handed and right-handed spin states. Weak forces act on one or the other. For example, left-handed down quark can become up quark, making neutron into proton, electron, and neutrino {radioactive beta decay}. Right-handed down quarks have no change.
antiparticles
Particles that are right-handed or left-handed have weak nuclear force, but their antiparticles do not have weak force.
bosons
Weak-force W-particle and Z-particle bosons can come from vacuum with no conservation, except at very high energy.
distance
Weak nuclear force acts over less than 10^-16 centimeters.
strength
Weak nuclear force is 10^34 times stronger than gravity.
Relation between weak and electromagnetic forces became constant at 10^-12 seconds after universe origin, when U(2) symmetry broke at a rotation angle {electroweak mixing angle} {Weinberg angle}.
Motions {simple harmonic motion}| can oscillate along lines, with acceleration proportional to distance from center point. Molecule-bond vibrations, springs, pendulums, rigid-bar vibrations, rotations, guitar-string vibrations, bridge vibrations, and tall-building sway have simple harmonic motion.
force
Restoring-force strength depends on material type and distance from center. For molecule bonds, spring constant depends on electrical forces between atoms. Restoring force F equals negative of spring constant k expressing restoring force strength times displacement x: F = -k*x. Restoring force is negative because it opposes displacement.
amplitude
Amplitude depends on input energy, which causes more or less displacement.
period
One oscillation takes one time period. Period depends on material restoring force. Period and amplitude are independent. Spring period T is 360 degrees expressed in radians 2*pi times square root of mass m divided by spring constant k: T = 2 * pi * (m/k)^0.5. Higher mass makes longer period. Higher spring constant makes shorter period.
energy
Potential energy PE equals half spring constant k times displacement x squared, which is average force, k*x/2, times distance x: PE = 0.5 * k * x^2. At center, force equals zero, and potential energy equals zero. At maximum displacement amplitude, force and potential energy are highest. At maximum displacement, kinetic energy equals zero, because motion stops as direction reverses. At center, velocity and kinetic energy maximize, because potential energy is zero.
velocity
Maximum velocity v is maximum displacement A times square root of spring constant k divided by object mass m: v = A * (k/m)^0.5. Average velocity is 4*A/T, where A is amplitude and T is period. Average velocity is 2 * v / pi, where v is maximum velocity.
friction
If friction damps simple harmonic motion, amplitude decreases, but frequency stays the same, because material is the same.
When pulled sideways and released, weight {pendulum} hanging by string or wire from point starts oscillating motion.
force
Pendulum restoring force is gravity. Gravity g pulls pendulum-bob mass m back toward center with force F from distance x, depending on displacement angle A: F = m * g * sin(A) = m * k * x.
distance
If pendulum displacement is small, displacement-angle sine equals displacement angle: sin(A) = A. For small displacement, displacement x is displacement angle, expressed in radians, times pendulum length L: x = A*L. For small displacement, constant k is gravity acceleration g divided by pendulum length L: k = g/L.
period
Pendulum period T is 360 degrees, expressed in radians 2*pi, times square root of gravitational-constant reciprocal 1/g: T = 2 * pi * (1/g)^0.5. Longer pendulums have longer periods. Weaker gravity makes longer period. Pendulum mass does not affect period.
Spring oscillation time T {period, oscillation}| is 360 degrees, expressed in radians 2*pi, times square root of mass m divided by spring constant k: T = 2 * pi * (m/k)^0.5. Higher mass makes longer period. Higher spring constant makes shorter period.
Springiness {spring constant, force} depends on length, cross-sectional area, and force strength between molecules. Stiff springs {spring, metal}| have high spring constant.
Motion can cause force in opposite direction {friction}|, from surface-bump collisions as surfaces slide over each other or from electrical forces between close surfaces.
process: sliding
As one surface slides over another surface, surface molecules collide, forcing object backward and upward {kinetic friction}.
process: rolling
If one surface rolls over another surface, collisions do not push surfaces upwards, because surfaces have no sliding. A continuously changing rolling-surface part contacts stationary surface. Bumps and grooves in the surfaces mesh deeper, so contact is greater {static friction}. Static friction is greater than kinetic friction.
factors: force direction
Friction force is directly proportional to force perpendicular to surface. If surface is horizontal, perpendicular force is gravity. If surface is not horizontal, perpendicular friction force F is gravity g times sine of incline-to-horizontal angle A: F = g * sin(A).
factors: area
Contact area between two surfaces affects friction force only slightly, because more area makes pressure and force less, and less area makes pressure and force more.
factors: speed
Faster speed makes less sliding friction, because moving surface rides higher over stationary surface.
factors: electric force
Friction depends on hydrogen bonds and van der Waals electric forces between surface molecules. Smooth surfaces can be in close contact, and so have high electrical attractions and high friction. For example, two smooth glass plates or plastic pieces stick together tightly. Rough-surface molecules are farther apart on average, have smaller electrical attractions, and have less friction.
factors: lubricant
Oil, grease, and graphite can fill surface depressions and keep two surfaces separated, so surfaces have little hydrogen bonding or van der Waals forces, and friction is small. Heavier weight oil flows less easily and sticks to surfaces more, but stickiness causes the objects to have more friction. Lighter weight oil flows more easily and sticks to surfaces less but has less friction. Best-weight oil balances ability to stay in bumps with ability to flow easily. Multi-weight oil flows easily when cold and gets thicker as it gets warmer.
tires
When cornering, radial tires keep more tread on road, so force goes more into road, instead of going into tire side. Radial tires act the same as bias-ply tires while going straight.
Tires with greater radius keep more tread area on road, but this has little effect. Tires with greater radius have larger sidewalls, which can flex more and so become hotter and weaker. Tires with stiff and narrow sides stay cooler and stronger but have harsher ride.
Wider tires have increased area, but more area does not affect friction much. Wider tires can find dry or debris-free road parts, while smaller tires contact only sand or water. Wider tires can aquaplane more on water or snow, because fluid cannot leave treads fast enough.
Friction force knocks molecules from surface through collisions and wears away surface {abrasion, surface}|.
Simple machines {simple machine} use small force exerted over long distance or large force exerted over short distance.
lever
Rods, such as crowbars, can have balance point near one end. Short radius moves short distance and exerts large force, as long radius moves long distance and needs only small force.
inclined plane
Ramps can lift heavy objects short vertical distances, by using small forces over long horizontal distances.
screw
Inclined planes can wind around axis, to allow lifting water short vertical distances using small forces over long horizontal distances.
Wheels on axles {wheel and axle} {axle and wheel} allow objects to move horizontally with no sliding friction. Pulley wheels on axles can hold rope, which can lift weight short vertical distances by pulling rope using small forces over long horizontal distances. Pulleys can alter force direction. One rope can wrap around two pulleys to make a block and tackle, to lift heavy objects short distances with small forces applied over long distances.
Construction levers {crane, machine} {derrick} lift and lower objects using counterbalancing weight.
Forces can redirect using movable connections {joint, connector}|.
Force on ladder steps transmits to both sides and then to ground. Ladders {ladder} have angle to ground. If angle is too large, frictional torque is too small to balance weight torque, and ladder feet slip.
Forces can redirect using connections {strut}| that exert force inward or outward.
Double inclined planes {wedge, machine}|, such as axes and knives, use small force over long distance to exert large force over short distance, to split things or push grooves into softer material.
Actual mechanical-advantage to ideal mechanical-advantage ratio {mechanical advantage}| {efficiency, dynamics} inversely relates to friction. Actual mechanical advantage is lower than ideal mechanical advantage. Actual force Fi exerted on machine divides into force Fo exerted on object-to-move {actual mechanical advantage}: Fo / Fi. Distance moved by output force divides into distance si moved by input force {ideal mechanical advantage}: si / so.
A dynamic quantity {momentum}| depends on mass and velocity. Momentum p equals mass m times velocity v: p = m*v. Constant force F times time change (tf - ti) makes momentum p: p = F * (tf - ti) = m * a * (tf - ti) = m*v. Force F is momentum change dp per time change dt: F = dp / dt.
Momentum {angular momentum}| can be through angles around rotation axes or points. Momentum p equals mass m times velocity v: p = m*v. Tangential momentum pt equals mass m times tangential velocity vt, which equals angular velocity w times circle curvature radius r: pt = m * vt = m*w*r. Angular momentum L equals tangential momentum pt times radius r: L = pt * r.
Moment of inertia I equals mass m times radius r squared: I = m * r^2. Angular momentum L equals moment of inertia I times angular velocity w: L = pt * r = (m*w*r) * r = (m * r^2) * w = I*w.
torque
Force F equals mass m times acceleration a: F = m * a. Tangential force Ft equals mass m times tangential acceleration at: Ft = m * at = m * dv / dt = m * r * dw / dt, where dv is velocity change, dt is time change, dw is angular-velocity change, and r is curvature radius. Torque T equals tangential force Ft times radius r: T = Ft * r = (m * at) * r = m * r * (dw / dt) * r = (m * r^2) * dw / dt = I * (dw / dt) = I * aa, where m is mass, at is tangential acceleration, r is curvature radius, dw is angular-velocity change, dt is time change, I is moment of inertia I, and aa is angular acceleration.
Force F equals momentum change dp over time change dt: F = dp / dt. Tangential force Ft equals tangential momentum change dp over time change dt: Ft = dp / dt = m * r * dw / dt, where m is mass, r is curvature radius, dw is angular-velocity change, and dt is time change. Torque T equals angular momentum change dL over time change dt: T = dL / dt = (m * r^2) * dw / dt, where m is mass, r is curvature radius, and dw is angular-velocity change.
Force acts over time {impulse}|, to change object motion.
Force tends to alter object structure. Force can apply over area {pressure, physics}|. Pressure P equals force F per unit area A: P = F/A. Pressure P is energy E divided by volume V: P = F/A = (F*s) / (A*s) = E/V. Pressure is energy density: P = F/A = F / s^2 = F*s / s^3 = E/V.
Force causes structure change {strain}| in force direction. Pressure changes length. For example, gravity pulls feet into floor and tends to flatten feet, shoes, and floor. Strain is proportional to stress.
Force and pressure {stress, physics}| tend to alter object structure. I-beam, twisting torsion bars, and standing up use material stresses, strains, and strengths.
Mechanical forces {compression}| can be pushes on materials.
Mechanical forces {tension, force}| can be pulls on materials.
Mechanical forces {torsion, force}| can be twists on materials. Torsion is simultaneous push in one direction and pull in another direction.
Stress S is proportional to length change dL divided by length L {Hooke's law} {Hooke law}: S = k * (dL / L). Pressure compresses material against restoring force. Restoring force F varies with material type k and distance compressed x: F = -k * x. Pressure is compression force F divided by area A: P = F/A. Pressure equals restoring force -k*x divided by area A: P = F/A = -k * x / A. Constant k depends on Young's or other elasticity modulus u, cross-sectional area A, and object length L: k = A * u / L.
If pressure exceeds limiting value {elastic limit}|, material inelastically changes shape permanently. Higher temperature lowers elastic limit.
If pressure is less than elastic limit, material returns to original shape after removing pressure {elasticity, physics}|. Elasticity can be bulk strength, shear strength, or tensile strength. Elasticity varies for different material axes.
If pressure exceeds elastic limit, material changes shape permanently {inelasticity}|.
Elastic modulus reciprocal {compliance}| can indicate shape-changing ease.
Elasticity {Young's modulus, elasticity} {Young modulus, elasticity} varies for different material types. Steel is hard and bouncy and has high Young's modulus. Wood is soft and has low Young's modulus.
Elasticity varies for compression {bulk strength}|.
Elasticity varies for torsion {shear strength}|.
Elasticity varies for tension {tensile strength}|.
All forces on objects result in net force {resultant, forces}|. Force-vector sum equals resultant-force vector. Vector resultants explain motions in sailing, billiard balls colliding, planes flying, crowbar leveraging, wrenches twisting, gears turning, wrecking balls breaking, roller coaster riding, and car rocking to get out of mud or snow.
Parallel forces {couple, physics}| can act in opposite directions on axis ends, to tend to cause rotation.
All system forces can add to zero {equilibrium, physics}|. All system velocities can add to zero {static equilibrium}. Objects have gravity centers. When objects move, gravity centers can rise {stable equilibrium}, fall {unstable equilibrium}, or stay the same {neutral equilibrium}.
Pressure, temperature, concentration, or force can change over distance {gradient, physics}|. Gradients are like forces and cause flow in resistant medium. Flow increases until reaching terminal velocity. Terminal velocity determines flux.
Force pushes or pulls objects along line between interacting-object centers. When one object exerts force in a direction on another object, second object exerts same force in opposite direction on first object {reaction, forces}. Relation is symmetric.
examples
Rockets burn fuel. Hot gas pushes rocket forward, while rocket pushes gas backward in opposite direction.
For objects on coil springs, coil spring pushes object up, while object pushes coil spring down.
When walking, foot pushes back against ground, and ground pushes back on person, to send body forward.
An hourglass on a balance has dropping sand, but balance stays still, because weight is constant. Downward collisions balance upward force.
Objects can spin or orbit or rotate around balance point {rotation, force}. Rotations rotate around point {center of rotation} {rotation center} {balance point}. Forces or weights are at distances {radius, rotation} from balance point or rotation center. On levers, forces can act perpendicular, or at an angle, to radius. Weighing balances and seesaws are levers. Weights on strings can orbit, so string provides centripetal force, and spin provides centrifugal force.
Spinning objects, including Earth, bulge at equator {equatorial bulge} and flatten at poles, by centrifugal force.
Just as mass traveling in straight line has inertia that tends to keep velocity constant, mass rotating around axis has inertia {moment of inertia}| {inertia moment} that tends to keep angular velocity constant.
mass
Inertia depends directly on mass. Moment of inertia substitutes for mass when quantities use angular velocity instead of velocity. When mass rotates around rotation center at radius r, tangential momentum pt is mass m times tangential velocity vt: pt = m * vt. Angular momentum L is moment of inertia I times angular velocity w: L = I * w. Tangential velocity vt varies directly with angular velocity w: vt = w * r. Tangential momentum pt varies directly with angular momentum L: L = pt * r. Moment of inertia I depends on mass m and radius r: L = I * w = pt * r = m * w * r * r = (m * r^2) * w. Moment of inertia I is mass m times square of distance r from axis or point: I = m * r^2.
summation
Masses have volume, so object points have different radii from rotation center. Total moment of inertia is sum of moments of inertia at each radius.
Thin-ring moment of inertia equals total mass m times square of distance r from ring center to ring middle: m * r^2.
Disk or cylinder moment of inertia equals half total mass m times square of distance r from disk or cylinder center to outer edge: 0.5 * m * r^2.
Pipe or doughnut moment of inertia equals half total mass m times sum of squares of distances from pipe or doughnut center to inner edge ri and outer edge ro: 0.5 * m * (ri^2 + ro^2).
radius
Objects with moment of inertia around rotation center have moment of inertia around any axis parallel to rotation axis. New moment of inertia Inew is old moment of inertia Iold plus total object mass m times square of distance d between axes: Mnew = Mold + m * d^2.
Forces {torque}| can tend to cause motions around rotation centers.
acceleration
Torque causes angular acceleration. For example, force can act on a rigid rod that can turn around a balance point. Force can act perpendicular to rod or at another angle. Torque T is force F times radius r from balance point times sine of force-to-radius angle A: T = F * r * sin(A), which is cross product of force and radius vectors: T = F X r. Torque-vector direction is perpendicular to both force vector and radius vector and parallel to axis.
moment of inertia
Tangential force Ft equals mass m times tangential acceleration at, which equals angular acceleration aa times radius r: Ft = m * at = aa * r. If torque acts perpendicular to radius, torque T equals moment of inertia (I = m*r^2) times angular acceleration aa: T = Ft * r = m * at * r = m * aa * r * r = m * (r^2) * aa = I * aa.
examples
Frisbees and yo-yos have torques. Torque causes car front to fall when car stops. Torque causes car front to rise when car accelerates. To open door, push farthest from hinge to apply least force, because radius is greatest. Spins in ice-skating begin with torque. Gymnasts and divers apply torque. Torque causes spin on footballs, bullets, bicycle wheels, helicopter blades, and propellers. Scales use opposing torques to weigh objects.
equilibrium
When lever is not moving around balance point {equilibrium}, right Tr and left Tl side torques, F * r * sin(A), are equal: Tl = F1l * r1l * sin(A1l) + F2l * r2l * sin(A2l) + ... = F1r * r1r * sin(A1r) + F2r * r2r * sin(A2r) + ... = Tr.
In curved motion, force can go away from curvature center {centrifugal force}|, along radius direction. For example, Moon is in orbit around Earth.
In curved motion, force can go toward curvature center {centripetal force}|, along radius direction. For example, Moon is in orbit around Earth.
Moon falls toward Earth center by Earth gravity {free fall}|. Gravity is centripetal force. Orbital speed moves Moon tangentially in orbit. Tangential movement accelerates Moon away from Earth along radius. This acceleration is centrifugal force. Centrifugal force and centripetal force are equal, and motion rates away from and toward center are equal, so Moon maintains approximately same distance from Earth.
weightless
Moon in orbit has no weight, because centrifugal force equals centripetal force, just as astronauts in orbit are weightless. If jumping from height, in free fall, one feels weightless, because no force is opposing fall.
Kepler formulated three planetary-motion laws {Kepler's laws} {Kepler laws}.
first law
Radius from orbiting body to Sun sweeps out equal areas in equal times, because velocity is slow at large radius and fast at small radius.
second law
If object in orbit moves closer, speed increases as potential energy changes to kinetic energy and moves object back outward. If object in orbit moves farther away, speed decreases as kinetic energy changes to potential energy and moves object inward.
If spinning object becomes more compact, radius decreases and speed increases as potential energy changes to kinetic energy. If spinning object expands, radius increases and speed decreases as kinetic energy changes to potential energy.
third law
Acceleration cubed is directly proportional to time squared, because acceleration is highest at greatest curvature point, where velocity is highest.
Two objects in different orbits interact by gravity or electromagnetism to make torque that changes object spin axis {nutation}|.
Earth spins on an axis that is at an angle to axis of Earth orbit around Sun. Sun gravity causes torque on Earth axis and causes it to rotate {precession}| {precession of the equinoxes, Earth}, as angular velocity around axis interacts with angular velocity around orbit. Object spin and orbital motion interact to cause spin-axis precession.
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Date Modified: 2022.0225