Fun With Formulas

Mathematical formulas and the interesting stories behind them

Period of a Pendulum

This formula allows the period of a pendulum to be calculated.

The motion of the pendulum was studied and documented by Ibn Yunus al-Masri during the 10th century. It was re-descovered by Galileo during the 17th century. The story goes that one day while attending Sunday mass in Pisa Cathedral, he dozed off and noticed the chandelier above him swinging to and fro. By using his pulse as a clock he noticed that the period of the swing was independent of how far it swung. Only the length of the pendulum made any difference to the time required for a swing.

This eventually led to the invention of the pendulum clock, the most accurate time piece yet invented. Accurate clocks lead to Europeans being able to navigate accurately around the world.

Distance under Gravity

This formula shows how far a body will fall under gravity if air resistance is ignored.

Galileo also discovered this formula.

The ancients believed that if two object were dropped, the heavier one would fall faster (and hence travel further in a given time). By dropping cannon balls from the leaning tower at Pisa, Galileo showed that all objects will travel the same distance and speed regardless of the mass / weight of the body.

Force of Gravity

This formula calculates the force of gravity between two bodies.

This formula was discovered by Isaac Newton. It applies to the attraction between the Earth and an apple falling from a tree, the Earth and the Moon, or the Sun and a planet. It is the basis of celestial mechanics and can be used to calculate the path of a probe to a distant planet or the motion of a comet.

Escape Velocity

This formula calculates the escape velocity from a planet or star.

This formula was also discovered by Isaac Newton. It allows the calculation of Escape Velocity from a planet or star. If a body is moving with this velocity it will escape from the gravitational pull of the planet or star.

Acceleration of Gravity for Star / Planet

This formula calculates the acceleration of gravity on the surface of a planet or star.

Yet another formula derived by Isaac Newton. By entering the mass and radius of a planet or star it is possible to calculate the acceleration of gravity on the surface.

Kinetic Theory of Matter

This formula calculates average velocity of a molecule, atom or ion in a gas at a given temperature.

This formula was discovered by Ludwig Boltzmann. It assumes that gases are made of particles. These particles will have an average velocity dependent on the masses of the particles and the temperature.

For a planet, this formula can be combined with the formula for Escape Velocity. If the average velocity of particles is greater than the planet's escape velocity, then the planet will slowly lose its atmosphere to space.

Stellar Luminosity

This formula calculates the luminosity of a star.

This formula arises from Stefan's Law and allows the luminosity of a star to be calculated if its radius and temperature is known. Once the luminosity is known, it can be compared with the brightness as seen from Earth. This will then give the star's distance.

Time Dilation

This formula calculates Einstein's famous time dilation effect for a moving body.

According to Albert Einstein, when a body is in motion its time slows down. This formula allows the time (as measured by the moving body) to be compared with the rest time. For low velocities the effect is negligible. It is only when the body moves at a velocity comparable to that of light that these effects become noticeable.

Matter Waves

This formula calculates the wavelength of a moving body.

Louis De Broglie came out with the extraordinary idea that moving matter could behave as waves. The wavelength of the body can be calculated from this formula. Only very small bodies will give a measurable effect.

Radiation Law

This formula describes the radiation profile for a body at a particular temperature.

This formula was discovered by Max Planck. A body that is glowing at a given temperature will radiate in a particular way depending on the temperature. For example, heated bodies glow red hot then white hot. This formula describes the exact way that the body will radiate.

The body is assumed to be a perfect absorber and radiator of energy, the so-called black body.

© 1999, 2009 KryssTal

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All the above formulas assume that units are in the Metric System. This reference defines all the units used.

This table lists the values of the constants used in the above formulas.