Science Tools

Thirty-nine calculators, and every one of them says what it rests on

Electronics first, because it is where the arithmetic is shortest and the mistakes are most expensive: Ohm's law solved whichever way round you have it, a resistor's colours read in four bands or five and the value spelt out, the series resistor a diode wants with the nearest one you can actually buy, a divider, a capacitor's three-digit code, an RC corner, a bill from a list of appliances and how long a battery lasts. Then the wiring: voltage drop down a real cable, a transformer's turns, decibels either way round, an antenna cut to a band, and a wavelength from a frequency.

Then the physics and the chemistry — speed, acceleration, a thrown ball, a dropped one, work and energy; density, the pressure at a depth, the ideal gas law, molarity, a dilution and pH; a half-life table, a crystal's unit cell, Beer-Lambert, and uncertainties combined in quadrature. Then the workshop: torque against power and rpm, a gear train, a belt, a loaded beam and the drill a tap wants. Last the weather — wind chill, heat index, dew point, the pressure at altitude, the Beaufort scale and what a roof collects when it rains.

None of them rounds a limitation into a feature. A fitted formula is refused outside the range it was fitted in rather than extrapolated quietly; a result that is arithmetic rather than a design says which authority decides what may actually be built; and every figure carries the formula it came from, so you can check it against the book rather than against this page.

Circuits

Ohm's law
Voltage, current, resistance and power: any two of them give the other two

Resistor colour code
The value from the bands, with the band count asked rather than guessed

LED resistor
The resistor a diode needs, the heat it makes, and the nearest part somebody sells

Voltage divider
The output from two resistors, the current it costs, and the nearest real pair

Capacitor code
Three digits into a capacitance and back, with the tolerance letter read too

RC filter
The corner frequency of a resistor and a capacitor, with the response either side of it

Power draw
What a set of parts draws, the supply it wants, and what a year of it costs

Battery life
How long a cell lasts at a draw, and how long it takes to fill again

Wiring and waves

Voltage drop
What a run of cable loses, and whether that is inside the limit somebody set

Transformer ratio
The turns, the voltages and the currents, with the losses named rather than modelled

Decibels
A ratio as decibels and back, with ten for a power and twenty for an amplitude

Antenna length
The wavelength a frequency has, and the four lengths cut from it

Frequency and wavelength
Either half of the pair, the photon's energy, and which part of the spectrum it is

Motion

Speed, distance, time
Any two of the three, and the same speed in every unit anybody asks for

Acceleration
What a steady acceleration does to speed and distance, step by step

Projectile motion
Where a throw lands and how high it goes, in a vacuum and saying so

Free fall
How long a drop takes and how fast it arrives, with terminal velocity named

Work and energy
Whichever of work, power, kinetic and potential energy the numbers given can make

Matter

Density
Mass, volume and density, and how the answer compares with fifteen everyday things

Pressure at depth
What a column of fluid weighs on whatever is under it, in every unit that matters

Ideal gas law
Any three of pressure, volume, amount and temperature give the fourth, in kelvin

Molarity
How much to weigh out, how much to make up to, and what the concentration comes to

Dilution
What to take of a stock solution and what to add, from C₁V₁ = C₂V₂

pH
The pH of a strong or a weak solution, solved by the quadratic rather than approximated

In the laboratory

Half-life
What is left after a time, or how long since there was more, with the mean life too

Crystal unit cell
The density a lattice gives, with the atoms per cell and the packing that follow from it

Beer–Lambert law
Absorbance, concentration or path length, with the linear range named

Error propagation
What the uncertainties of the inputs do to the answer, and which one dominates

Machines

Torque, power, rpm
Any two of the three, with both horsepowers named because they are different numbers

Gear ratio
What a train of gears does to speed and torque, stage by stage

Belt and chain length
How long a belt has to be for two pulleys at a distance, open or crossed

Beam load
Reactions, bending moment and deflection for the three standard cases

Tap drill size
The hole to drill before cutting a thread, at whatever engagement you want

Weather and air

Wind chill
What the wind does to cold skin, inside the range the index was defined for

Heat index
What the humidity does to heat, with the risk band the weather service publishes

Dew point
Where the air lets go of its water, and whether that is above the cold surface

Altitude and pressure
The pressure, the temperature, the air's density and where water boils, up to the troposphere

Beaufort scale
A wind speed as a force, and what that force does to the sea and the trees

Rainwater harvest
How much water a roof catches in a year, from the one fact that a millimetre is a litre

Questions

Why does a calculator show the formula it used?: Because there is usually more than one, and the one a page used is the only thing that makes its answer checkable. Two of these are named rather than written out — the heat index is Rothfusz's 1990 regression and the dew point is Magnus-Tetens — for the same reason: you can look them up and find the same numbers.
Why is there no periodic table or component database?: Because a table of every element's molar mass, every transistor's pinout and every cable's rating is larger than the whole application, and a short one is worse than none — it says "not found" about things that exist. The tools take the figure from you and say which figure they took.
What about the constants?: The defined ones are exact: standard gravity is 9.806 65 m/s² by definition, the speed of light 299 792 458 m/s, standard atmosphere 101 325 Pa. The Earth's *real* gravity runs from 9.78 at the equator to 9.83 at the poles, and the tool that uses it says so rather than implying the defined figure is a measurement.
Which tools from the group are missing?: The ones that are a running interface rather than a function from one text to one result — an oscilloscope trace, a live logger, a circuit drawn on a canvas. This is where they are named rather than left for you to look for.

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