Physics curriculum

SI builds every unit from a small set of base units, such as the meter, kilogram, second and ampere. A derived unit is a combination of them: a newton is a kilogram meter per second squared. Read the unit as part of the answer, because a result in joules per second is a power whatever the question called it. Kilo means a thousand and milli a thousandth.

Write each conversion factor as a fraction equal to one, such as 1 km over 1,000 m, and multiply by as many as you need. Arrange each so the unit you want to remove appears once on top and once on the bottom. If the units left over are not the ones the question asks for, the setup is wrong, and you find that out before doing any arithmetic.

A scalar has size only, like mass, time or speed. A vector also has a direction, like displacement, velocity or force. Walk 50 m east and 50 m back west and you have covered 100 m of distance but made zero displacement. Check whether a question says speed or velocity, distance or displacement, because mixing them is the commonest early error.

Any vector can be replaced by two perpendicular parts that add back to it. For a vector of size A at angle θ from the x-axis, the parts are A cos θ along x and A sin θ along y. To rebuild it, use the Pythagorean theorem for the size and the inverse tangent of y over x for the angle, then check which quadrant the signs of the parts put it in.

Both sides of a valid equation must have the same dimensions. If you remember that a pendulum's period depends on its length and on g, the only combination of those with units of seconds is the square root of length over g. Dimensions catch a missing square or a flipped ratio, but they cannot catch a wrong pure-number factor such as 2π.

A result is only as precise as its least precise input. Multiply a length known to two significant figures by one known to five, and the product earns two. Extra calculator digits are noise, not information. When adding or subtracting, round instead to the least precise decimal place among the terms.

Before computing, bound the answer to the nearest power of ten. Round every input to one significant figure, multiply, and see whether the result is about 10, 1,000 or 1,000,000. A unit slip or a misplaced decimal usually shifts an answer by a factor of 10 or 1,000, which a quick estimate exposes and a careful calculation hides.

Add vectors by adding their components, x with x and y with y; add their sizes only when they point the same way. The dot product equals AB cos θ, so it measures how much one vector lies along the other and is zero for perpendicular vectors. The cross product has size AB sin θ and points perpendicular to both, in the direction the right-hand rule gives.

Displacement is final position minus initial position, one arrow from start to finish. Distance counts every meter traveled along the way. Run a full lap of a track and your displacement is zero while your distance is the lap length. Average velocity uses displacement; average speed uses distance.

Average velocity is total displacement divided by total time, so a trip with a long stop still has a single average. Instantaneous velocity is the value at one moment, which is what a speedometer shows. On a position-time graph, the average is the slope of the straight line between two points, and the instantaneous value is the slope of the tangent at one point.

Acceleration is the rate at which velocity changes. A negative acceleration means the change points in the negative direction, not that the object is slowing down. An object moving in the negative direction with a negative acceleration speeds up. It slows only when its velocity and acceleration point opposite ways.

With constant acceleration, four equations link displacement, initial velocity, final velocity, acceleration and time, and each one leaves out one of those five. List the three quantities you know and the one you want, then pick the equation that omits the quantity you neither know nor need. If the acceleration changes, none of them applies.

On a position-time graph, the slope is velocity. On a velocity-time graph, the slope is acceleration and the area under the line is displacement. A common trap is reading the height of a velocity graph as a position. A flat line on a velocity graph means constant velocity, not that the object is standing still.

In free fall, with air resistance ignored, every object accelerates downward at g, about 9.8 m/s², whatever its mass. At the top of a vertical toss the velocity is zero for an instant, but the acceleration is still g downward. If the acceleration were zero there, the object would simply hang in the air.

Velocity is always measured relative to something. Walk forward at 1.5 m/s inside a train moving at 25 m/s and you move at 26.5 m/s relative to the ground; walk toward the back and it is 23.5 m/s. Choose the frame first, then add or subtract velocities as signed numbers in that frame.

Horizontal and vertical motion run independently, linked only by time. A ball rolled off a table and one dropped from the same height land together, because both start with zero vertical velocity. Solve each direction with its own equations, then use the shared flight time to connect them.

A projectile keeps a constant horizontal velocity while accelerating downward at g, so its path is a parabola. Split the launch velocity into components first. The time of flight comes from the vertical motion alone; the range is the horizontal speed times that time, and the peak height comes where the vertical velocity reaches zero.

On level ground with no air resistance, range is greatest at a 45-degree launch. Angles that add to 90 degrees, such as 30 and 60, give the same range but different heights and flight times. Launching from above the landing point lowers the best angle below 45 degrees.

An object moving in a circle at constant speed is still accelerating, because its direction keeps changing. That acceleration points toward the center and equals v²/r, so doubling the speed quadruples it. Nothing pushes the object outward: without an inward pull it would carry straight on along the tangent.

The period T is the time for one full turn, and the frequency f = 1/T counts turns per second. In one period an object covers one circumference, so its speed is 2πr/T. Points farther out on a spinning object share the same period but move faster, because they cover a bigger circle in the same time.

If the speed also changes, the acceleration has two parts. The radial part, v²/r, points to the center and turns the velocity. The tangential part points along the path and changes the speed. The total acceleration is their vector sum, so it no longer points straight at the center.

To cross a river or allow for wind, add velocities as vectors: your velocity relative to the ground is your velocity relative to the water plus the water's velocity relative to the ground. To land straight across, aim partly upstream. The crossing time depends only on the component of your velocity straight across the river.

A force is a push or pull that one object exerts on another, so every force has a source you can name. Contact forces, such as friction, tension and the normal force, need touching surfaces. Field forces, such as gravity, act at a distance. If you cannot name what exerts a force, it probably is not there.

An object keeps its velocity, including staying at rest, unless a net force acts on it. Motion needs no force; only a change in motion does. A sliding puck slows because friction acts on it, not because its force runs out. Whenever velocity is constant, the forces on the object balance exactly.

The net force on an object equals its mass times its acceleration, F = ma, and the acceleration points the same way as the net force. Apply it one axis at a time: add the forces along x to get the x-acceleration, then do the same for y. Using a single force where the net force belongs is the most common error.

Mass measures how hard an object is to accelerate, and it is the same on Earth, on the Moon or in orbit. Weight is the gravitational force on that mass, W = mg, so it changes wherever g changes. Kilograms measure mass and newtons measure weight; an astronaut in orbit keeps all of their mass.

When A pushes on B, B pushes back on A with a force equal in size and opposite in direction. The two forces act on different objects, so they can never cancel each other. A horse can still pull a cart forward, because the cart's motion depends only on the forces acting on the cart.

The normal force is whatever a surface must supply to stop an object passing through it, so it equals mg only on a level surface with no other vertical forces. Push down on a box and it grows; tilt the surface and it shrinks. An ideal light string has the same tension along its whole length, and an ideal pulley only changes that tension's direction.

Draw one object on its own and include only the forces acting on it, each from a source you can name. Leave out the forces it exerts on other things. Never add a separate force of motion or a centripetal force, because neither has a source. Then choose axes along the direction the object will accelerate.

A scale reads the normal force it supplies, not your weight. In an elevator accelerating upward the floor must push harder than mg, so the scale reads more; accelerating downward it reads less, and in free fall it reads zero. At constant velocity in either direction the reading is normal, because the acceleration is zero.

Static friction holds surfaces together, and it adjusts to match whatever push it opposes, up to a maximum of μs times the normal force. Below that limit, its value equals the applied push, so μsN is the ceiling and not the actual friction. The object starts sliding only when the push exceeds that ceiling.

Once surfaces slide, kinetic friction equals μk times the normal force and opposes the sliding. In the standard model it does not depend on speed or on the area in contact. The friction force drops once sliding starts, which is why a heavy crate becomes easier to keep moving than to start.

Centripetal force is a job, not a new kind of force. Some real force, such as tension, gravity, friction or a normal force, points toward the center and supplies the needed mv²/r. In a free-body diagram you draw that real force and never add a separate arrow labeled centripetal.

For objects joined by strings or in contact, first treat them as one system: the net external force divided by the total mass gives their shared acceleration. Internal forces, such as the tension between two carts, cancel at that stage. Then isolate one object and use its own free-body diagram to find an internal force.

On a slope, tilt your axes so x runs along the surface. Gravity then splits into mg sin θ down the slope and mg cos θ into it. The normal force balances only the into-slope part, so it is mg cos θ, less than mg. With no friction, the acceleration down the slope is g sin θ for any mass.

On a banked curve, part of the normal force points toward the center, so a car can take the turn with no friction at one design speed. In a vertical loop, gravity and the track's push together supply the inward force. At the top, the slowest safe speed is the one where gravity alone does the whole job, which gives v² = gr.

Air drag grows as an object speeds up. A falling object accelerates until drag equals its weight; the net force is then zero, and it falls at a constant terminal speed. A heavier object of the same shape needs more drag to balance its weight, so it reaches a higher terminal speed. A parachute raises drag and lowers that speed.

Work is the force times the displacement times the cosine of the angle between them, W = Fd cos θ. A force perpendicular to the motion does no work, so carrying a box across a level room does no work against gravity. Work is negative when the force opposes the motion, as friction usually does, and its unit is the joule.

Kinetic energy is one-half mv². Because speed is squared, doubling your speed quadruples the energy, and with the same braking force the stopping distance quadruples too. Kinetic energy is a scalar and is never negative, so it carries no direction.

Power measures how fast work gets done, P = W/t, in watts, which are joules per second. Two people who climb the same stairs do the same work, but the faster one develops more power. For a force pushing an object along at speed v, the power is P = Fv.

A watt is one joule per second, so a kilowatt-hour, 1,000 watts for one hour, is 3.6 million joules. Efficiency is the useful energy out divided by the total energy in. Real machines always stay below 100 percent, because some of the input ends up as heat you did not want.

When a force changes along the path, the work is the area under the force-versus-position graph. A spring's force grows in a straight line with stretch, so that area is a triangle, and the work to stretch it by x is one-half kx². Stretching it twice as far takes four times the work.

Add up the work every force does on an object and you get exactly its change in kinetic energy. It often beats a kinematics route: to find a speed after a push, you need the forces and the distance, not the time. Count the net work from every force, friction included, not the work of one force.

Kinetic friction turns mechanical energy into thermal energy, and the amount is the friction force times the length of path actually traveled, not the straight-line displacement. Drag a box around a loop back to where it started and friction has still done negative work along the whole path, even though the displacement is zero.

Energy is stored in several forms: kinetic, gravitational and elastic potential, chemical, thermal, radiant and nuclear. Most problems are about a conversion between them. Name the starting and ending forms before you write an equation: a falling ball turns gravitational energy into kinetic, and an engine turns chemical energy into kinetic and thermal.

Near Earth's surface, raising a mass m through a height h stores mgh of gravitational potential energy. Only changes in it matter, so you may set the zero level wherever is convenient, such as the floor or the lowest point of a swing, as long as you keep it fixed for the whole problem.

If no friction or other nonconservative force does work, kinetic plus potential energy stays constant. A roller coaster's speed at any point then depends only on how far it has dropped from the start, not on the shape of the track. Set the initial and final totals equal and solve; time never enters.

A spring obeys Hooke's law: its force is proportional to how far it is stretched or compressed, F = -kx. The energy it stores is one-half kx², the area under that straight-line force graph. Because x is squared, compressing a spring twice as far stores four times the energy.

A force is conservative if the work it does between two points is the same along every path. Gravity and ideal springs pass that test, so each has a potential energy. Friction fails it, because a longer path means more work against it, so friction has no potential energy and instead turns mechanical energy into heat.

With friction present, mechanical energy is not conserved, but total energy still is. Write the initial kinetic plus potential energy as equal to the final kinetic plus potential energy plus the thermal energy produced, which is the friction force times the path length. Leaving that term out predicts a final speed that is too high.

On a graph of potential energy against position, draw a horizontal line at the total energy. Motion is possible only where the curve lies below that line, and the object turns around where the line meets the curve. A valley in the curve is a stable equilibrium and a hilltop is an unstable one.

Force is the negative slope of the potential energy curve, F = -dU/dx. A steep slope means a strong force, and the force points downhill on the graph, toward lower potential energy. Where the curve is flat the force is zero, which is why the bottom of a valley and the top of a hill are both equilibrium points.

Momentum is mass times velocity, p = mv, a vector pointing the same way as the velocity. A slow truck can carry more momentum than a fast ball. Momentum and kinetic energy rank objects differently: of two objects with equal momentum, the lighter one has more kinetic energy.

If no net external force acts on a system, its total momentum stays constant, even in a crash that wrecks the objects and turns energy into heat. The forces the colliding objects exert on each other are internal and cancel in pairs. Set total momentum before equal to total momentum after, with signs for direction.

Impulse is a force multiplied by how long it lasts, it matches the momentum change it produces, and on a force-time graph it is the area under the curve. Airbags and crumple zones do not reduce your change in momentum in a crash; they stretch out the stopping time, which lowers the peak force on you.

In an elastic collision, both momentum and kinetic energy are conserved. When equal masses collide head-on and one starts at rest, they swap velocities: the moving one stops and the other leaves at its speed. A light object hitting a much heavier one at rest bounces back with nearly its original speed.

In an inelastic collision, momentum is conserved but kinetic energy is not. When the objects stick together the collision is perfectly inelastic, and it loses the most kinetic energy that momentum conservation allows. The missing energy becomes heat, sound and permanent deformation.

In two dimensions, momentum is conserved separately along each axis, so write one equation for x and one for y. An object at rest that explodes must produce pieces whose momenta add to zero, which is why a rifle recoils backward as the bullet goes forward.

The center of mass is the mass-weighted average position of a system. It moves as if all the mass were there and every external force acted there. When a firework bursts in flight, its pieces scatter, but their center of mass keeps following the original arc until the first piece lands.

Rotation uses angular position θ, angular velocity ω and angular acceleration α. Measure angles in radians: one radian is the angle at which the arc length equals the radius. Radians keep the link to straight-line motion clean, since arc length is simply rθ with no conversion factor.

With constant angular acceleration, the rotational equations have exactly the form of the linear ones: put θ for displacement, ω for velocity and α for acceleration. Choose among them the same way, by finding the quantity you neither know nor need.

A point at distance r from the axis has speed v = rω and tangential acceleration a = rα. Every point on a rigid spinning object shares the same ω, but points farther from the axis move faster. That is why the edge of a merry-go-round feels so much faster than its middle.

Torque is the turning effect of a force: the force times its lever arm, the perpendicular distance from the axis to the force's line of action. Pushing a door near its hinges needs far more force for the same torque. A force aimed straight through the axis produces no torque at all, however large it is.

Moment of inertia measures how hard an object is to spin up, and it depends on where the mass sits, not just on how much there is. Each bit of mass counts as mass times distance from the axis squared, so mass far from the axis dominates. The same object has different values about different axes.

Net torque equals moment of inertia times angular acceleration, τ = Iα, the rotational form of F = ma. For a weight hanging from a pulley, write F = ma for the weight and τ = Iα for the pulley, then link the two with a = rα.

A spinning object stores rotational kinetic energy, one-half Iω². A rolling object has both translational and rotational energy, so part of its energy goes into the spin. The work done by a torque is the torque times the angle turned, and power is the torque times ω.

When a wheel rolls without slipping, its center moves at v = rω and the point touching the ground is momentarily at rest. Static friction acts at that point, but since the point does not slide, that friction does no work, so mechanical energy can still be conserved.

Released from rest on the same slope, the shape with the smallest moment of inertia relative to its mass and radius wins, because less of its energy goes into spinning. A solid sphere beats a solid cylinder, which beats a hoop, whatever their masses and sizes.

A particle's angular momentum about a point is L = r × p, with size r times p times the sine of the angle between them. Its value depends on the point you choose. Even a particle moving in a straight line has angular momentum about any point that is not on its path.

With no net external torque, angular momentum stays constant. A spinning skater who pulls in their arms lowers their moment of inertia, so their spin rate rises to keep Iω the same. A collapsing star and a diver tucking into a ball spin faster for the same reason.

Tilt a spinning top and it does not fall over: the torque from gravity changes the direction of its angular momentum instead. Its spin axis sweeps slowly around in a circle, called precession. Spin it faster and it precesses more slowly, because the same torque now turns a larger angular momentum.

An object in static equilibrium has zero net force and zero net torque. Put your pivot at the point where an unknown force acts: that force then has no lever arm, drops out of the torque equation, and leaves you fewer unknowns.

An object stays upright while the vertical line through its center of gravity falls inside its base of support, and it tips once that line passes outside. A low center of gravity and a wide base let it lean much further first, which is why racing cars are built low and wide.

Stress is force per unit area, and strain is the fractional change in length. Below the elastic limit the two are proportional, and their ratio is Young's modulus, a property of the material rather than of the particular object. Past the elastic limit the material stays permanently deformed.

Every two masses attract with a force F = Gm₁m₂/r². Double the distance and the force drops to a quarter. G is tiny, so the pull between everyday objects is far too weak to notice; gravity matters in practice only when at least one of the masses is the size of a moon or planet.

Surface gravity comes from a planet's mass and radius, g = GM/R². A planet with twice the mass and the same radius has twice the g, while the same mass packed into half the radius has four times the g. Climbing to a higher altitude moves you farther from the center, so g gets slightly smaller.

For any separation, gravitational potential energy is U = -GMm/r: zero at infinite distance and negative everywhere closer. The familiar mgh is the change in this quantity over a small height near the surface, so it is accurate only when the height is tiny compared with the planet's radius.

In a circular orbit, gravity supplies exactly the centripetal force, which gives an orbital speed v = √(GM/r). The speed does not depend on the satellite's mass. A higher orbit is slower and has a longer period, so a satellite close to Earth laps it many times a day while a distant one takes far longer.

Astronauts in orbit float not because gravity is missing; at the space station's altitude, gravity is still about 90 percent of its strength at the surface. The astronauts and their station are falling around Earth together, so nothing presses against them. Free fall, not zero gravity, is the cause.

Escape speed is the launch speed whose kinetic energy just equals the energy needed to climb to infinite distance, v = √(2GM/R). The escaping object's own mass cancels out, so a pebble and a rocket need the same speed. It is √2 times the speed of a circular orbit at the same distance.

Kepler found three rules for planets: each orbit is an ellipse with the Sun at one focus; a planet sweeps out equal areas in equal times, so it moves fastest when closest; and the square of the orbital period is proportional to the cube of the orbit's semi-major axis. Newton's law of gravitation explains all three.

Tides come from the difference in the Moon's pull across Earth. The side facing the Moon is pulled harder than Earth's center, and the center harder than the far side, so the ocean bulges both toward the Moon and away from it. As Earth turns through both bulges, most coasts get two high tides a day.

Density is mass per unit volume. Specific gravity is a material's density divided by the density of water, so it has no units. Anything with a specific gravity below 1 floats in water and anything above 1 sinks, whatever its size or shape.

Pressure is force per unit area, and in a fluid it pushes equally in every direction. The extra pressure at depth h is ρgh, which depends only on depth and not on the container's shape. A gauge reads pressure above atmospheric; absolute pressure adds the atmosphere back in.

Pressure applied to an enclosed fluid is passed on undiminished to every part of it. In a hydraulic lift, a small force on a small piston creates a pressure that pushes a large piston with a proportionally larger force. The price is distance: the small piston must travel much farther than the large one rises.

Anything in a fluid feels an upward buoyant force equal to the weight of the fluid it pushes aside. It floats if it can displace its own weight of fluid before it is fully submerged, sinks if it cannot, and hovers when its average density matches the fluid's.

For a liquid flowing through a pipe, the same volume must pass every point each second, so the cross-sectional area times the speed stays constant. Where the pipe narrows to half the area, the fluid moves twice as fast. That is why covering part of a hose's opening makes the water shoot farther.

Along a streamline, pressure plus kinetic energy per unit volume plus potential energy per unit volume stays constant. Where a fluid speeds up, its pressure drops. That drop is what pulls liquid up an atomizer tube. The equation holds only under its assumptions, such as steady flow with negligible friction.

Viscosity is a fluid's resistance to flowing. In smooth laminar flow, layers slide past one another; at higher speeds the flow turns turbulent, full of swirling eddies. The Reynolds number, built from speed, size, density and viscosity, predicts which: low values mean laminar flow and high values mean turbulent flow.

Simple harmonic motion happens whenever the restoring force is proportional to the displacement and points back toward equilibrium. That rule forces a sine-shaped motion with a fixed period. The object moves fastest as it passes the equilibrium point and stops for an instant at each end.

A mass m on a spring of stiffness k oscillates with period T = 2π√(m/k). The amplitude does not appear: pull it back farther and it moves faster, covering the extra distance in the same time. A heavier mass or a softer spring gives a longer period.

In simple harmonic motion, energy swaps back and forth between kinetic and potential while the total stays constant. At each end the energy is all potential and the speed is zero; at the center it is all kinetic and the speed is greatest. For a spring the total energy is one-half kA², set by the amplitude A.

Watch a point moving at steady speed around a circle from the side, and its shadow moves back and forth in simple harmonic motion. That is why the equations of an oscillator use sine and cosine, and why the oscillation's period equals the time for one trip around the circle.

For small swings, a pendulum's period is T = 2π√(L/g), so it depends only on the length and on g, not on the mass or the width of the swing. The approximation works well at small angles; for large swings the real period grows longer than the formula predicts.

Friction or drag drains energy from an oscillator, so its amplitude shrinks. An underdamped system keeps oscillating as it dies away. A critically damped one returns to rest in the shortest time without overshooting, which is what a car's shock absorbers aim for. An overdamped one creeps back slowly with no oscillation.

Push an oscillator periodically and it responds most strongly when your pushing frequency matches its natural frequency. That is resonance, and damping limits how large the amplitude can grow. Pushing a child on a swing at the same point in every swing is resonance at work.

In a transverse wave the medium moves at right angles to the direction of travel, as on a plucked string; in a longitudinal wave it moves along the direction of travel, as in sound. Amplitude is the largest displacement, wavelength the length of one cycle, period the time for one cycle, and frequency the cycles per second.

A wave's speed equals its frequency times its wavelength, v = fλ. The medium sets the speed. If a source raises its frequency in the same medium, the speed stays put and the wavelength shortens. When a wave passes into a new medium, its frequency stays fixed while its speed and wavelength change.

Waves on a string travel faster when it is tighter and slower when it is heavier per unit length: v = √(T/μ), where T is the tension and μ the mass per length. Tightening a guitar string raises its pitch, and the thick low strings are heavier, so they vibrate more slowly.

When waves overlap, their displacements add. Crest meeting crest builds a larger wave, which is constructive interference; crest meeting trough cancels, which is destructive. For two sources in step, a path difference of a whole number of wavelengths reinforces and an odd number of half wavelengths cancels.

Waves reflecting back and forth on a string can form a standing wave, with nodes that never move and antinodes between them. A string fixed at both ends fits only whole numbers of half wavelengths, so it vibrates only at its fundamental frequency and whole-number multiples of it, called harmonics.

Sound is a pressure wave, so its speed is set by the medium, not by how loud or how high the sound is. In a gas it depends on the temperature: warmer air carries sound faster. In air at sea level on a standard day it is about 761 miles per hour, roughly a kilometer every three seconds.

Sound intensity from a small source falls with the square of the distance, so twice as far means a quarter of the intensity. The decibel scale is logarithmic: each 10 decibels more is ten times the intensity, so a sound 20 decibels louder is a hundred times as intense.

Two tones with slightly different frequencies drift in and out of step, so their combined sound swells and fades. The number of swells per second, the beat frequency, equals the difference between the two frequencies. Musicians tune by adjusting a string until the beats slow down and vanish.

When a source moves toward you its waves bunch up and you hear a higher frequency; moving away, they spread out and the pitch drops. The source's own frequency never changes. If the source moves faster than the waves, they pile up into a cone-shaped shock wave.

An echo is sound reflected from a boundary. Time the echo, multiply by the speed of sound, and halve the result, because the sound traveled out and back. Sonar, bat echolocation and medical ultrasound all work this way, and higher frequencies reveal smaller details.

Two objects in contact exchange heat until they reach the same temperature, called thermal equilibrium. The zeroth law says that if A and B are each in equilibrium with C, they are in equilibrium with each other, which is exactly what lets a thermometer work.

Celsius places water's freezing and boiling points at 0 and 100 degrees at standard pressure, and Fahrenheit at 32 and 212. Kelvin uses Celsius-sized steps but starts at absolute zero, so 0 K is -273.15 °C. To convert Celsius to Fahrenheit, multiply by 9/5 and add 32.

Most materials expand when heated, by an amount proportional to their size and the temperature change. Bridges have expansion joints for this, and a bimetallic strip bends because its two metals expand by different amounts. Water is the exception near freezing: it is densest at about 4 °C, so colder water floats on top.

The heat needed to change the temperature of a mass m by ΔT is Q = mcΔT, where c is the specific heat. Water's high specific heat lets it absorb a lot of heat with only a small temperature rise. In a mixing problem, the heat lost by the hot part equals the heat gained by the cold part.

During melting or boiling, added heat goes into the change of phase, and the temperature holds steady until the change is complete. That energy is the latent heat. Steam at 100 °C burns worse than water at 100 °C, because it also releases its latent heat of vaporization as it condenses on your skin.

Conduction moves heat through direct contact, convection moves it with a flowing fluid, and radiation carries it as electromagnetic waves, even across empty space. A body radiates in proportion to the fourth power of its absolute temperature, so doubling its kelvin temperature multiplies its radiated power by 16.

An ideal gas obeys PV = nRT, or PV = NkT counting molecules. Its pressure comes from molecules striking the container walls. Heat the gas at fixed volume and the molecules move faster and strike harder and more often, so the pressure rises. Temperatures in these laws must be in kelvins.

The average kinetic energy of a gas molecule is three-halves kT, so temperature measures molecular motion. At the same temperature, lighter molecules move faster than heavier ones. Molecular speeds are not all equal; they spread out in a Maxwell–Boltzmann distribution.

A state variable, such as pressure, volume, temperature or internal energy, depends only on the system's present condition. Heat and work do not: they depend on the path between states. A quasi-static process runs slowly enough that the system stays close to equilibrium, so its state is defined at every step.

A gas does work when it expands, equal to the area under its path on a pressure-volume graph. Different paths between the same two states enclose different areas, so they involve different amounts of work. Taking W as work done by the gas, expansion gives positive W and compression negative.

The first law is energy conservation for a system: ΔU = Q - W, where Q is the heat added and W is the work done by the system. Added heat raises the internal energy unless the system spends it doing work. Check which sign convention a problem uses, because some texts take W as work done on the system.

The internal energy of an ideal gas depends only on its temperature. Any process that ends at the starting temperature leaves the internal energy unchanged. In an isothermal process ΔU is therefore zero, and the heat added equals the work done.

An isothermal process holds the temperature constant, so ΔU is zero and its path on a pressure-volume graph is a hyperbola. An isobaric process holds the pressure constant, so its path is a horizontal line and the work done is simply PΔV.

An isochoric process holds the volume constant, so no work is done and all the heat changes the internal energy. An adiabatic process exchanges no heat, so any work comes entirely out of internal energy. On a PV graph an adiabatic curve is steeper than an isothermal one, because PV^γ stays constant.

Heating a gas at constant pressure takes more energy than at constant volume, because a gas held at constant pressure expands as it warms and spends some of the energy as work. So Cp is larger than Cv, and for an ideal gas the difference is R. Molecules with more ways to store energy, such as rotation, have larger heat capacities.

A reversible process could run backward and leave no trace on the system or its surroundings. Real processes are irreversible: friction, heat flowing across a temperature difference and sudden expansion all spoil reversibility. The reversible case is an ideal limit used to find the best possible performance.

A heat engine takes heat from a hot source, turns part of it into work and rejects the rest to a cold sink. Its efficiency is the work out divided by the heat in. The second law forbids rejecting no heat at all, so no engine can turn all of its heat input into work.

A refrigerator or heat pump spends work to move heat from cold to hot, against its natural direction. Its coefficient of performance is the heat moved divided by the work supplied, and it can be well above 1. That is how a heat pump can deliver more heat to a house than the electrical energy it uses.

The Kelvin statement says no cyclic engine can turn heat from a single reservoir entirely into work. The Clausius statement says heat cannot flow by itself from a colder body to a hotter one. The two are equivalent: a device that broke either one could be combined with ordinary machines to break the other.

No engine working between a hot reservoir at temperature Th and a cold one at Tc can beat the Carnot efficiency, 1 - Tc/Th, with both temperatures in kelvins. Plugging in Celsius is the classic error. Raising the hot temperature or lowering the cold one is the only way to raise that ceiling.

For a reversible process, the change in entropy is the heat transferred divided by the absolute temperature, ΔS = Q/T. Entropy is a state function, so its change between two states is the same along any path, even though heat is not. The total entropy of an isolated system never decreases.

Entropy also counts how many microscopic arrangements match a system's large-scale state. A mixed or spread-out state has vastly more arrangements than an ordered one, so systems drift toward it by sheer probability. The second law is statistical: a reversal is not forbidden, just overwhelmingly unlikely for anything large.

Charge comes in two kinds, positive and negative: like charges repel and unlike charges attract. Charge comes in whole multiples of the elementary charge e, carried by the electron and the proton, and the total charge of an isolated system never changes. Charging an object moves electrons around; it does not create charge.

In a conductor some electrons move freely; in an insulator they stay bound to their atoms. Rubbing two materials together transfers electrons between them. Touching a charged object shares its charge by conduction, while holding it near a conductor without touching shifts the charges inside, which is induction. Grounding lets charge flow to or from the earth.

The force between two point charges is F = kq₁q₂/r², repulsive for like charges and attractive for unlike ones. Like gravity it falls with the square of the distance, but it is enormously stronger and can repel. With several charges, find each force on its own and add them as vectors.

The electric field at a point is the force a small positive test charge would feel there, divided by its charge. The field exists whether or not any test charge is present. A positive charge is pushed along the field and a negative charge against it.

Field lines begin at positive charges and finish at negative ones. Their direction shows which way the field points, and how closely they crowd together shows how strong it is. Lines never cross, because the field at any point has only one direction.

Spread-out charges have standard field patterns. A long straight line of charge has a field that falls off as 1/r, and a large flat sheet has a uniform field that does not weaken with distance. Seen from far enough away, any finite charge looks like a point, and its field returns to the inverse-square law.

Gauss's law says the total electric flux through any closed surface equals the charge enclosed divided by ε₀. It is always true, but it is useful for finding a field only when symmetry makes the field constant over the surface, as with spheres, long cylinders and flat sheets.

In a conductor carrying no current, the field inside is zero, any excess charge sits on the surface, and the field just outside points straight out from the surface. A closed metal shell therefore shields everything inside it from outside electric fields.

Electric potential energy is the work needed to assemble charges into an arrangement. For like charges it is positive, because you must push them together; for unlike charges it is negative, because they pull together on their own. Released charges move toward lower potential energy.

Electric potential is potential energy per unit charge, measured in volts, or joules per coulomb. Only differences in potential matter, so where you put zero is a choice. A charge q moved through a potential difference ΔV gains or loses energy qΔV.

A point charge q creates a potential kq/r, and the potentials from several charges simply add as signed numbers. That makes potential easier to find than the field, which must be added as vectors. Halfway between equal and opposite charges the potential is zero, even though the field there is not.

The electric field points in the direction in which the potential falls fastest, and its strength is how quickly the potential changes with distance. Equipotential surfaces always cross field lines at right angles, so moving a charge along one takes no work.

Capacitance is charge stored per volt, C = Q/V, measured in farads. For parallel plates, C = ε₀A/d: bigger plates store more, and so do plates moved closer together. Capacitance depends on the geometry and the material between the plates, not on how much charge happens to be on them.

Capacitors in parallel simply add, because each adds plate area. In series their reciprocals add, so the total is less than the smallest one. These rules are the reverse of the resistor rules, which is exactly why they are easy to mix up.

A charged capacitor stores energy one-half CV², which is the same as one-half QV. The energy lives in the electric field between the plates. Doubling the voltage across the same capacitor quadruples the energy stored.

Putting an insulator called a dielectric between the plates raises the capacitance by its dielectric constant: its molecules polarize and partly cancel the field inside. With a battery still connected, the voltage stays fixed and the charge rises; with the capacitor isolated, the charge stays fixed and the voltage falls.

Current is the rate at which charge passes a point, measured in amperes, or coulombs per second. Conventional current is taken to flow from positive to negative, the opposite of the electrons' actual motion in a wire. The electrons drift slowly, but the electrical signal travels almost instantly, which is why a lamp lights the moment you switch it on.

A wire's resistance is R = ρL/A: it grows with length, shrinks with cross-sectional area, and depends on the material through its resistivity ρ. For most metals, resistance also rises with temperature, a change described by the temperature coefficient of resistance.

Ohm's law, V = IR, says current is proportional to voltage for an ohmic material at constant temperature. It describes certain materials, not every component: a diode, or a lamp filament heating up as it glows, does not follow it.

Electrical power is P = IV, and with Ohm's law it can also be written I²R or V²/R; pick the form that uses what you know. Resistors in series carry the same current, so I²R shows the larger one heats more. In parallel they share a voltage, so V²/R shows the smaller one heats more.

A battery's emf is the energy it gives each coulomb of charge. Every real battery has some internal resistance r, so when a current I flows the terminal voltage falls below the emf by Ir. The bigger the current you draw, the bigger the drop, which is why lights dim while a car's starter motor runs.

Resistors in series carry the same current, and their resistances add. Resistors in parallel share the same voltage, and their reciprocals add, so the total is less than the smallest branch. Adding another parallel branch lowers the total resistance and raises the current drawn from the source.

Kirchhoff's junction rule says the current flowing into a junction equals the current flowing out, because charge is conserved. The loop rule says the voltage changes around any closed loop add to zero, because energy is conserved. Pick a direction around each loop and keep every sign consistent with it.

A capacitor charging through a resistor gains voltage quickly at first and then more slowly, along an exponential curve. The time constant, RC, is the time to reach about 63 percent of the final value. After about five time constants the capacitor is, for practical purposes, fully charged.

Magnetic field lines leave a magnet's north pole and return to its south pole, and they always form closed loops. Cut a magnet in half and you get two smaller magnets, each with its own north and south pole. The SI unit of magnetic field is the tesla.

A charge q moving at speed v through a magnetic field B feels a force qvB sin θ, at right angles to both the velocity and the field, in a direction the right-hand rule gives. Because the force always points across the motion, it changes the charge's direction but never its speed, and it does no work.

A charged particle moving across a uniform magnetic field follows a circle of radius r = mv/(qB), so faster or heavier particles curve less. The time to go once around does not depend on the speed, which is what makes a cyclotron work. Any motion along the field adds a steady drift, turning the circle into a helix.

A wire of length L carrying a current I in a field B feels a force BIL sin θ, the sum of the forces on all the moving charges inside it. When the wire runs parallel to the field there is no force at all.

A current loop in a magnetic field feels a torque that turns it to line up with the field. An electric motor keeps that torque turning the same way by reversing the current every half turn, using a commutator or alternating current.

When current flows through a conductor in a magnetic field, the magnetic force pushes the moving charge carriers toward one side. That builds a small voltage across the conductor, the Hall voltage. Its sign shows whether the carriers are positive or negative, and its size measures the carrier density or the field.

A long straight wire carrying a current makes circular magnetic field lines around itself. The field is strongest close to the wire and falls off as 1/r. Point your right thumb along the current and your curled fingers show the field's direction.

Inside a long coil, a solenoid, the magnetic field is nearly uniform and proportional to the current and to the number of turns per unit length. An iron core multiplies that field greatly, because iron is ferromagnetic: its small magnetic domains line up with the applied field.

Magnetic flux through a loop is the field strength times the area times the cosine of the angle between the field and the loop's perpendicular. You can change it in three ways: change the field, change the area, or turn the loop.

Faraday's law says the emf induced in a loop equals the rate at which the magnetic flux through it changes. A large flux that holds steady induces nothing. Moving a magnet faster, or adding more turns of wire, gives a larger emf.

Lenz's law gives the direction: the induced current flows so that its own magnetic field opposes the change in flux. If it reinforced the change instead, energy would appear from nowhere. Eddy currents in a moving metal plate obey the same rule, which is how magnetic brakes slow it down.

A conductor of length L moving at speed v through a field B has an emf BLv across it. When a current flows, the field pushes back on the moving conductor, so you must keep doing work to keep it moving. That work is the electrical energy a generator delivers.

A coil turning in a magnetic field produces an emf that rises and falls like a sine wave, which is how generators make alternating current. A running motor also acts as a generator, producing a counter-emf that opposes the supply. When a motor stalls, that counter-emf vanishes, the current surges and the windings can overheat.

An inductor opposes changes in the current through it by inducing an opposing emf, with a strength set by its inductance L. It stores energy one-half LI² in its magnetic field. In a circuit with resistance, the current rises toward its final value along an exponential curve with time constant L/R.

Alternating current swings between plus and minus a peak value. Its effective, or rms, value is the peak times about 0.707 for a sine wave, and it produces the same heating as a steady current of that size. A capacitor's reactance falls as the frequency rises, while an inductor's rises.

In a circuit with resistance, inductance and capacitance, the two reactances cancel at the resonant frequency, and the current peaks. A transformer changes an AC voltage by the ratio of its coil turns: more turns on the output side step the voltage up and the current down, keeping the power nearly the same.

Maxwell's four equations sum up electricity and magnetism: charges create electric fields; there are no isolated magnetic poles; a changing magnetic field creates an electric field; and currents and changing electric fields create magnetic fields. That last effect is what lets an electromagnetic wave sustain itself.

An electromagnetic wave is a pair of oscillating electric and magnetic fields, at right angles to each other and to the direction of travel. Each changing field regenerates the other, so the wave needs no medium and crosses empty space. Light, radio waves and X-rays are all the same kind of wave.

In a vacuum, every electromagnetic wave travels at c, exactly 299,792,458 meters per second. Its value follows from the electric and magnetic constants. Every observer measures the same c, whatever their own motion, and that fact leads directly to special relativity.

Electromagnetic waves carry energy, and their intensity is power per unit area. From a point source, intensity falls with the square of the distance, because the same power spreads over a sphere whose area grows four times each time the distance doubles.

Light carries momentum as well as energy, so it pushes on whatever absorbs or reflects it. A mirror feels twice the push an absorbing surface does, because the light's momentum reverses. The force is tiny but steady, enough to drive a solar sail or to hold small particles in optical tweezers.

From low to high frequency, the spectrum runs radio, microwave, infrared, visible light, ultraviolet, X-rays and gamma rays. Wavelength shrinks as frequency rises, and the energy carried by each photon grows, which is why ultraviolet light and X-rays can damage cells while radio waves cannot.

Red, green and blue light added together look white; pigments work the other way, each absorbing some colors, so mixing them gives darker results. An object looks a given color because it reflects those wavelengths and absorbs the rest. Under pure red light, a blue object has nothing to reflect and looks black.

Sunlight is scattered by the gases in air, and short blue wavelengths scatter far more than long red ones. That scattered blue light reaches you from every direction, so the sky looks blue. At sunset the light crosses much more air, most of the blue is scattered away, and what is left looks red and orange.

When light reflects, the angle of reflection equals the angle of incidence, both measured from the normal, a line perpendicular to the surface. A smooth surface sends a beam off in one direction and can form images. A rough surface scatters it every way, which is why paper shows no reflection of your face.

Light bends when it crosses into a medium where it travels at a different speed, and Snell's law, n₁ sin θ₁ = n₂ sin θ₂, gives the angles. Entering a medium with a higher index, light bends toward the normal. Because the index changes slightly with wavelength, a prism spreads white light into colors.

Light passing from a higher-index medium into a lower-index one bends away from the normal. Beyond a critical angle, where sin θc = n₂/n₁, none of it escapes and all of it reflects back. Optical fibers carry light this way, bouncing it along their length, and a cut diamond sparkles for the same reason.

Light's electric field can vibrate in many directions; polarized light vibrates in just one. A polarizing filter passes one direction, and Malus's law says a second filter at angle θ to the first transmits cos²θ of the light reaching it. Glare reflected from water is partly polarized, which is why polarized sunglasses cut it.

A plane mirror makes an upright virtual image the same distance behind the glass as the object stands in front of it. A concave mirror focuses parallel rays at a point half its radius of curvature away and can form real images. A convex mirror always gives a smaller upright virtual image, which is why it offers a wide view.

A converging lens bends parallel rays to a focal point; a diverging lens spreads them as if they came from one. The thin-lens equation, 1/f = 1/do + 1/di, links the focal length to the object and image distances. Most mistakes come from signs, so fix your sign convention before you start.

Magnification is image height over object height, equal to minus the image distance over the object distance. In the usual convention, a positive image distance means a real image, which can be projected onto a screen and is inverted. A negative one means a virtual image, which is upright.

Huygens' principle treats every point on a wavefront as a source of small wavelets, and the new wavefront is the surface touching them all. It explains reflection and refraction, and it predicts that waves spread around edges and through openings, which is diffraction.

Light passing through two narrow slits makes a pattern of bright and dark bands on a screen. The bright bands fall where the paths from the two slits differ by a whole number of wavelengths. Particles alone could not produce the dark bands, which is why the experiment showed that light behaves as a wave.

For two slits a distance d apart, bright fringes appear where d sin θ equals a whole number of wavelengths, with dark fringes halfway between. On a distant screen the spacing between fringes is about λL/d, so slits closer together or a longer wavelength spread the fringes out.

Light reflected from the top and bottom of a thin film travels different distances, and reflection off a higher-index layer also flips the wave by half a cycle. Together these decide which colors reinforce, which is why soap bubbles shimmer and how a thin coating on a lens reduces glare.

Light through a single narrow slit spreads into a broad central band with dimmer bands on either side. The narrower the slit, the wider the pattern. The first dark band appears where the slit width times sin θ equals one wavelength.

A grating has many evenly spaced slits. Its bright lines appear at the same angles as for two slits, d sin θ = mλ, but they are far sharper and brighter. Because the angle depends on wavelength, a grating separates light precisely into its colors, which makes it the heart of a spectrometer.

Diffraction blurs every image, so any lens or mirror has a smallest angle it can resolve. By the Rayleigh criterion, that angle is about 1.22 λ/D for an opening of diameter D. A larger telescope mirror resolves finer detail, and so does a shorter wavelength.

Interference needs waves that keep a steady phase relationship, called coherence. A laser produces it through stimulated emission: an incoming photon triggers an excited atom to emit an identical photon. The result is light of nearly one wavelength, in step, in a narrow beam, unlike the jumbled light of an ordinary lamp.

Special relativity rests on two postulates. The laws of physics are the same in every inertial frame, meaning every frame that is not accelerating. And the speed of light in a vacuum is the same for every observer, whatever the motion of the source or of the observer.

Two events that happen at the same time in one frame can happen at different times in another frame moving relative to it. Most relativity paradoxes dissolve once you stop assuming that 'at the same time' means the same thing for everyone.

A clock moving relative to you runs slow by the Lorentz factor, γ = 1/√(1 - v²/c²). Muons created high in the atmosphere reach the ground in large numbers because their clocks run slow in our frame. GPS satellites must correct for this effect, and for gravity's effect on time, to give accurate positions.

An object moving relative to you is shorter along its direction of motion by the same factor γ; its width and height do not change. Its proper length is the length measured in the frame where it is at rest, and that is always the longest value anyone measures.

The Lorentz transformation converts an event's position and time from one inertial frame to another, and it mixes space and time together. At speeds far below light speed it reduces to the everyday Galilean rule, in which you simply add or subtract velocities.

Velocities do not simply add at high speed. If a ship moving at 0.6c fires a probe forward at 0.6c relative to itself, an observer sees the probe moving at about 0.88c, not 1.2c. No combination of speeds below c ever produces a result above c.

At high speed, momentum and energy grow by the factor γ, which rises without limit as the speed approaches c, so no massive object can reach light speed. Even at rest an object has energy E = mc², so mass and energy are equivalent and can be converted into each other.

A hot object glows across a spread of wavelengths, and its peak shifts to shorter wavelengths as it gets hotter, which is Wien's law. Classical physics predicted endless ultraviolet output, the ultraviolet catastrophe. Planck resolved it by assuming energy is emitted in packets of size hf.

Light shining on a metal ejects electrons only if its frequency is above a threshold; below it, none come out however bright the light. Each photon carries energy hf, and an electron escapes with that energy minus the metal's work function. Brighter light ejects more electrons, not faster ones.

When an X-ray photon scatters off an electron, it comes away with a longer wavelength, having handed some of its energy and momentum to the electron. The size of the shift depends on the scattering angle. The effect showed that photons carry momentum, just as particles do.

Bohr pictured hydrogen's electron in orbits with fixed energies. When it drops from a higher level to a lower one, it emits a photon whose energy equals the difference, so hydrogen gives off only certain wavelengths. Those lines group into series, such as the visible Balmer series.

De Broglie proposed that matter also behaves as a wave, with wavelength λ = h/p. For everyday objects the wavelength is far too small to notice. For electrons it is comparable to the spacing of atoms in a crystal, and electron beams diffracting from crystals confirmed it.

In quantum mechanics a particle is described by a wave function. The square of its size at a point gives the probability density of finding the particle there. Since the particle must be somewhere, the probabilities over all space add up to 1, a condition called normalization.

The uncertainty principle says a particle's position and momentum cannot both be known exactly: the product of their uncertainties has a minimum set by Planck's constant. It is a property of nature, not a flaw in instruments. A similar limit links energy and time.

Confine a particle to a small region and only certain energies are allowed, because its wave has to fit. Shrink the region and those energy levels spread farther apart. A particle can also tunnel through a barrier it could not cross classically, with a chance that falls off sharply as the barrier gets thicker.

A nucleus holds protons and neutrons, together called nucleons. Protons repel each other electrically, but the strong nuclear force, much stronger at very short range, binds the nucleons together. A nucleus's radius grows roughly with the cube root of its number of nucleons.

A nucleus has less mass than its separate protons and neutrons, and that mass defect is its binding energy, through E = mc². Binding energy per nucleon peaks around iron. Fusing light nuclei or splitting heavy ones both move toward that peak, which is why each can release energy.

Alpha decay emits a helium nucleus, which a sheet of paper can stop. Beta decay emits an electron or a positron, which a thin sheet of metal such as aluminum stops. Gamma rays are high-energy photons that need thick, dense material such as lead or concrete to reduce them.

Each nucleus decays at random, but a large sample is predictable: in every half-life, half of the remaining nuclei decay, so after three half-lives an eighth is left. Activity, the number of decays per second, is measured in becquerels, one decay per second, or in curies.

In fission, a heavy nucleus such as uranium-235 absorbs a neutron, splits and releases more neutrons, which can split further nuclei in a chain reaction. A reactor uses a moderator to slow the neutrons so they cause fission more readily, and control rods that absorb neutrons to keep the reaction steady.

Fusion joins light nuclei into heavier ones and releases energy; it is what powers the Sun. The nuclei must get very close despite their electric repulsion, which takes extreme temperatures, so the fuel becomes a plasma. On Earth that plasma must also be held together long enough, using magnetic fields or lasers.

Absorbed dose, measured in grays, is the energy radiation deposits per kilogram of tissue. Equivalent dose, in sieverts, weights that by the type of radiation, because some kinds do more biological harm per unit of energy than others. Ionizing radiation damages cells by breaking chemical bonds, including those in DNA.

The Standard Model builds matter from quarks and leptons, such as the electron; a proton or neutron is made of three quarks. Forces are carried by particles: photons for electromagnetism, gluons for the strong force, and W and Z bosons for the weak force. Gravity is not yet part of it.

A renewable resource is replenished naturally on a human timescale, such as sunlight, wind, flowing water and biomass. Nonrenewable resources, such as coal, oil, natural gas and uranium, exist in limited amounts that form far more slowly than they are used.

Coal, oil and natural gas formed from the remains of ancient plants and organisms, buried and transformed over millions of years. Burning them releases their stored chemical energy, along with carbon dioxide.

Most power plants share one chain: a heat source boils water into steam, the steam spins a turbine, and the turbine turns a generator that produces electricity. Coal, gas and nuclear plants differ mainly in the heat source, while wind and hydropower skip the steam and turn the turbine directly.

Photovoltaic cells turn sunlight directly into electricity, and wind turbines turn the kinetic energy of moving air into rotation that drives a generator. Both depend on weather and time of day, so their output varies, and the grid must balance that variation.

Hydropower releases water held behind a dam, turning its gravitational potential energy into turbine motion. Tidal power taps the rise and fall of the tides. Geothermal power uses heat from deep inside the Earth to make steam.

A nuclear plant uses heat from fission in place of a furnace, and its spent fuel stays radioactive, so it must be stored securely. A biomass plant burns wood, crops or waste, materials that captured their energy from sunlight recently, and it produces ash and smoke.

A thermal power plant turns only part of its fuel's energy into electricity and rejects the rest as waste heat, as the second law requires. More energy is lost as heat in transmission lines, and since that loss grows with the square of the current, power is sent at high voltage and low current.

Wind and solar output does not always match demand, so energy is stored for later, for example by pumping water uphill to release through turbines or by charging large batteries. Using less energy, through better insulation and more efficient appliances, cuts how much has to be generated in the first place.

A light-year is a distance, not a time: how far light travels in one year, about 9.46 trillion kilometers. Because light takes time to arrive, you see a distant object as it was when its light set out, so looking farther into space means looking further back in time.

Distant galaxies recede from us at speeds proportional to their distance, v = H₀d, which is Hubble's law. Space itself is expanding, so an observer in any galaxy would see the same pattern. There is no center: every galaxy sees the others moving away from it.

As light crosses expanding space, its wavelength stretches toward the red, and the farther the source, the greater the stretch. This redshift comes from space expanding while the light travels, which is different from an ordinary Doppler shift caused by a source moving through space.

The universe began about 13.8 billion years ago in an extremely hot, dense state, and it has been expanding and cooling ever since. Within the first fractions of a second the basic particles formed, and within minutes the lightest atomic nuclei had formed too.

About 380,000 years after the big bang, the universe cooled enough for atoms to form and light to travel freely. That light still fills space, stretched into microwaves at a temperature of about 2.7 kelvins, and it arrives almost equally from every direction.

In its first few minutes, the young universe was hot enough to fuse protons and neutrons into the lightest nuclei: deuterium, helium, and a trace of lithium. Stars alone could not have produced all the helium observed today, which is strong evidence for the big bang.

Galaxies spin too fast for their visible matter to hold them together, pointing to dark matter that has gravity but gives off no light. Separately, the expansion of the universe is speeding up, which is attributed to dark energy. Together the two make up most of the universe's content.

A household circuit has three wires. The hot wire carries the alternating voltage, and the neutral wire carries the current back and sits near ground voltage. The ground wire normally carries no current at all; it connects metal parts to the earth so that a fault has a safe path.

A fuse or circuit breaker opens the circuit when too much current flows, before the wiring can overheat: a fuse melts and a breaker trips. It belongs in the hot wire, so that when it opens, the appliance is cut off from the voltage instead of being left connected to it.

Grounding connects an appliance's metal case to the earth, so a fault sends a large current down the ground wire and trips the breaker instead of leaving the case live. Double-insulated tools and appliances use two layers of insulation instead, so they need no ground connection.

A ground-fault circuit interrupter, or GFCI, compares the current going into a device with the current coming back. If they differ by more than about 5 milliamperes, some current is leaking elsewhere, possibly through a person, and it cuts the power in as little as 1/40 of a second.

A shock harms you through the current that flows through your body; voltage matters because it drives that current. How severe it is depends on the amount of current, its path through the body, how long it flows and its frequency. Wet skin conducts far better than dry skin, which raises the current.

An appliance's rating plate gives its power. Divide the power by the supply voltage to find the current it draws, from P = VI. Utilities bill energy in kilowatt-hours: an 800-watt heater running for 2.5 hours uses 2 kilowatt-hours.