[{"data":1,"prerenderedAt":1292},["ShallowReactive",2],{"subject:mechanics":3,"course-wordcounts":75,"nav:mechanics":987},{"id":4,"title":5,"blurb":6,"body":7,"brief":18,"category":56,"description":57,"draft":58,"extension":59,"meta":60,"module":15,"navigation":25,"path":61,"practice":62,"rawbody":63,"readingTime":64,"seo":69,"sources":70,"status":71,"stem":72,"summary":15,"topics":73,"__hash__":74},"course\u002F01.mechanics\u002Findex.md","Mechanics & Dynamics","How forces change motion, how conservation laws simplify many-body systems, and\nhow the same mechanics extends from projectiles and rotation to fluids, waves,\nand thermal processes.\n",{"type":8,"value":9,"toc":14},"minimark",[10],[11,12,13],"p",{},"Mechanics asks a single question in many forms: given how a system interacts with\nits surroundings, how does it move? The sequence starts with kinematics — the pure\ngeometry of position, velocity, and acceleration — then introduces Newton's laws,\nwhich turn forces into equations of motion. Because those equations are often\nawkward to solve head-on, the subject develops the conservation laws that\nshort-circuit them: energy, linear momentum, and angular momentum, each a quantity\nthat stays fixed while the details churn. With those tools in hand it moves through\nrigid-body rotation, gravitation and orbits, static equilibrium, the statics and\ndynamics of fluids, and finally oscillations, waves, and thermodynamics. Each topic\nreuses the same handful of principles, so the reach of the framework grows while\nits foundations stay small, following Tipler & Mosca, Parts I–III.",{"title":15,"searchDepth":16,"depth":16,"links":17},"",2,[],[19,21,26,30,32,34,38,40,42,46,48,52,54],{"p":20},"Mechanics is the study of how interactions change motion. Specify the forces\non a system and you can, in principle, predict its entire future — where it\ngoes, how fast, and when.\n",{"fig":22,"n":23,"caption":24,"large":25},"mech-projectile","001","Projectile motion: constant horizontal velocity, gravity bending the path\ninto a parabola, the velocity vector tangent at every point.\n",true,{"fig":27,"n":28,"caption":29},"mech-pendulum","002","A pendulum: gravity's restoring torque gives simple harmonic motion for\nsmall swings.\n",{"p":31},"Kinematics comes first — position, velocity, and acceleration as the\ngeometry of motion, before any mention of what causes it. Vectors carry\nthat geometry into two and three dimensions.\n",{"p":33},"Then \u003Cstrong>Newton's laws\u003C\u002Fstrong> supply the cause. Force equals mass\ntimes acceleration is a differential equation for the trajectory, and the\nart is learning to read the forces off a situation and write it down.\n",{"fig":35,"n":36,"caption":37},"mech-freebody","003","The free-body diagram: isolate the object and draw only the forces acting\non it — weight, normal, friction.\n",{"p":39},"Solving the equations directly is often hard, so mechanics leans on\n\u003Cem>conservation laws\u003C\u002Fem>. Energy is the first: work done on a body\nchanges its kinetic energy, and stored potential energy converts back into\nmotion.\n",{"p":41},"\u003Cstrong>Momentum\u003C\u002Fstrong> is the second, and it makes collisions and\nmany-body problems tractable. Whatever the internal forces, the total\nmomentum of an isolated system is unchanged.\n",{"fig":43,"n":44,"caption":45},"mech-collision","004","An elastic collision of equal masses: the velocities simply swap, momentum\nand energy both conserved.\n",{"p":47},"Extend the same ideas to spinning bodies and you get rotational inertia,\ntorque, and \u003Cstrong>angular momentum\u003C\u002Fstrong> — the conserved quantity\nbehind everything from a spinning top to a planet's orbit.\n",{"fig":49,"n":50,"caption":51},"mech-orbit","005","A Keplerian orbit: an ellipse with the star at one focus, the radius\nsweeping equal areas in equal times.\n",{"p":53},"Gravitation ties it together. One inverse-square law explains falling\napples and planetary orbits alike, and Kepler's empirical rules fall out of\nNewton's mechanics as a consequence.\n",{"p":55},"From there the same framework reaches static equilibrium, fluids at rest\nand in flow, oscillations and travelling waves, and the thermal behaviour of\nmatter — one language for the mechanical world.\n","physics","Mechanics connects a system's interactions to its motion. Kinematics supplies the\ngeometry; Newton's laws supply the local equations; energy, momentum, and angular\nmomentum expose conserved structure. The same framework reaches rigid bodies,\ngravitation, equilibrium, fluids, oscillations, waves, and thermodynamics, following\nTipler & Mosca, Parts I–III.\n",false,"md",{},"\u002Fmechanics",[],"---\ntitle: Mechanics & Dynamics\nstatus: available\ncategory: physics\nblurb: |\n  How forces change motion, how conservation laws simplify many-body systems, and\n  how the same mechanics extends from projectiles and rotation to fluids, waves,\n  and thermal processes.\ndescription: |\n  Mechanics connects a system's interactions to its motion. Kinematics supplies the\n  geometry; Newton's laws supply the local equations; energy, momentum, and angular\n  momentum expose conserved structure. The same framework reaches rigid bodies,\n  gravitation, equilibrium, fluids, oscillations, waves, and thermodynamics, following\n  Tipler & Mosca, Parts I–III.\nbrief:\n  - p: |\n      Mechanics is the study of how interactions change motion. Specify the forces\n      on a system and you can, in principle, predict its entire future — where it\n      goes, how fast, and when.\n  - fig: mech-projectile\n    n: \"001\"\n    caption: |\n      Projectile motion: constant horizontal velocity, gravity bending the path\n      into a parabola, the velocity vector tangent at every point.\n    large: true\n  - fig: mech-pendulum\n    n: \"002\"\n    caption: |\n      A pendulum: gravity's restoring torque gives simple harmonic motion for\n      small swings.\n  - p: |\n      Kinematics comes first — position, velocity, and acceleration as the\n      geometry of motion, before any mention of what causes it. Vectors carry\n      that geometry into two and three dimensions.\n  - p: |\n      Then \u003Cstrong>Newton's laws\u003C\u002Fstrong> supply the cause. Force equals mass\n      times acceleration is a differential equation for the trajectory, and the\n      art is learning to read the forces off a situation and write it down.\n  - fig: mech-freebody\n    n: \"003\"\n    caption: |\n      The free-body diagram: isolate the object and draw only the forces acting\n      on it — weight, normal, friction.\n  - p: |\n      Solving the equations directly is often hard, so mechanics leans on\n      \u003Cem>conservation laws\u003C\u002Fem>. Energy is the first: work done on a body\n      changes its kinetic energy, and stored potential energy converts back into\n      motion.\n  - p: |\n      \u003Cstrong>Momentum\u003C\u002Fstrong> is the second, and it makes collisions and\n      many-body problems tractable. Whatever the internal forces, the total\n      momentum of an isolated system is unchanged.\n  - fig: mech-collision\n    n: \"004\"\n    caption: |\n      An elastic collision of equal masses: the velocities simply swap, momentum\n      and energy both conserved.\n  - p: |\n      Extend the same ideas to spinning bodies and you get rotational inertia,\n      torque, and \u003Cstrong>angular momentum\u003C\u002Fstrong> — the conserved quantity\n      behind everything from a spinning top to a planet's orbit.\n  - fig: mech-orbit\n    n: \"005\"\n    caption: |\n      A Keplerian orbit: an ellipse with the star at one focus, the radius\n      sweeping equal areas in equal times.\n  - p: |\n      Gravitation ties it together. One inverse-square law explains falling\n      apples and planetary orbits alike, and Kepler's empirical rules fall out of\n      Newton's mechanics as a consequence.\n  - p: |\n      From there the same framework reaches static equilibrium, fluids at rest\n      and in flow, oscillations and travelling waves, and the thermal behaviour of\n      matter — one language for the mechanical world.\n---\n\nMechanics asks a single question in many forms: given how a system interacts with\nits surroundings, how does it move? The sequence starts with kinematics — the pure\ngeometry of position, velocity, and acceleration — then introduces Newton's laws,\nwhich turn forces into equations of motion. Because those equations are often\nawkward to solve head-on, the subject develops the conservation laws that\nshort-circuit them: energy, linear momentum, and angular momentum, each a quantity\nthat stays fixed while the details churn. With those tools in hand it moves through\nrigid-body rotation, gravitation and orbits, static equilibrium, the statics and\ndynamics of fluids, and finally oscillations, waves, and thermodynamics. Each topic\nreuses the same handful of principles, so the reach of the framework grows while\nits foundations stay small, following Tipler & Mosca, Parts I–III.\n",{"text":65,"minutes":66,"time":67,"words":68},"1 min 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and Dimensions","\u002Fmechanics\u002Ffoundations\u002Fmeasurement-and-dimensions",[989],"Every physical quantity is a number attached to a unit, and that pairing is what lets you check an equation before computing anything, since terms that add together must carry the same dimensions. We build the SI base units and the notion of dimension, then use dimensional analysis to test a proposed relation and form scaling groups — a method that fixes a formula's shape but never its numerical constants. The lesson also sets how precisely a result may be stated, through significant figures, propagated uncertainty, and order-of-magnitude checks that catch errors a raw calculator answer hides.\n",{"title":999,"path":1000,"lessonNumber":16,"topics":1001,"summary":1002},"Vector Algebra","\u002Fmechanics\u002Ffoundations\u002Fvector-algebra",[989],"Force, velocity, and displacement all carry a direction, so mechanics needs an arithmetic that respects it; adding magnitudes alone gives the wrong answer the moment two arrows point different ways. We set up vectors and their components in a chosen basis, then build the two products that carry most of the physics — the dot product, which extracts the part of one vector along another and yields work and power, and the cross product, which measures oriented area and yields torque and angular momentum. Rotating the axes changes the components while leaving the vector itself untouched, and the same component method resolves a force along whatever directions a constraint picks out.\n",{"module":1004,"moduleNumber":16,"slug":1005,"lessons":1006},"Kinematics","kinematics",[1007,1012,1017,1023,1029],{"title":1008,"path":1009,"lessonNumber":990,"topics":1010,"summary":1011},"One-Dimensional Motion","\u002Fmechanics\u002Fkinematics\u002Fone-dimensional-motion",[1004],"Motion along a line already forces the two questions the whole of kinematics repeats: how fast is the object moving now, and where will it be next? Velocity and acceleration answer the first as derivatives of position; integrating them back — the signed area under a graph — answers the second. We derive the constant-acceleration equations, mark exactly where the \"constant\" assumption is load-bearing, and see why sign, not magnitude, is what carries direction.\n",{"title":1013,"path":1014,"lessonNumber":16,"topics":1015,"summary":1016},"Motion Graphs","\u002Fmechanics\u002Fkinematics\u002Fmotion-graphs",[1004],"Draw a motion as a graph and its two most useful facts turn geometric: the slope of the position curve is the velocity, and the area under the velocity curve is the displacement. We read motion in both directions — differentiating a graph for the next rate, integrating it back to recover position — and handle the curved, piecewise, and noisy graphs that real measurements produce. Along the way we see why a velocity estimated from two positions belongs to the midpoint of their interval, not its end.\n",{"title":1018,"path":1019,"lessonNumber":1020,"topics":1021,"summary":1022},"Projectile Motion","\u002Fmechanics\u002Fkinematics\u002Fprojectile-motion",3,[1004],"Throw an object and it seems to trace one curved path, but the motion is really two independent one-dimensional motions running at once: constant velocity across the ground and free fall in the vertical. Splitting it that way turns every projectile question — how long it stays up, how far it lands, how high it climbs, whether it clears an obstacle — into a pair of equations you already know. We derive the parabolic trajectory, work both the forward and the inverse problems, and show why the familiar $45^\\circ$ range-maximizing angle holds only when launch and landing heights match.\n",{"title":1024,"path":1025,"lessonNumber":1026,"topics":1027,"summary":1028},"Relative Motion","\u002Fmechanics\u002Fkinematics\u002Frelative-motion",4,[1004],"A velocity is only ever measured relative to some observer, so a boat's speed through the water, over the ground, and as seen from another boat are three different vectors. Choosing the right frame — and subtracting one motion from another — collapses river crossings, crosswind headings, pursuit, and closest-approach problems into a single vector equation. We build the relative-velocity and relative-position relations for uniformly moving frames, show why acceleration is the one quantity all such observers agree on, and note where rotating frames break the simple subtraction.\n",{"title":1030,"path":1031,"lessonNumber":1032,"topics":1033,"summary":1034},"Circular Motion","\u002Fmechanics\u002Fkinematics\u002Fcircular-motion",5,[1004],"An object going around a circle at a steady speed is still accelerating, because its velocity is forever changing direction — the fact that governs everything from a car on a curve to a satellite in orbit. We tie the angular description (angle, angular velocity, angular acceleration) to the linear one through $v=r\\omega$, split the acceleration into an inward part that turns the velocity and a tangential part that changes its speed, and extend the inward $v^2\u002Fr$ result to any curved path through its local radius of curvature. Constant angular acceleration then mirrors straight-line motion equation for equation.\n",{"module":1036,"moduleNumber":1020,"slug":1037,"lessons":1038},"Dynamics","dynamics",[1039,1044,1049,1054,1059],{"title":1040,"path":1041,"lessonNumber":990,"topics":1042,"summary":1043},"Newton's Laws","\u002Fmechanics\u002Fdynamics\u002Fnewtons-laws",[1036],"What makes a body change its motion, and in which frames does the answer take its simplest form? Newton's three laws settle both: inertial frames are the ones where a force-free body coasts, force is whatever changes momentum, and every interaction pushes back on its source. We write the second law as $\\sum\\vec F=\\d\\vec p\u002F\\d t$, reduce it to $m\\vec a$ at constant mass, and separate what a scale actually reads — the support force — from the weight $m\\vec g$ it is so often mistaken for.\n",{"title":1045,"path":1046,"lessonNumber":16,"topics":1047,"summary":1048},"Free-Body Diagrams","\u002Fmechanics\u002Fdynamics\u002Ffree-body-diagrams",[1036],"Once several forces act on a body at once, the reliable way to predict its motion is to isolate that one body and draw every external push and pull on it — nothing more, nothing less. The free-body diagram is that discipline. We fix a system boundary, resolve $\\sum\\vec F=m\\vec a$ into components along axes chosen to fit the geometry, and solve for the unknowns a problem hands us — normal forces, tensions, friction, and the acceleration a constraint permits — seeing why internal forces drop out only when the boundary encloses both bodies that share them.\n",{"title":1050,"path":1051,"lessonNumber":1020,"topics":1052,"summary":1053},"Friction and Curved Motion","\u002Fmechanics\u002Fdynamics\u002Ffriction-and-curved-motion",[1036],"Real surfaces grip before they slip, fluids push back harder the faster you move through them, and anything rounding a bend must be pulled toward the inside of the curve by something. This lesson supplies the force laws for those three cases. We bound static friction by $|f_s|\\leq\\mu_sN$ and switch to kinetic friction $\\mu_kN$ once sliding starts, model drag as a speed-dependent resistance that levels off at a terminal speed, and show that circular motion demands an inward net force $mv^2\u002Fr$ furnished by real interactions — friction, a banked normal force, tension — never by an invented outward one.\n",{"title":1055,"path":1056,"lessonNumber":1026,"topics":1057,"summary":1058},"Numerical Dynamics","\u002Fmechanics\u002Fdynamics\u002Fnumerical-dynamics",[1036],"Most force laws — quadratic drag, coupled oscillators, anything nonlinear — admit no closed-form trajectory, so we advance the motion one small time step at a time and let arithmetic do what algebra cannot. This lesson turns $\\d\\vec y\u002F\\d t=f(t,\\vec y)$ into a marching rule. We derive the Euler, Euler--Cromer, midpoint, and Verlet updates, weigh their accuracy and stability, watch a drifting energy expose a bad scheme, and use step-halving and conserved quantities to separate the error of the method from the error of the model.\n",{"title":1060,"path":1061,"lessonNumber":1032,"topics":1062,"summary":1063},"Center-of-Mass Systems","\u002Fmechanics\u002Fdynamics\u002Fcenter-of-mass-systems",[1036],"A firework bursts into a dozen fragments, yet one point keeps gliding along the original parabola as though nothing had happened. That point is the centre of mass, and following it collapses a many-body tangle into a single equation of motion. We define $\\vec R=\\frac1M\\sum_i m_i\\vec r_i$ and its continuous form, show that internal forces cancel so that only external ones move it, $M\\vec A_{\\rm cm}=\\sum\\vec F_{\\rm ext}$, and put the result to work on recoil, collisions viewed from the centre-of-mass frame, and rocket propulsion, where mass leaving the boundary carries momentum with it.\n",{"module":1065,"moduleNumber":1026,"slug":1066,"lessons":1067},"Energy","energy",[1068,1073,1078,1083,1088],{"title":1069,"path":1070,"lessonNumber":990,"topics":1071,"summary":1072},"Work and Kinetic Energy","\u002Fmechanics\u002Fenergy\u002Fwork-and-kinetic-energy",[1065],"A constant push along a straight path is trivial to score, but real forces vary and bend along curved trajectories, and only the component along the motion transfers any energy. Work captures exactly that transfer as the line integral $W=\\int\\vec F\\cdot\\d\\vec r$, and the work-kinetic-energy theorem turns it into a statement about speed: the net work on a particle equals the change in its $\\tfrac12 mv^2$. We build work up from the dot product to the signed area under a force curve, derive the theorem from Newton's second law, and read power as its instantaneous rate $P=\\vec F\\cdot\\vec v$.\n",{"title":1074,"path":1075,"lessonNumber":16,"topics":1076,"summary":1077},"Potential Energy","\u002Fmechanics\u002Fenergy\u002Fpotential-energy",[1065],"When a force does the same work no matter which path a particle takes, that work can be stored as a function of position alone, and solving for the motion becomes bookkeeping instead of integration. We single out the forces that qualify — the conservative ones, for which $\\oint\\vec F\\cdot\\d\\vec r=0$ — define their potential energy through $\\vec F=-\\nabla U$, and use conservation of mechanical energy to read speeds, turning points, and equilibria straight off a potential curve. Friction breaks the shortcut, so we also track where mechanical energy leaks away as heat.\n",{"title":1079,"path":1080,"lessonNumber":1020,"topics":1081,"summary":1082},"Multiparticle Work","\u002Fmechanics\u002Fenergy\u002Fmultiparticle-work",[1065],"A single particle has one velocity and one kinetic energy; a system of many can spin, deform, explode, and warm up while its centre of mass glides along as if nothing happened. Splitting the motion into a centre-of-mass part and an internal part separates the energy that momentum already fixes from the energy left free for relative motion, $K=\\tfrac12MV_{\\rm cm}^2+K'$. We derive the centre-of-mass work theorem, see why an explosion or a released spring can raise total kinetic energy with no external work at all, and use the reduced-mass and centre-of-mass frames to make collisions and internal transfers clean.\n",{"title":1084,"path":1085,"lessonNumber":1026,"topics":1086,"summary":1087},"Mass-Energy and Binding","\u002Fmechanics\u002Fenergy\u002Fmass-energy-and-binding",[1065],"Relativity puts rest itself on the energy ledger: a mass $m$ carries energy $mc^2$ even when it sits still, so weighing a system's separated pieces and weighing the assembled whole give different answers, and the gap is binding energy. We convert freely between mass units and MeV, compute the energy that holds a nucleus together, and read the binding-energy-per-nucleon curve that explains why fusing light nuclei and splitting heavy ones both release energy. Reaction $Q$ values, thresholds, and recoil then follow from the same mass-difference accounting, once the frame and mass convention are fixed.\n",{"title":1089,"path":1090,"lessonNumber":1032,"topics":1091,"summary":1092},"Photons and Quantization","\u002Fmechanics\u002Fenergy\u002Fphotons-and-quantization",[1065],"Light delivers its energy in indivisible lumps: a photon of frequency $f$ carries exactly $hf$, and this one fact explains why a dim blue lamp ejects electrons that an intense red one cannot. We fix a photon's energy and momentum from its wavelength, follow the quanta through emission, absorption, and the photoelectric threshold $K_{\\rm max}=hf-\\phi$, and watch energy and momentum conservation together produce the Compton wavelength shift when a photon scatters from an electron. The recurring discipline is unit and frame care, where a stray factor of $10^9$ or a forgotten rest energy quietly ruins an answer.\n",{"module":1094,"moduleNumber":1032,"slug":1095,"lessons":1096},"Momentum","momentum",[1097,1102,1107],{"title":1098,"path":1099,"lessonNumber":990,"topics":1100,"summary":1101},"Momentum and Collisions","\u002Fmechanics\u002Fmomentum\u002Fmomentum-and-collisions",[1094],"When two objects collide, the forces between them are too brief and too tangled to integrate directly, yet the result is fixed by one conserved quantity. Linear momentum $\\vec p=m\\vec v$ turns Newton's second law into the impulse-momentum theorem $\\vec J=\\Delta\\vec p$, and for an isolated system into a conservation law that holds through any internal collision, however dissipative. We use it to separate elastic from inelastic collisions, follow the centre of mass, and read impulse as the signed area under a force-time curve — always tracking which external impulses the chosen system and interval let us drop.\n",{"title":1103,"path":1104,"lessonNumber":16,"topics":1105,"summary":1106},"Center-of-Mass Collisions","\u002Fmechanics\u002Fmomentum\u002Fcenter-of-mass-collisions",[1094],"A two-body collision that looks asymmetric in the laboratory becomes almost trivial in the frame that rides along with the centre of mass, where the total momentum is zero and the two momenta stay equal and opposite. We build that frame, reduce the pair to a single relative coordinate carrying the reduced mass $\\mu$, and show that an elastic collision there only rotates one momentum vector while its length holds fixed. Transforming back to the laboratory then handles elastic and inelastic collisions, scattering angles, and reaction thresholds with the same construction — and shows why relative speed, not laboratory kinetic energy, measures what a collision can convert.\n",{"title":1108,"path":1109,"lessonNumber":1020,"topics":1110,"summary":1111},"Rocket Propulsion","\u002Fmechanics\u002Fmomentum\u002Frocket-propulsion",[1094],"A rocket speeds up by throwing mass backward, so its own mass drops as it flies and $\\vec F=m\\vec a$ no longer applies to a fixed body. Tracking the momentum the exhaust carries across the vehicle boundary gives thrust $T=Ru_e$ and, for a force-free burn, the rocket equation $\\Delta v=u_e\\ln(m_i\u002Fm_f)$ — a logarithm that makes large velocity changes expensive in propellant and forces staging. We then add the forces a real ascent cannot ignore, gravity, drag, and steering, and show how thrust and mass-flow records are cross-checked to infer the exhaust speed.\n",{"module":1113,"moduleNumber":1114,"slug":1115,"lessons":1116},"Rotation",6,"rotation",[1117,1122,1127,1132,1137,1142],{"title":1118,"path":1119,"lessonNumber":990,"topics":1120,"summary":1121},"Rotational Inertia","\u002Fmechanics\u002Frotation\u002Frotational-inertia",[1113],"Push a wheel and a merry-go-round with the same force and they speed up at wildly different rates: the same mass resists rotation differently depending on where it sits relative to the axis. That single fact is the moment of inertia, $I=\\int r_\\perp^2\\,\\d m$, and this lesson builds it from the ground up. We tie angular motion to linear through $s=r\\theta$, $v=r\\omega$, and $a_t=r\\alpha$, derive $I$ for rods, disks, and spheres, and use the parallel- and perpendicular-axis theorems to move between axes — always naming the axis, because the same body has as many moments of inertia as it has lines to spin about.\n",{"title":1123,"path":1124,"lessonNumber":16,"topics":1125,"summary":1126},"Rotational Dynamics","\u002Fmechanics\u002Frotation\u002Frotational-dynamics",[1113],"A force applied to a wheel does nothing unless it acts off the axis: what turns a rigid body is torque, force times lever arm. This lesson makes that precise and turns it into the rotational Newton's second law, $\\sum\\tau=I\\alpha$ about a fixed axis, the exact analogue of $\\sum F=ma$. From there we get rotational work $W=\\int\\tau\\,\\d\\theta$ and power $P=\\tau\\omega$, size a motor to a load, and solve pulleys and Atwood machines where the pulley's own inertia can no longer be ignored — always insisting that every torque be measured about the same axis.\n",{"title":1128,"path":1129,"lessonNumber":1020,"topics":1130,"summary":1131},"Rolling Motion","\u002Fmechanics\u002Frotation\u002Frolling-motion",[1113],"A rolling wheel is doing two things at once — translating and spinning — but the no-slip condition $v_{cm}=R\\omega$ locks them together, and that single constraint is what makes rolling tractable. We use it to split the kinetic energy into $\\tfrac12Mv_{cm}^2+\\tfrac12I\\omega^2$, find how fast a cylinder reaches the bottom of an incline, and show why the contact point is instantaneously at rest. The static friction that enforces rolling does no work; we track its direction from the tendency to slip, and mark exactly where the model breaks once the required friction exceeds $\\mu_sN$.\n",{"title":1133,"path":1134,"lessonNumber":1026,"topics":1135,"summary":1136},"Angular Momentum","\u002Fmechanics\u002Frotation\u002Fangular-momentum",[1113],"A skater pulls in her arms and spins faster, with no torque acting: that is angular momentum conservation, and it lets us answer questions that would be hopeless force by force. We build $\\vec L=\\vec r\\times\\vec p$, show it obeys $\\vec\\tau_{ext}=\\d\\vec L\u002F\\d t$, and use its conservation under zero external torque to link before and after in collisions, reconfigurations, and coupled rotors without ever resolving the internal forces. The catch is bookkeeping: the origin, the system boundary, and the frame must be fixed first, and a change in total $\\vec L$ always points to an external impulse someone forgot.\n",{"title":1138,"path":1139,"lessonNumber":1032,"topics":1140,"summary":1141},"Rolling Resistance","\u002Fmechanics\u002Frotation\u002Frolling-resistance",[1113],"Ideal rolling should coast forever, yet every real wheel slows down. The reason is that a deformable tire and road do not press through a single point: the contact patch spreads, the normal-force resultant shifts ahead of the axle, and that offset is a resisting moment even with no gross sliding. We package it as an equivalent force $F_{rr}=C_{rr}N$, tie the coefficient to load, surface, speed, and temperature, and use coast-down, towing, and traction tests to separate this contact loss from aerodynamic drag, bearing friction, and the adhesion limit where rolling gives way to skidding.\n",{"title":1143,"path":1144,"lessonNumber":1114,"topics":1145,"summary":1146},"Gyroscopic Precession","\u002Fmechanics\u002Frotation\u002Fgyroscopic-precession",[1113],"A spinning top leans over but does not fall — it swings its axis in a slow horizontal circle instead. The paradox dissolves once torque is read as the rate of change of a vector: gravity's torque is perpendicular to the spin angular momentum, so it turns $\\vec L$ rather than toppling it. We derive the steady precession rate $\\Omega\\simeq Mgr\u002F(I_s\\omega_s)$ in the fast-top limit, state the assumptions it leans on — dominant spin, slow tilt, negligible bearing torque — and read nutation, support motion, and a decaying spin as the ways real gyroscopes depart from it.\n",{"module":1148,"moduleNumber":1149,"slug":1150,"lessons":1151},"Gravitation and Matter",7,"gravity-and-matter",[1152,1157,1162,1167,1172,1177,1182],{"title":1153,"path":1154,"lessonNumber":990,"topics":1155,"summary":1156},"Keplerian Orbits","\u002Fmechanics\u002Fgravity-and-matter\u002Fkeplerian-orbits",[1148],"Why do the planets trace ellipses rather than any other curve? Newton's inverse-square law collapses the two-body problem onto a single conic section, and the answer falls out of two conserved quantities: a central force can exert no torque, so angular momentum is fixed, and gravity is conservative, so energy is fixed. We read an orbit's size and shape straight off those invariants, recover all three of Kepler's laws, and derive escape speed, the vis-viva relation, and the timing of a pass. We also mark where the ideal ellipse breaks down — drag, oblateness, and a third body slowly move a real orbit.\n",{"title":1158,"path":1159,"lessonNumber":16,"topics":1160,"summary":1161},"Gravitational Fields","\u002Fmechanics\u002Fgravity-and-matter\u002Fgravitational-fields",[1148],"Instead of tracking the force between every pair of masses, we attach a field to the source and ask a test mass to read it off locally. That move pays off because gravity is conservative: the field is the gradient of a single scalar potential, and potentials from many sources simply add. We build the field-potential picture, use spherical symmetry and the shell theorem to get the point-mass exterior field and the zero interior field of a shell, and read tides straight out of the field's gradient. Along the way we mark exactly when the constant-$g$ and point-mass shortcuts hold and when a shape correction is needed.\n",{"title":1163,"path":1164,"lessonNumber":1020,"topics":1165,"summary":1166},"Static Equilibrium","\u002Fmechanics\u002Fgravity-and-matter\u002Fstatic-equilibrium",[1148],"What does it take for a loaded structure to stay put? A body at rest needs its forces to cancel and its turning effects to cancel — $\\sum\\vec F=0$ and $\\sum\\vec\\tau=0$ about any point — and almost all of statics is the craft of turning a physical setup into those equations. We build free-body diagrams, replace supports, cables, friction, couples, and distributed loads with their idealized reactions, and locate the centre of gravity that decides whether a body tips. We also count equations against unknowns to separate a determinate problem from one that needs the material's deformation to resolve, and read every negative or inconsistent reaction as a sign that a contact or a boundary was chosen wrong.\n",{"title":1168,"path":1169,"lessonNumber":1026,"topics":1170,"summary":1171},"Fluid Statics","\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-statics",[1148],"A fluid at rest cannot support a shear, so the only stress it carries is a pressure that must grow with depth to hold up the fluid above it. That single balance, $\\d p\u002F\\d z=-\\rho g$, runs the whole subject: it sets manometer readings, the force on a dam, and — integrated over a submerged boundary — Archimedes' buoyant force $F_B=\\rho g V_{\\rm disp}$. We derive these, use them to decide when a body floats and whether it floats upright, and mark where acceleration, rotation, compressibility, or capillarity forces a richer pressure model.\n",{"title":1173,"path":1174,"lessonNumber":1032,"topics":1175,"summary":1176},"Fluid Flow","\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-flow",[1148],"Two accounting rules carry most of steady flow: mass cannot pile up, so the same volume crosses every section each second, and mechanical energy is conserved along a streamline when the fluid is ideal. From those we get continuity, Bernoulli's relation between pressure, speed, and height, and the results that follow — Torricelli's efflux speed, the Venturi meter, the Pitot tube. We then let go of the ideal assumptions one at a time: viscosity adds wall shear and head loss, Reynolds number decides laminar versus turbulent, and Mach number marks where a gas stops behaving as incompressible.\n",{"title":1178,"path":1179,"lessonNumber":1114,"topics":1180,"summary":1181},"Orbital Motion","\u002Fmechanics\u002Fgravity-and-matter\u002Forbital-motion",[1148],"A circular orbit is nothing but free fall with enough sideways speed to keep missing the ground, and setting gravity equal to the centripetal requirement fixes that speed and the period once and for all. From the same energy bookkeeping we read off escape speed, sort orbits into bound, parabolic, and hyperbolic by the sign of their specific energy, and see why a tangential burn is the efficient way to change an orbit. We build the Hohmann transfer and its launch window, work the numbers for a geostationary orbit and an escape burn, and mark where finite thrust, perturbations, and an uncertain initial state pull a real trajectory off the ideal.\n",{"title":1183,"path":1184,"lessonNumber":1149,"topics":1185,"summary":1186},"Stress and Elasticity","\u002Fmechanics\u002Fgravity-and-matter\u002Fstress-and-elasticity",[1148],"Rigid bodies are a fiction; every real material stretches, shears, or squeezes under load, and the useful question is how much. We define stress as force per area and strain as fractional deformation, then find that for small deformations the two are simply proportional — Hooke's law — with Young's, shear, and bulk moduli as the constants for stretch, twist, and volume change. From these we compute extensions, torsional twist, and stored elastic energy, and read a tensile curve for the yield, ultimate, and fracture points where linear elasticity ends. We also mark the practical limits: stress concentrations, fatigue, and the multiaxial states a single uniaxial modulus cannot capture.\n",{"module":1188,"moduleNumber":1189,"slug":1190,"lessons":1191},"Oscillations and Waves",8,"oscillations-waves",[1192,1198,1204,1209,1214,1219,1224,1229,1234,1240,1246,1252],{"title":1193,"path":1194,"lessonNumber":990,"topics":1195,"summary":1197},"Damped Oscillators","\u002Fmechanics\u002Foscillations-waves\u002Fdamped-oscillators",[1196],"Oscillations","Every real oscillator eventually stops: friction, drag, and internal loss drain its energy, so free motion is a decay rather than a permanent swing. Adding a velocity-proportional resistance to the spring-mass equation produces one dimensionless number, $b\u002F(2\\sqrt{mk})$, that decides whether the mass rings down through many cycles, returns once without overshoot, or crawls back slowly. We solve the three regimes, tie the observed decay to the power balance $b\\dot x^2$, and turn a measured ring-down into the decay rate and quality factor of the apparatus — reading damping off the data instead of assuming it.\n",{"title":1199,"path":1200,"lessonNumber":16,"topics":1201,"summary":1203},"Travelling Waves","\u002Fmechanics\u002Foscillations-waves\u002Ftravelling-waves",[1202],"Waves","A wave carries a shape, not the material: each element of a rope or air column oscillates in place while the disturbance travels through it. Writing that shape as $f(x\\mp vt)$ turns \"the pattern moves\" into a statement about the cosine's argument, and a local force balance on one string segment fixes the speed at $v=\\sqrt{T\u002F\\mu}$ — restoring stiffness over inertia, with amplitude nowhere in it. We build the sinusoidal wave and its phase, derive the wave equation from Newton's second law, and follow the energy a travelling wave transports, then check speed and power against those predictions.\n",{"title":1205,"path":1206,"lessonNumber":1020,"topics":1207,"summary":1208},"Wave Superposition","\u002Fmechanics\u002Foscillations-waves\u002Fwave-superposition",[1202],"When two waves cross the same point, what does a probe read? In a linear medium the answer is arithmetic: the displacements add, $y=y_1+y_2$, and the pulses pass through each other unchanged. That one rule produces interference — reinforcement where the signs agree, cancellation where they oppose — and it guards against a common mistake, since displacement can vanish at an instant while the energy sits in transverse motion instead. We work out the signed sum, the phase bookkeeping for equal-frequency components, and why a null in the record is not a null in the wave.\n",{"title":1210,"path":1211,"lessonNumber":1026,"topics":1212,"summary":1213},"Standing Waves","\u002Fmechanics\u002Foscillations-waves\u002Fstanding-waves",[1202],"Clamp a string at both ends and only certain frequencies survive: the ends must be nodes, and that single geometric demand quantizes the wave into a discrete set of modes $f_n=nv\u002F(2L)$. The travelling wave becomes a fixed pattern of nodes and antinodes — standing, not moving — because equal waves running in opposite directions superpose. We build the standing wave from its counter-propagating pieces, read the harmonic sequence off the boundary conditions (half-wavelengths for a fixed-fixed string, odd quarter-wavelengths for a closed pipe), and test the ideal model against node scans and resonance peaks.\n",{"title":1215,"path":1216,"lessonNumber":1032,"topics":1217,"summary":1218},"Sound Waves","\u002Fmechanics\u002Foscillations-waves\u002Fsound-waves",[1202],"Sound is a pressure wave so small that a loud tone displaces air molecules by less than the width of an atom, yet a microphone reads it easily — because pressure, not displacement, is what the ear and the instrument sense. The acoustic impedance $Z=\\rho c$ ties pressure, density, and particle velocity together, fixes the intensity a wave carries, and sets the reference for the decibel, a logarithm that tames a $10^{12}$ range in power. We derive the sound speed from the gas's stiffness, convert between pressure and intensity levels, and treat the measurement itself — calibration, geometry, background, averaging — as part of the physics.\n",{"title":1220,"path":1221,"lessonNumber":1114,"topics":1222,"summary":1223},"Doppler Effect","\u002Fmechanics\u002Foscillations-waves\u002Fdoppler-effect",[1202],"A passing siren drops in pitch not because the source changes but because motion repacks the wavefronts: an approaching source crowds its crests, a receding one stretches them, and a moving listener samples them at a different rate. For mechanical waves every velocity is measured against the medium, and one signed ratio $f_r=f_s(v-u_r)\u002F(v-u_s)$ captures both effects at once. We separate source motion, which sets crest spacing, from receiver motion, which sets arrival rate, invert the shift to recover radial velocity, and mark where the model breaks — supersonic sources, moving air, and reflected paths that carry two shifts, not one.\n",{"title":1225,"path":1226,"lessonNumber":1149,"topics":1227,"summary":1228},"Wave Packets","\u002Fmechanics\u002Foscillations-waves\u002Fwave-packets",[1202],"No real signal is a single frequency: a disturbance that starts and stops is built from a band of wave numbers, and the width of that band is what makes it local. We ask how such a packet moves — carrier crests at the phase velocity $v_\\mathrm p=\\omega\u002Fk$, the envelope at the group velocity $v_\\mathrm g=\\d\\omega\u002F\\d k$ — and why the two differ once a medium is dispersive. Curvature $\\d^2\\omega\u002F\\d k^2$ spreads and chirps the packet as it travels, and the Fourier reciprocity that ties bandwidth to duration explains why a finite record, aliasing, or a coarse probe can imitate that spreading unless the sampling limits are respected.\n",{"title":1230,"path":1231,"lessonNumber":1189,"topics":1232,"summary":1233},"Beats and Coupling","\u002Fmechanics\u002Foscillations-waves\u002Fbeats-and-coupling",[1196],"Add two tones a few hertz apart and the sum swells and fades at their difference frequency — a beat — though neither source is changing. We work out that envelope, then ask the mechanical version of the same question: join two oscillators and a single resonance splits into normal modes, with energy sloshing between the coordinates at their frequency difference. The lesson identifies when a slow amplitude envelope signals genuine coupling rather than two independent sources, drift, or deliberate modulation, reading it from envelope timing, spectral sidebands, and the mode shapes.\n",{"title":1235,"path":1236,"lessonNumber":1237,"topics":1238,"summary":1239},"Simple Harmonic Motion","\u002Fmechanics\u002Foscillations-waves\u002Fsimple-harmonic-motion",9,[1196],"Any system pushed back toward equilibrium by a force proportional to its displacement obeys one equation, $\\ddot x+\\omega_0^2x=0$, and so moves sinusoidally at $\\omega_0=\\sqrt{k\u002Fm}$ whatever the amplitude. We derive that motion, follow its energy $E=mv^2\u002F2+kx^2\u002F2$ trading between kinetic and potential form at constant total, and read the elliptical phase-space orbit Hooke's law implies. Period, amplitude, velocity, and acceleration then supply redundant checks: an amplitude-dependent period or a curved force residual is the signature that the linear model has failed, and mass-loading and offset tests separate a calibration error from a real frequency shift.\n",{"title":1241,"path":1242,"lessonNumber":1243,"topics":1244,"summary":1245},"Pendulum Motion","\u002Fmechanics\u002Foscillations-waves\u002Fpendulum-motion",10,[1196],"A pendulum keeps time only because, for small swings, gravity supplies a restoring torque proportional to the angle — and $T=2\\pi\\sqrt{L\u002Fg}$ then follows without the mass appearing at all. We derive that result, mark exactly which assumptions carry it (small angle, negligible pivot loss, a rigid support), then relax them: finite amplitude lengthens the period through an elliptic integral, and an extended body replaces $L$ with the ratio of its moment of inertia to its center-of-mass distance. How the period drifts with amplitude or pivot position is what diagnoses the geometric, damping, and distributed-mass corrections.\n",{"title":1247,"path":1248,"lessonNumber":1249,"topics":1250,"summary":1251},"Driven Oscillators","\u002Fmechanics\u002Foscillations-waves\u002Fdriven-oscillators",11,[1196],"Drive a damped oscillator at a frequency you control and it eventually forgets its own: $m\\ddot x+b\\dot x+kx=F_0\\cos\\omega t$ settles into a steady response whose amplitude and phase depend sharply on how close the drive sits to resonance. We solve for that response, show how damping alone fixes the resonance width, the peak power, and the settling time, and treat base excitation as the same problem with a different input. The steady-state formulas hold only for constant $m$, $b$, and $k$; level-dependent peaks or hysteresis between up- and down-sweeps are how nonlinearity or an extra mode announces itself.\n",{"title":1253,"path":1254,"lessonNumber":1255,"topics":1256,"summary":1257},"Wave Boundaries","\u002Fmechanics\u002Foscillations-waves\u002Fwave-boundaries",12,[1202],"A pulse traveling along a string does something abrupt where the string's properties change: part reflects, part transmits, and which is which is set by the impedance mismatch alone. We impose continuity of displacement and transverse force at the join to get the reflection and transmission coefficients in terms of $Z=\\sqrt{T\\mu}$, fix their signs and the polarity flip, and balance the energy. The clean result assumes linear, nondispersive segments meeting at a localized join; pulse polarity, return timing, and energy ratios are the measurements that expose a real connector's mass, loss, or distributed transition.\n",{"module":1259,"moduleNumber":1237,"slug":1260,"lessons":1261},"Thermodynamics","thermodynamics",[1262,1267,1272,1277,1282,1287],{"title":1263,"path":1264,"lessonNumber":990,"topics":1265,"summary":1266},"Kinetic Theory of Ideal Gases","\u002Fmechanics\u002Fthermodynamics\u002Fkinetic-theory-of-ideal-gases",[1259],"A gas has no springs and no gears, yet it pushes on its container with a definite pressure and stores energy in a lawful way. Kinetic theory explains both from the motion of the molecules alone: pressure is the accumulated recoil of countless elastic impacts, and temperature is the average translational kinetic energy each molecule carries. We derive $pV=\\tfrac13Nm\\overline{v^2}$ from momentum transfer, read off $\\overline{K}_{\\rm tr}=\\tfrac32kT$, and use the Maxwell–Boltzmann distribution to separate the most probable, mean, and rms speeds — each the right average for a different question — while marking where the dilute, classical assumptions stop holding.\n",{"title":1268,"path":1269,"lessonNumber":16,"topics":1270,"summary":1271},"First Law of Thermodynamics","\u002Fmechanics\u002Fthermodynamics\u002Ffirst-law-of-thermodynamics",[1259],"Heat a gas and it may warm, expand, or both; compress it and the same energy can reappear as a temperature rise. The first law settles the bookkeeping: internal energy is a state property whose change equals the heat added plus the work done on the system, $\\Delta E_{\\rm int}=Q_{\\rm in}+W_{\\rm on}$. We fix a system boundary and one sign convention, compute boundary work as $\\int p\\,\\d V$ along a path, and use calorimetry to measure heat and heat capacities. The recurring point is that heat and work are path-dependent transfers while their sum is not, so an energy ledger closes only once every boundary crossing is named.\n",{"title":1273,"path":1274,"lessonNumber":1020,"topics":1275,"summary":1276},"Entropy and the Second Law","\u002Fmechanics\u002Fthermodynamics\u002Fentropy-and-the-second-law",[1259],"The first law lets energy flow either way; it never says which way heat actually goes. The second law supplies the missing arrow. Entropy, defined through the reversible transfer $\\d S=\\delta Q_{\\rm rev}\u002FT$, can only increase in an isolated system, and that single inequality fixes the direction of heat flow and caps every engine, refrigerator, and heat pump at its Carnot value. We build entropy ledgers for reservoirs and working substances, separate the entropy carried by heat from the entropy generated by irreversibility, and read the sign of the total as a hard check on any proposed thermal machine.\n",{"title":1278,"path":1279,"lessonNumber":1026,"topics":1280,"summary":1281},"Thermal Processes","\u002Fmechanics\u002Fthermodynamics\u002Fthermal-processes",[1259],"Heat rarely sits still: it stretches solids, pushes real gases off their ideal isotherms, and leaks across walls by conduction, convection, and radiation. Each behavior becomes a number a designer can use. Thermal expansion sets the gaps in a bridge and the stress in a clamped rod; the van der Waals equation and a phase diagram fix when $pV=nRT$ or a latent-heat term applies; Fourier's law, Newton cooling, and Stefan–Boltzmann radiation give the rate of heat flow. We assemble these into thermal-resistance networks and transient time constants, then mark where contact resistance, phase change, or a hidden thermal bridge breaks the simple model.\n",{"title":1283,"path":1284,"lessonNumber":1032,"topics":1285,"summary":1286},"Phase Changes","\u002Fmechanics\u002Fthermodynamics\u002Fphase-changes",[1259],"Add heat to ice and its temperature climbs — until it reaches $0\\ ^\\circ\\mathrm C$, where the thermometer stalls while the ice melts. That plateau is the whole subject: at a phase boundary the energy rearranges molecules, $Q=mL$, instead of raising temperature, which resumes only once one phase is gone. We stage a heating path into sensible-heat legs ($Q=mc\\Delta T$) and latent plateaus, use the Clausius–Clapeyron relation to track how a boiling point moves with pressure, and solve calorimetry by testing each coexistence endpoint — so a melt fraction that lands outside $[0,1]$ flags a wrong final-state guess rather than a real state.\n",{"title":1288,"path":1289,"lessonNumber":1114,"topics":1290,"summary":1291},"Thermal Machines","\u002Fmechanics\u002Fthermodynamics\u002Fthermal-machines",[1259],"An engine, a refrigerator, and a heat pump are one machine read three ways: each shuttles heat between a hot and a cold reservoir while trading work at the boundary, and only the flow you call useful separates them. A heat engine turns part of $Q_h$ into work, $W=Q_h-Q_c$; a refrigerator spends work to pull $Q_c$ from the cold side; a heat pump counts the warm-side delivery instead. We measure each with its own ratio — efficiency or coefficient of performance — bound them all by the Carnot limit that reservoir temperatures alone set, and track how finite temperature differences, throttling, and friction generate entropy and pull real machines below that bound.\n",1786059306723]