Unit 20 · Physical Science: Forces, Energy and Waves

Picture a Chicago Transit Authority train leaving a station. To describe where it is, you need a starting point. Physicists call that point the origin, and they call the distance and direction from it the position. A train 2 kilometers east of the station has a different position from one 2 kilometers west, even though both are the same distance away. Every description of motion starts by choosing an origin and a direction to call positive.

Speed is how far something travels in a certain time. If the train covers 6 kilometers in 10 minutes, its average speed is 0.6 kilometers per minute, or 36 kilometers per hour. Speed alone does not say which way the train is going. Velocity does. Velocity is speed with a direction attached, such as 36 kilometers per hour east. Two trains at the same speed in opposite directions have different velocities. This matters, because a change in direction is a change in velocity, even if the speed stays the same.

Scientists use meters and seconds for most calculations, so a speed of 36 kilometers per hour becomes 10 meters per second. The formula is simple: average velocity equals the change in position divided by the time it took. If you walk 100 meters north in 80 seconds, your average velocity is 1.25 meters per second north. Your speed at any single instant might be more or less than that, which is why we say average.

Here is a test of the idea. Suppose a runner circles a 400 meter track in 80 seconds and ends where she started. Her average speed is 5 meters per second, since she covered 400 meters. But her change in position is zero, because she ended at her origin. So her average velocity for the whole lap is zero. That sounds strange until you remember that velocity is about where you ended up compared with where you began, not about the path you took.

A graph can hold a whole trip in one picture. A position-time graph puts time on the horizontal axis and position on the vertical axis. A flat line means the object is not moving: time passes but position stays the same. A straight, tilted line means steady motion. The steeper the line, the faster the object. The slope of the line, the rise divided by the run, is exactly the velocity. A line going downhill means the object is moving back toward the origin.

A velocity-time graph tells a different story about the same trip. Now the vertical axis is velocity. A flat line means constant velocity, not standing still. A line sloping upward means the object is speeding up. The slope of this graph is acceleration, the rate at which velocity changes. A line at zero means the object is stopped. The area between the line and the time axis gives the distance traveled, because velocity multiplied by time equals distance.

Try the crash test from the story. On a velocity-time graph, the car’s line rises from zero as the cable pulls it, holds steady at about 15.6 meters per second, then plunges to zero in about a tenth of a second when it hits the barrier. That steep drop is a huge negative acceleration. On the dummy’s graph, the drop comes a moment later and is not as steep, because the belt and airbag stretch his stop over a longer time. The whole point of safety design is to make that line less steep.

Students often mix up the two kinds of graph. A curved position-time line does not mean a curved path; it means the velocity is changing. A velocity-time line that crosses zero does not mean the object is back at the start; it means it has reversed direction. Always check the label on the vertical axis before you read a motion graph. The same shape means different things on different graphs.

Drop a ball from a balcony and it does not fall at a steady speed. It starts from rest and moves faster and faster. Near Earth’s surface, a falling object gains about 9.8 meters per second of speed every second, if air resistance is small. Physicists write this as 9.8 meters per second squared and call it g, the acceleration of gravity. After 1 second the ball moves at about 9.8 meters per second; after 2 seconds, about 19.6 meters per second. Galileo showed in the early 1600s that heavy and light objects fall with the same acceleration when air does not interfere.

Acceleration is not only speeding up. Any change in velocity counts. A car braking at a red light is accelerating, in the negative direction. A car rounding a curve at a steady 40 kilometers per hour is also accelerating, because its direction is changing. Your body notices: you press against the door on a curve and lean forward when the driver brakes. Whenever you feel that push, velocity is changing.

The formula is average acceleration equals the change in velocity divided by the time it took. A sprinter who goes from rest to 10 meters per second in 2 seconds has an average acceleration of 5 meters per second squared. The crash-test car goes from 15.6 meters per second to zero in about 0.1 seconds, an acceleration of about negative 156 meters per second squared, roughly 16 times g. That is why a crash at city speed can injure a person who would be fine after a fall from a chair.

Air resistance changes the story for real objects. A feather falls slowly because air pushes up on it almost as hard as gravity pulls down. In 1971 an Apollo 15 astronaut dropped a hammer and a feather on the Moon, where there is no air, and they landed together. On Earth, a skydiver speeds up until air resistance balances gravity, and then falls at a steady terminal velocity. The rule that acceleration is 9.8 meters per second squared is really a rule about gravity alone.

Why did the crash-test dummy keep going when the car stopped? Because nothing stopped him yet. In 1687 Isaac Newton stated a rule now called the first law of motion: an object at rest stays at rest, and an object in motion keeps moving at the same velocity, unless a net force acts on it. The tendency of an object to keep doing what it is doing is called inertia. The dummy’s inertia carried him forward at 15.6 meters per second until the belt, the airbag or the dashboard applied a force.

This idea was not obvious. For almost two thousand years, most thinkers followed Aristotle, who taught that a moving object needs a push to keep moving. It looks that way: a rolling ball slows and stops. Galileo argued that the ball stops because of friction, not because motion naturally runs out. Roll a ball on ice and it goes much farther. Remove friction entirely, he reasoned, and the ball would roll forever. Newton built his first law on Galileo’s insight.

The key phrase is net force, the total of all forces acting on an object. A book on a table has gravity pulling down and the table pushing up with equal strength. The forces are balanced, the net force is zero, and the book stays put. A car cruising at a steady 100 kilometers per hour on a highway also has zero net force: the engine’s push forward exactly balances air resistance and friction. Constant velocity does not require a net force. Changing velocity does.

You can feel inertia in everyday life. When a bus starts, you lurch backward, not because something pushed you back, but because your body tried to stay at rest while the floor moved forward. When the bus stops, you lurch forward, because your body tries to keep moving. Seat belts exist because of the first law. Without one, in a crash you keep moving at the car’s old speed until something inside the car stops you.

Push an empty shopping cart and it leaps ahead. Push a full one with the same effort and it moves grudgingly. Newton’s second law puts this into numbers: net force equals mass times acceleration, or F = ma. Mass is the amount of matter in an object, measured in kilograms. The unit of force is the newton, written N. One newton is the force that gives a 1 kilogram mass an acceleration of 1 meter per second squared. It is roughly the weight of a small apple.

The law works in both directions. If you know the force and the mass, you can predict the acceleration. A 1,000 kilogram car with a net forward force of 3,000 newtons accelerates at 3 meters per second squared. If you know the mass and can measure the acceleration, you can find the force. That is what the crash dummy’s sensors do. A dummy head of about 4.5 kilograms that stops at 1,000 meters per second squared felt a force of about 4,500 newtons, more than the weight of a small car.

Weight is a force, not a mass. Your weight is the force of gravity on your mass, equal to mass times g. A 60 kilogram student weighs about 588 newtons on Earth but only about 96 newtons on the Moon, where g is about 1.6 meters per second squared. Her mass is 60 kilograms in both places. Bathroom scales in the United States report pounds, which are units of force. Doubling an object’s mass doubles its weight, but it also doubles its inertia, which is why all objects fall with the same acceleration.

The second law also explains why safety design is about time. The same change in velocity can happen quickly or slowly. Acceleration is that change divided by time, so a longer stop means a smaller acceleration, and a smaller acceleration means a smaller force on the same mass. A crumple zone that doubles the stopping time cuts the average force on the passengers in half. Engineers cannot change the mass of a person or the speed of the crash, but they can stretch out the time.

Stand on a skateboard and push against a wall. The wall does not move, but you roll backward. You pushed the wall, and the wall pushed you. Newton’s third law says that forces always come in pairs: if object A pushes on object B, then B pushes on A with a force of equal size in the opposite direction. The two forces act on different objects, which is why they do not cancel. Your push moved nothing, but the wall’s push moved you.

This law explains how anything moves at all. When you walk, your foot pushes backward on the ground, and the ground pushes forward on you. A swimmer pushes water backward, and the water pushes the swimmer forward. A rocket does not push against the air; it throws hot gas out the back, and the gas pushes the rocket forward. That is why rockets work in the vacuum of space, where there is nothing to push against except their own exhaust.

The pairs are always equal, even when the results look very different. When a truck hits a mosquito, the force the truck puts on the mosquito is exactly the same size as the force the mosquito puts on the truck. The difference is mass. By the second law, the same force gives the tiny mosquito an enormous acceleration and the massive truck an acceleration too small to notice. In the crash test, the barrier pushes the car exactly as hard as the car pushes the barrier.

A common mistake is to pair the wrong forces. A book resting on a table has gravity pulling it down and the table pushing it up. Those are balanced forces, but they are not a third-law pair, because both act on the book. The third-law partner of Earth pulling down on the book is the book pulling up on Earth. The partner of the table pushing up on the book is the book pushing down on the table. Third-law pairs always involve two objects acting on each other.

A bowling ball rolling slowly and a baseball flying fast can be equally hard to stop. What they share is momentum, the product of mass and velocity. A 7 kilogram bowling ball at 3 meters per second has 21 kilogram-meters per second of momentum. A 0.15 kilogram baseball at 40 meters per second has 6. Momentum has a direction, the same as the velocity. Two cars with equal momentum heading toward each other have total momentum of zero.

To change an object’s momentum, you apply a force for some time. The product of force and time is called impulse, and impulse equals the change in momentum. This is really Newton’s second law written differently. A catcher stopping a fastball can use a big force for a short time, with a stiff glove, or a smaller force for a longer time, by letting the glove give as the ball arrives. Either way the impulse is the same, because the ball’s momentum goes to zero either way.

Now the crash test makes full sense. The dummy’s momentum, about 75 kilograms times 15.6 meters per second, or roughly 1,170 kilogram-meters per second, must drop to zero. That impulse is fixed. The only question is how long the stop takes. Hitting a dashboard, the stop might last 0.01 seconds, requiring an average force of about 117,000 newtons. Stretched over 0.1 seconds by a belt and airbag, the force drops to about 11,700 newtons. Ten times the time means one tenth the force.

You already use the impulse idea without knowing it. You bend your knees when you land from a jump, lengthening the stop. Gymnasts land on padded mats. Cars have bumpers that squash. Eggs shipped to Illinois grocery stores ride in cartons that crush a little before the egg does. In every case, the change in momentum is unavoidable, so the design spreads it over more time to cut the force.

Roll a marble into a line of marbles and one pops out the far end at almost the same speed. Something was handed on. That something is momentum. When objects push on each other, the third law guarantees that the forces are equal and opposite and act for the same time, so the impulses are equal and opposite too. Whatever momentum one object loses, the other gains. The total stays the same. Physicists say momentum is conserved.

This rule lets you predict collisions without knowing anything about the forces. Suppose a 1,000 kilogram car moving at 10 meters per second rear-ends a 1,000 kilogram car at rest, and the bumpers lock so they move together. Before, the total momentum is 10,000 kilogram-meters per second. After, the same 10,000 is shared by 2,000 kilograms, so the pair moves at 5 meters per second. Investigators use this idea backward, working out how fast cars were going from where they ended up.

Conservation only holds for a closed system, a set of objects with no outside force acting on them. During the brief moment of a collision, the forces between the cars are so large that friction from the road hardly matters, so the two cars together are nearly a closed system. Over a longer time, friction from the road takes momentum away and hands it to Earth. Earth is so massive that the change in its motion is far too small to measure.

Momentum conservation shows up everywhere. When a gun fires, the bullet gains forward momentum and the gun gains equal backward momentum, which the shooter feels as recoil. When a figure skater pushes off a partner, both glide apart, the lighter one faster. When a rocket burns fuel, the exhaust carries momentum one way and the rocket the other. And in a total momentum of zero, as in a head-on crash of equal cars at equal speeds, both cars stop dead, which is the worst possible outcome for the people inside.

In 1959 a Swedish engineer named Nils Bohlin designed the three-point seat belt, the lap-and-shoulder strap in every modern car. His company, Volvo, let other carmakers use the design for free. Before belts, a person in a crash kept moving until the steering wheel, windshield or road stopped them in a few thousandths of a second. A belt stops the body over a longer time and spreads the force across the strong bones of the hips and chest instead of the head.

An airbag goes further. A sensor detects the sudden deceleration of a crash and fires a small chemical charge that fills the bag with gas in about 30 thousandths of a second, faster than a person can blink. The head sinks into the bag, which vents gas through small holes as it squashes, so the stop is gradual. The bag must inflate before the head arrives but not so hard that it becomes a hazard itself. Getting that timing right took decades of tests with dummies.

The car’s body does its part too. A crumple zone is a front or rear section built to fold in a controlled way. Folding steel takes time and absorbs energy, so the passenger compartment, which is built stiff, decelerates more gently. It seems backward that a car designed to crush is safer than a rigid one, but the physics is clear. A rigid car stops in a shorter distance and time, and everything inside feels a larger force.

Engineers judge these designs with numbers from the dummy: how much the head accelerated, how far the chest compressed, how much force went through the neck and legs. In the United States, the National Highway Traffic Safety Administration and the insurance-funded Insurance Institute for Highway Safety run crash tests and publish ratings. Deaths per mile driven in the United States have fallen sharply since the 1960s, and belts, airbags and better car bodies are a large part of the reason. Illinois, like every state except one, requires belt use by law.

The same force that pulls an apple down keeps the Moon in orbit. That was Newton’s great leap. He proposed that every object with mass attracts every other object. The force between two masses is proportional to the product of the masses and inversely proportional to the square of the distance between their centers. Double the distance and the force drops to one quarter. Triple it and the force drops to one ninth. This is the law of universal gravitation.

Written as a formula, the force is F = G times m1 times m2 divided by r squared, where r is the distance between centers and G is a constant. G is tiny, about 6.67 times 10 to the minus 11 newton meters squared per kilogram squared. That is why you do not feel the pull of your desk. Two 60 kilogram students one meter apart attract each other with roughly 0.00000024 newtons. Only when one mass is planet-sized does the force become obvious.

Newton could not measure G. In 1798 Henry Cavendish did, using a delicate balance with lead balls hanging from a wire inside a shed, watching the wire twist as the large balls pulled the small ones. From his measurement, people could calculate Earth’s mass, about 5.97 times 10 to the 24 kilograms. Cavendish is sometimes said to have weighed the Earth. In a sense he did, with a few balls of lead and a very patient eye.

The inverse-square rule explains why g is 9.8 meters per second squared at the surface but smaller far above it. At the height of the International Space Station, about 400 kilometers up, Earth’s pull is still about 90 percent of surface gravity. Astronauts float not because gravity is gone but because they and their station are falling together around Earth. If you climbed to twice Earth’s radius from its center, about 6,400 kilometers up, your weight would be one quarter what it is now.

Rub a balloon on your hair and it sticks to a wall. Pull off a wool sweater in a dry Illinois winter and you hear crackles and see tiny sparks. These are signs of electric charge. Matter contains protons with positive charge and electrons with negative charge. Rubbing moves electrons from one material to another, leaving one positive and the other negative. Like charges repel and opposite charges attract. Charge is measured in coulombs, and an electron carries a very small charge, about 1.6 times 10 to the minus 19 coulombs.

In 1785 the French physicist Charles-Augustin de Coulomb measured the force between charged spheres with a twisting balance much like the one Cavendish later used. He found that the force is proportional to the product of the two charges and inversely proportional to the square of the distance between them. This is Coulomb's law: F = k times q1 times q2 divided by r squared. The constant k is about 9 times 10 to the 9 newton meters squared per coulomb squared.

Compare that with gravity’s G, about 6.67 times 10 to the minus 11. The electric force is enormously stronger. Two charges of one coulomb each, one meter apart, would push with about 9 billion newtons. Yet you do not feel electric forces from your desk, because ordinary objects have almost exactly equal amounts of positive and negative charge, and the effects cancel. Gravity, weak as it is, only attracts, so it adds up over a whole planet. Electricity is what holds atoms and molecules together; gravity is what holds the solar system together.

The two laws have the same shape, and that is not an accident of history. Both are inverse-square laws, and both describe forces that act across empty space without any touching. In the crash test, every force you can see, the belt on the chest, the bag on the face, the steel folding, is at bottom electric. The atoms in the belt are pushing on the atoms in the dummy through the electric force between their electrons. Contact forces are electric forces in disguise.

How does the Sun pull on Earth across 150 million kilometers of empty space? Physicists answer with the idea of a field. A gravitational field fills the space around any mass, and it tells you what force a kilogram placed at any point would feel. Near Earth’s surface the field is about 9.8 newtons per kilogram, pointing down. An electric field does the same for charge, and magnets are surrounded by a magnetic field. The field is not just a picture; it stores energy and can carry waves, as the next chapter shows.

Newton explained orbits with a thought experiment. Fire a cannonball horizontally from a tall mountain, and it curves down and hits the ground. Fire it faster and it goes farther before landing. Fire it fast enough and the ground curves away beneath it as fast as it falls, so it never lands. It is in orbit. An orbit is not the absence of gravity; it is gravity doing exactly what it always does, bending a straight path into a circle or an ellipse. Take gravity away and the satellite would fly off in a straight line.

Near Earth’s surface that speed is about 7.9 kilometers per second. The International Space Station, about 400 kilometers up, moves at roughly 7.7 kilometers per second and circles Earth about every 90 minutes. Higher orbits are slower, because gravity is weaker there. At about 36,000 kilometers up, a satellite takes exactly one day to circle Earth and so hangs over the same spot; many weather and television satellites sit there. The Moon, at about 384,000 kilometers, takes about 27 days.

The first artificial satellite, Sputnik 1, was launched by the Soviet Union on October 4, 1957, and circled Earth every 96 minutes, beeping to anyone with a radio. Today thousands of satellites carry phone calls, weather images, television and the GPS signals that locate a bus in Chicago to within a few meters. Every one of them follows the same law Newton wrote down for the apple and the Moon. Engineers who plan a launch are, in the end, solving F = G m1 m2 divided by r squared.

A car at 30 kilometers per hour dents a bumper. The same car at 60 kilometers per hour can destroy the front end. Speed doubled, but the damage grew far more than double. The reason is kinetic energy, the energy an object has because it is moving. Its formula is one half times mass times speed squared. Because speed is squared, doubling the speed multiplies the kinetic energy by four. Tripling it multiplies the energy by nine.

Energy is measured in joules, written J. One joule is the energy needed to push with a force of one newton through one meter. It is a small amount, about what it takes to lift an apple one meter. A 1,000 kilogram car at 10 meters per second has one half times 1,000 times 100, or 50,000 joules of kinetic energy. At 20 meters per second it has 200,000 joules. A crumple zone must absorb all of that, which is why crashes at highway speed are so much worse than crashes in a parking lot.

Energy and momentum are different things, though both grow with mass and speed. Momentum is mass times velocity and has a direction. Kinetic energy is a plain number with no direction; a car moving east and a car moving west can have the same kinetic energy. And because energy depends on speed squared, a fast light object can carry more energy than a slow heavy one. A 0.05 kilogram bullet at 400 meters per second has 4,000 joules, more than a 100 kilogram football player at 8 meters per second, who has 3,200 joules.

When work is done on an object, its kinetic energy changes. Work is force times the distance moved in the direction of the force, and it is also measured in joules. Push a cart with 20 newtons for 5 meters and you have done 100 joules of work, and the cart’s kinetic energy rises by 100 joules if friction is ignored. Brakes work the other way: friction pushes backward over the stopping distance and takes the kinetic energy away, mostly as heat in the brake pads.

A roller coaster car at the top of the first hill is barely moving, yet it is about to reach its fastest speed of the ride. The energy is there, stored in the car’s height. This is gravitational potential energy. Near Earth’s surface it equals mass times g times height, or mgh. Lift a 2 kilogram book 1.5 meters onto a shelf and you have stored 2 times 9.8 times 1.5, about 29 joules. Let it fall and that stored energy turns into kinetic energy on the way down.

Potential energy is really energy stored in a field. Lift the book and you are pulling it against Earth’s gravitational field; the energy lives in the arrangement of book and Earth. The same idea works for electric fields. Pull two opposite charges apart and you store electric potential energy that returns as they snap back. A battery stores energy in the arrangement of its chemicals, which is electric potential energy at the scale of atoms. A stretched spring or a drawn bow stores elastic potential energy in the stretched bonds between atoms.

Height is measured from wherever you choose. A book on a shelf has potential energy relative to the floor, more relative to the basement, and none relative to the shelf itself. Only changes in potential energy matter. As the roller coaster drops from a 60 meter hill, it loses about 60 times 9.8, or 588 joules for every kilogram, and if friction is small, gains that much kinetic energy. Setting one half v squared equal to 588 gives a speed of about 34 meters per second at the bottom, regardless of the car’s mass.

Hydroelectric dams are potential energy on an industrial scale. Water behind a dam sits high; as it drops through pipes it spins turbines that turn Faraday’s generators. Pumped storage plants run this backward, pumping water uphill at night when electricity is cheap and letting it fall during the day when demand is high. A lake behind a dam is a battery, storing energy as height rather than chemistry.

Where does the kinetic energy go when brakes stop a car? Into the brake pads and discs, which get hot enough to glow on a race track. The energy did not vanish. It became thermal energy, the kinetic energy of trillions of atoms jiggling faster. Every object is made of particles in constant motion. In a solid they vibrate in place; in a gas they fly around and collide. Thermal energy is the total of all that microscopic motion.

Temperature is different. It measures the average kinetic energy of the particles, not the total. A bathtub of warm water has far more thermal energy than a cup of boiling water, because it has so many more particles, but the cup has the higher temperature. Scientists measure temperature in Celsius or in kelvin. The kelvin scale starts at absolute zero, minus 273.15 degrees Celsius, the temperature at which particles have the least possible motion. Water freezes at 273 kelvin and boils at 373.

Different materials need different amounts of energy to warm up. The energy needed to raise one gram of a substance by one degree Celsius is its specific heat. Water’s is high, about 4.18 joules per gram per degree; iron’s is about 0.45. That is why a cast-iron pan heats fast on the stove while the water in it takes minutes, and why Lake Michigan stays cool into June and warm into October, softening Chicago’s weather near the shore compared with towns farther inland.

Heat, in the strict sense, is thermal energy in transit from a warmer object to a cooler one. It moves three ways. Conduction passes energy through direct contact, as when a spoon in soup warms at the handle. Convection carries energy in a moving fluid, as warm air rises from a radiator. Radiation carries it as electromagnetic waves, which is how the Sun warms your face across empty space. Understanding all three is the difference between a house that is cheap to heat and one that is not.

In the 1840s an English brewer’s son named James Joule spent years on a stubborn experiment. He let falling weights spin a paddle wheel inside a tub of water and measured the water’s temperature with a thermometer he could read to a fraction of a degree. The water warmed. A known amount of mechanical work had become a known amount of heat, every time in the same ratio. Joule had shown that heat is not a mysterious fluid but a form of energy, and that energy can change form without being lost.

This is the law of conservation of energy: the total energy of a closed system stays the same. Energy can move from one object to another and change from one form to another, but it is never created or destroyed. A system is simply whatever you draw a boundary around: a roller coaster car, a car plus the road, a whole power plant. Choose the boundary and then keep track of every joule that crosses it. If more goes in than out, the energy inside has grown.

The law is a bookkeeping tool. A roller coaster at the top of a 50 meter hill has potential energy. At the bottom, if the track were frictionless, it would all be kinetic. In reality the car reaches the bottom a little slower than that. The missing joules are not gone; they have gone into warming the wheels, the track and the air, and into sound. Add up the kinetic energy, the thermal energy and the sound, and the total matches the starting potential energy.

Conservation is one of the most tested ideas in science. Whenever energy has seemed to vanish, careful measurement has found where it went. In the 1930s, physicists noticed that a certain kind of radioactive decay seemed to lose energy. Rather than give up the law, Wolfgang Pauli proposed an unseen particle carrying it away. The neutrino was detected in 1956. The law held. The mathematician Emmy Noether showed in 1918 that conservation of energy follows from a deep fact: the laws of physics are the same today as they were yesterday.

Set a hot cup of coffee on a desk and it cools. It never happens the other way: a room-temperature cup never draws heat from the desk and starts steaming. Energy would still be conserved if it did, so the first law does not forbid it. Something else does. The second law of thermodynamics says that heat flows on its own only from warmer to cooler, and that in any real process, energy tends to spread out and become less useful.

The reason is probability. Fast-moving particles in the coffee collide with slower ones in the air and share their motion. There are enormously more ways for the energy to be spread evenly than to stay bunched up in the cup, so spreading out is overwhelmingly likely. Physicists measure this spreading with a quantity called entropy. Entropy in an isolated system never decreases. You can move heat from cold to hot, as a refrigerator does, but only by spending energy, which spreads out even more heat elsewhere.

This is why no engine can turn heat completely into work. A car engine burns gasoline to make hot gas that pushes pistons, but most of the fuel’s energy leaves as hot exhaust and hot metal. Typical gasoline engines turn only about a quarter to a third of the fuel’s energy into motion. Power plants that boil water to spin turbines waste heat through their cooling towers. The waste is not carelessness. The second law sets a limit that no design can beat, though better designs get closer.

The second law also explains why energy can be conserved and still run short. When you use a battery, its chemical energy becomes light, sound and heat that spread into the room. Every joule is still there, but scattered among trillions of air molecules and useless for charging your phone. What we actually run out of is not energy but useful energy, energy concentrated enough to do work. The Sun, the wind, fuel and a charged battery are useful because they are concentrated. The warm air in a room is not.

Trace the electricity in your outlet backward and you reach Faraday’s generator, but what spins it? In the United States, most electricity comes from burning natural gas or coal to boil water into steam. About a fifth comes from nuclear plants, where splitting uranium atoms makes the heat instead. The rest comes from wind turbines, dams, solar panels and a few smaller sources. Illinois is unusual: its nuclear plants produce about half the state’s electricity, more nuclear power than any other state.

Each source is a different way of grabbing concentrated energy. Coal, oil and natural gas are fossil fuels, stored sunlight captured by plants and sea life hundreds of millions of years ago. They pack a great deal of energy into a small space and can be burned any time. Burning them releases carbon dioxide, which traps heat in the atmosphere, along with soot and other pollution. Their supply, though large, is finite and is not being replaced on any human timescale.

Renewable sources are replaced as fast as they are used. Wind and sunlight are free and clean at the point of use, but the wind does not always blow and the Sun sets, so they need storage or backup. Illinois is a major wind state, with turbines across its flat central prairie. Dams provide steady power but change rivers and flood land. Nuclear plants make no carbon dioxide while running and produce large amounts of power from little fuel, but their waste stays radioactive for thousands of years and accidents, though rare, can be severe.

There is no source without a cost, so choosing means weighing trade-offs: price, reliability, land, water, pollution, safety and climate. The mix has shifted quickly. Since the mid-2000s, coal’s share of U.S. electricity has fallen by more than half as natural gas, wind and solar have grown. Engineers and voters keep making these choices, and they work best when the numbers on all sides are honest.

Drop a pebble in a still pond and rings spread outward. A leaf floating nearby bobs up and down but does not travel with the rings. That is the key fact about a wave: it carries energy from place to place while the material it moves through stays put. Sound is a wave in air, ripples are waves on water, and a shake sent down a rope is a wave on the rope. In each case, a disturbance travels; the air, water or rope only moves back and forth.

Three numbers describe any wave. Wavelength is the distance from one crest to the next, measured in meters. Frequency is how many waves pass a point each second, measured in hertz, or cycles per second. Amplitude is how far the material moves from its rest position; a louder sound or a brighter light has a larger amplitude and carries more energy. A wave with a large amplitude can do more, which is why a big ocean wave knocks you down and a ripple does not.

Speed, frequency and wavelength are tied together by one equation: speed equals frequency times wavelength. In air, sound travels at about 343 meters per second. A note of 343 hertz has a wavelength of exactly one meter. A note of 686 hertz has a wavelength of half a meter. If the speed is fixed by the material, then raising the frequency must shorten the wavelength. Try it on the site’s oscilloscope and watch the crests crowd together as the frequency rises.

Waves also bend, bounce and combine. Sound reflects off a wall as an echo. Light bends as it enters water, which is why a straw in a glass looks broken. Two waves meeting can add up to a bigger wave or cancel to nothing, a behavior called interference. Noise-cancelling headphones make a sound wave that is the exact opposite of the noise, so the two cancel at your ear. Interference is how scientists know something is a wave at all.

Faraday’s fields turned out to be able to wave. In the 1860s the Scottish physicist James Clerk Maxwell wrote equations linking electricity and magnetism and found that a changing electric field makes a changing magnetic field, which makes a changing electric field, and so on, rippling outward through empty space. He calculated the ripple’s speed and got about 300,000 kilometers per second, the measured speed of light. Light, he concluded, is an electromagnetic wave. In 1887 Heinrich Hertz made and detected such waves in his lab, the first radio waves.

All electromagnetic waves travel at the same speed in a vacuum, about 3 times 10 to the 8 meters per second, so they differ only in frequency and wavelength. Lined up from longest to shortest wavelength they form the electromagnetic spectrum: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays. Visible light runs from about 700 nanometers for red to about 400 nanometers for violet, where a nanometer is a billionth of a meter. Your eye sees only that narrow band; everything else is invisible but just as real.

Each band has its uses. Radio waves carry broadcasts and phone signals because they pass through walls and travel far. Microwaves heat food by shaking water molecules and carry Wi-Fi. Infrared is felt as warmth and used in remote controls and night-vision cameras. Ultraviolet from the Sun tans and burns skin. X-rays pass through flesh but not bone, so a doctor can see a break. Gamma rays, the shortest and most energetic, come from nuclear reactions and are used to kill cancer cells.

Astronomy changed when telescopes learned to see beyond visible light. Radio telescopes map cold gas between stars, infrared telescopes see through dust to newborn stars, and X-ray telescopes find matter falling into black holes. The site’s telescope catches visible light, the same band Galileo used, but modern observatories cover the whole spectrum. Each band tells a different part of the story, because different temperatures and processes make different kinds of light.

Shine light on a clean metal surface and electrons can pop out. That should not have been surprising, since light carries energy. What puzzled physicists around 1900 was the pattern. Dim blue light kicked out electrons instantly, but bright red light, carrying far more total energy, kicked out none at all. If light were only a wave, a bright enough wave of any color should eventually shake electrons loose. It did not. Color, meaning frequency, mattered more than brightness.

In 1905 Albert Einstein proposed the answer. Light arrives in packets, later called photons, and each photon’s energy depends only on its frequency: energy equals a constant, h, times frequency. The constant is Planck’s constant, about 6.63 times 10 to the minus 34 joule-seconds, first introduced by Max Planck in 1900. A red photon has too little energy to free an electron from the metal, no matter how many red photons arrive. A single blue photon has enough. Brightness only changes how many photons arrive, and so how many electrons leave.

This is why the spectrum’s order matters for safety. Radio and visible photons carry too little energy to break the chemical bonds in your cells. Ultraviolet photons carry enough to damage DNA, which is why sunburn and skin cancer come from ultraviolet and not from the far brighter visible light. X-rays and gamma rays carry still more, which makes them useful for imaging and cancer treatment and dangerous in large doses. The energy of one photon, not the total brightness, decides what light can do to a molecule.

So is light a wave or a particle? The honest answer is that it is both, in a way nothing in everyday life prepares you for. Interference experiments show waves; the photoelectric effect shows particles. Each photon travels as a wave and arrives as a particle. This puzzle opened the door to quantum physics, and it is the reason a solar panel works: each photon of sunlight that has enough energy frees one electron in the silicon, and the flow of freed electrons is a current.

Faraday’s twitching needle became the law of electromagnetic induction: a changing magnetic field through a loop of wire creates a voltage that drives a current. Move a magnet into a coil and current flows one way; pull it out and current flows the other way; hold it still and nothing happens. Faster motion, a stronger magnet or more turns of wire all give a bigger push. The reverse works too. Current in a coil makes a magnetic field, which is how an electromagnet, a doorbell and an electric motor work.

A generator is a coil spun inside a magnetic field, or a magnet spun inside a coil. As it turns, the field through the coil grows, shrinks and reverses, so the current flows back and forth. This is alternating current, the kind in wall outlets, which in the United States reverses 60 times per second. A bicycle dynamo, a wind turbine and the generators at a nuclear plant are the same machine at different sizes. What differs is only what does the spinning.

Induction also makes the grid possible. A transformer is two coils on an iron ring, Faraday’s original 1831 setup. Alternating current in one coil makes a changing field that induces current in the other. With more turns on the second coil, the voltage rises; with fewer, it drops. Power lines carry electricity at hundreds of thousands of volts, because high voltage means low current and less energy lost as heat in the wires. Transformers step it down to 120 volts before it reaches your outlet.

Induction is everywhere once you know to look. An induction cooktop uses a changing field to induce currents directly in the steel of a pan, heating it while the glass surface stays cool. A wireless phone charger is a small transformer with an air gap. The card reader at a store’s checkout and the magnetic strip on a transit card both rely on it. Faraday was once asked what use his discovery was. The answer turned out to be almost everything.

On May 24, 1844, Samuel Morse sat in the United States Capitol and tapped a key. Forty miles away in Baltimore, an electromagnet clicked in the same pattern, spelling out the first official telegraph message in dots and dashes. It was the first time information had traveled faster than a person could carry it. The idea was simple: switch a current on and off in a code. Every message since, whether a phone call, a photo or a video stream, is the same idea with a faster switch.

Any wave can carry information if you change it in a pattern the receiver can read. An AM radio station varies the amplitude of its wave; FM varies the frequency. Your voice on a phone is turned into a pattern of electrical signals and back. A signal that copies the shape of the original sound, rising and falling smoothly, is called analog. Analog signals are easy to make, but every bit of static or fading along the way gets added to the message, and each copy is a little worse than the last.

Modern signals are digital. The sound or image is measured thousands of times per second and each measurement is written as a number in ones and zeros, the bits. A compact disc, for example, samples sound 44,100 times per second. The receiver only has to tell a one from a zero, so small static does not change the message, and a copy is perfect. Errors can be found and fixed with extra check bits. That is why a photo can cross the world through dozens of machines and arrive unchanged.

The bits ride on whatever wave is handy. Cell phones use microwaves. Wi-Fi uses microwaves at about 2.4 and 5 gigahertz. Most long-distance internet traffic travels as pulses of infrared light inside glass fibers thinner than a hair, bouncing along the inside of the glass for many kilometers before it needs a boost. Spacecraft send digital pictures home by radio from billions of kilometers away, so faint that the signal is weaker than a watch battery by the time it reaches Earth, and the bits still come through.