The basics of how things work

The four works here are “simplified teaching models” based on real phenomena. They leave out details such as friction and air resistance so that the one most important rule shows clearly.

Batteries and Bulbs (the site’s “Batteries and Bulbs”)

One rule covers how easily electricity flows: Ohm’s law, voltage = current × resistance. Georg Ohm wrote the relation between voltage, current and resistance in mathematical form in a book of 1827. The papers leading up to it date from 1825–26. He is said to have taken Fourier’s theory of how heat spreads as his model. His work was not rated highly at first, but he later received a prize from the Royal Society in London (the Copley Medal, 1841). The unit of resistance, the ohm, is said to have been adopted in the 1870s.

This site treats each battery as 1.5 volts and each bulb as 3 ohms. The site’s program checked the numbers below (circuit-basics/engine.test.ts).

• One battery, one bulb: 1.5 ÷ 3 = 0.5 amps. We call this 100% brightness.
• Two bulbs in series: the resistances add to 6 ohms and the current is 0.25 amps. Each bulb is 25% as bright.
• Two bulbs in parallel: each bulb gets the full battery voltage, so each stays at 100%. The battery supplies 1 amp in total.
• Two batteries in series: the voltage doubles to 3 volts. With one bulb the current is 1 amp and the brightness is 400%.
• With the switch open, there is no current and no light.

In series, resistances add (R = R1 + R2). In parallel, the reciprocals add (1/R = 1/R1 + 1/R2). Both follow from Ohm’s law.

A common explanation compares voltage to water pressure, current to the amount of water flowing, and resistance to how thin the pipe is. That is a way of teaching, not how Ohm found his law.

Seesaw Balance (the site’s “Seesaw Balance”)

A seesaw balances when weight × distance from the pivot is equal on both sides. This value is called a moment. For example, 2 kg at distance 3 on the left gives 2 × 3 = 6, and 3 kg at distance 2 on the right gives 3 × 2 = 6, so they balance. Heavy things go near the pivot, light things far away. This site’s program checked that this condition balances and that the heavier side goes down (lever-balance/engine.test.ts). The test also confirms that the answer to puzzle 3 is 2 kg at distance 3 on the left and 3 kg at distance 2 on the right.

Archimedes of ancient Greece (about 287 to 212 BC) is said to have proved the law of the lever with geometry in his book On the Equilibrium of Planes. Levers themselves had been used for ages. Archimedes did not “discover the lever”; he is said to be the first to prove its law in theory.

The line “Give me a place to stand and I will move the earth” is famous as Archimedes’s. But it is written in a book by Pappus, around the fourth century, more than 500 years after Archimedes died. The familiar English wording also comes from layers of translation. So the careful thing to say is that it is handed down as his saying, not that he spoke exactly those words.

Scissors, nail pullers and seesaws are all levers. They fall into three kinds, depending on how the pivot, the effort and the load are arranged.

Ball Throw Paths (the site’s “Ball Throw Paths”)

Without air resistance, a ball moves sideways at a steady speed while gravity pulls it down, tracing a curve. That curve is a parabola. In the “Fourth Day” of Two New Sciences, published in 1638, Galileo Galilei showed that combining a steady horizontal motion with a falling motion makes the path of a thrown object a parabola. Others had discussed the problem before him, so it cannot be said that Galileo was the first to notice a parabola. “He proved in the Fourth Day that the path is a parabola” is the safe way to say it.

This site’s program checked the following (projectile/engine.test.ts).

• At the same speed, 45 degrees goes the farthest.
• Two angles that add up to 90, like 30 and 60, land at the same spot.
• Thrown at 20 meters per second at 45 degrees, the ball goes about 40.8 meters (20 × 20 ÷ 9.8).
• The distance from the formula agrees within 0.1% with a simulation that follows the motion step by step.

Note that this is an idealization with no air resistance. Real baseballs and soccer balls feel air resistance, which bends their paths and makes the best angle lower than 45 degrees.

Gear Trains (the site’s “Gear Trains”)

Meshed gears pass the same number of teeth past the contact point, so a gear with fewer teeth spins faster. A 24-tooth gear driving an 8-tooth gear makes 24 ÷ 8 = 3 turns. An 8-tooth gear driving a 16-tooth gear makes half a turn. Each link reverses the direction, so two gears turn in opposite directions and three in the same direction. A gear in the middle does not change the output speed; it only sets the direction. This site’s program checked all this (gear-train/engine.test.ts). It also tests that the puzzle “three gears, three times faster, same direction” has the answer 24-8-8.

A famous gear device is the Antikythera mechanism: a bronze gear machine raised around 1900 from a shipwreck off the Greek island of Antikythera. It is said to have been a device for calculating eclipses, the phases of the moon and the calendar. Its date of manufacture is put around the second century BC, but scholars disagree and it is not settled. It is sometimes called the “oldest computer in the world”, but that depends on what you call a computer.

The faster a gear turns, the weaker its push. Speed and force are traded against each other.

What they share

• One rule explains many cases: voltage = current × resistance, weight × distance, the ratio of teeth.
• Know the idealization: decide to ignore air resistance and friction and the rule shows itself. The price is a small gap from reality.

All four are simplified teaching models. A battery’s internal resistance, the resistance of wires, the weight of the seesaw beam and slipping teeth are not considered here.