## Newton's Law of Universal Gravitation

Apples had a significant contribution to the discovery of gravitation. The English physicist Isaac Newton (1642-1727) introduced the term "gravity" after he saw an apple falling onto the ground in his garden. "Gravity" is the force of attraction exerted by the earth on an object. The moon orbits around the earth because of gravity too. Newton later proposed that gravity was just a particular case of gravitation. Every mass in the universe attracts every other mass. This is the main idea of Newton's Law of Universal Gravitation.

The law was published in Newton's famous work, the Principia ("Mathematical Principles of Natural Knowledge") in 1687. It states that every particle in the universe exerts a force on every other particle along the line joining their centers. The magnitude of the force is directly proportional to the product of the masses of the two particles, and inversely proportional to the square of the distances between them.

 In mathematical terms: By team C007571, ThinkQuest2000. where and are the masses of the two particles, r is the distance between the two masses, F is the gravitational force between them, and G is the universal gravitational, .

The above equation only calculates the gravitational force of the simplest case between two particles. What if there are more than two? In that case, we calculate the resultant gravitational force on a particle by finding the vector sum of all the gravitational forces acting on it:

By adding the unit vector to the equation, F now processes a direction!

Newton derived the relation F is proportional to m as the force on a falling body (remember the apple?) is directly proportional to its mass by Newton's 2nd law of Motion (F = ma = mg, so F is proportional to m.) When the earth exerts a force on the falling body, by Newton's 3rd law of Motion, the falling body exerts an equal and opposite force on the earth. Therefore, the gravitational force F is proportional to both the masses of the falling body and the earth, i.e. and . The inverse square relationship, was justified by observing the motion of the moon.

Newton's Law of Universal Gravitation has successfully explained the observation on planetary movements made by the German astronomer Kepler (1571-1630). It works perfectly well in the world of ordinary experience and has dominated for about 250 years. It, however, shows its shortcomings when explaining the unusual orbit of Mercury around the Sun. It breaks down when the gravitational forces get very strong or involving bodies moving at speeds near that of light. Einstein's General Theory of Relativity of 1915, which has overcome the limitations of Newton's Law, was able to demonstrate a better theory of gravitation.

## Gravitational Potential Energy

Let's study an example of an apple to see what's gravitational potential energy. As an apple falling out of a tree, it accelerates towards the earth. In other words, it gains kinetic energy. From the law of conservation of energy, the gain in one form of energy must be accompanied with the loss of energy in another form. We say that the "Gravitational potential energy" of an apple is being converted to its kinetic energy as it falls.

The gravitational potential energy of a mass, m at a distance r from another mass M is defined as the work done by the external force in moving m from r to infinity. (I'm sure you'll have a better idea of what I'm telling by seeing the figure below.)

By Team C007571, ThinkQuest 2000.

In the figure, the gravitational force and the external force are equal in magnitude and opposite in direction. By defining the gravitational potential energy at ground level as zero, we now derive a equation for calculating gravitational potential energy.

The work done by the external force in moving the mass m from r to infinity is given by the equation:

Thus by definition,

## Planck's equation

Science strides through revolutions, but people often refuse to accept revolutionary concepts at first place. This describes the emergence of the Planck's equation. In classical physics, energy of electromagnetic (EM) radiation was thought to be absorbed or emitted continuously. It wasn't until late 1900 the German scientist Max Planck (1858-1947) made a radical assumption in explaining the black body radiation spectrum, the idea of discrete energy arose.

In Planck's assumption, radiant energy is emitted in small bursts, known as "quanta". Each of the bursts called a "quantum" has energy E that depends on the frequency f of the electromagnetic radiation by the equation:

where h is a fundamental constant of nature, the "Planck constant".

This equation is later found to be true for all EM radiant energy emitted or absorbed.

Planck's equation implies the higher the frequency of a radiation, the more energetic are its quanta. It for example explains why you can never get brown from visible light (( to ), but from ultraviolet light (from to ). The quanta of visible light don't carry enough energy to start the chemical reaction in your skin!

Figure: Visible Spectrum. Courtesy of NASA.

The quantum energy is not to compare with the power of the light! The Power of light (Luminosity) is the total energy per second, that means the number of quanta per second times the quantum energy. Therefore even if visible light carrys a lot more Energy per second than UV-light, you won't get any browner from it.

The theoretical black body radiation spectra predicted by Planck's Radiation Law, with the assumption , agreed with the experimentally found spectra in all wavelengths and temperatures.

Many other scientists, including Wien, Rayleigh and Jean, attempted to explain the blackbody radiation spectra using classical wave theory and failed. However, the idea of quantized energy was too revolutionary for most scientists at the time (Even Planck puzzled his own conclusion). It was not generally accepted until 1905 when Einstein extended Planck's equation in deriving his formula for photoelectric emission. The idea of quantized energy led Einstein to postulate the particle-wave duality of light and other EM radiation. Planck's equation is essential to the formulation of quantum physics.

Before we discuss the term "black body radiation", let's first define what's a "black body". A "black body" is a theoretical perfect absorber, which absorbs radiation of all wavelengths falling on it. It reflects no light at normal temperatures and thus appears black. However, like ideal gas in kinetic theory, it is a theoretical model and we may find in reality only "Almost perfect black bodies".

It follows from Prévost's theory of exchanges of 1792 that the best radiation absorber - the black body, is also the best radiation emitter. The radiation emitted by a black body is called, you guess?... Black body radiation. (Straight forward isn't it?) It is also known as "full radiation" or "temperature radiation".

The intensities of the various wavelengths of radiation emitted by a black body depend only on its temperature. We may study the black body radiation spectrum with a suitable spectrometer. A thermopile or a bolometer can be used as the black body can be used as the black body radiation emitted usually consists of infra red, light and ultra violet which all produce a heating effect.

A black body radiation spectrum is shown below:

We may observe that the higher the temperature of a black body, the more energy is emitted in each band of wavelengths. The black body becomes "brighter". Moreover, the radiation emitted at the highest intensity, represented by the peak of the spectrum, doesn't fall in the visible region unless the temperature is very high, over 3700K.

 , whereis the wavelength whereat which the energy radiated is is a max.maximum and T is the temperature in Kelvin. This is stated in Wien's displacement law.

Therefore as implied by the formula, a hotter black body emits typical radiation with shorter wavelengths. This explains why black bodies at higher temperatures are blue, and those at lower temperatures are red.

## Einstein's Formula

Mass is a form of energy? In 1905, Albert Einstein (1873-1955) suggested that mass and energy are equivalent while developing his special theory of relativity. The famous mass-energy equivalence relation states that

 where E is the energy equivalent or mass energy, m is the mass of a body and is the speed of light.

It is difficult to show a simple logical path through which Einstein came to his equation. It was merely an hypothesis made by him from special relativity and Maxwell's equations.

Energy gain/loss in everyday examples can, however, hardly show any noticeable change in mass. This is because the total mass of an object changes only by a tiny fraction. To check Einstein's equation, we need something with tiny mass, so that an appreciable change in total mass can be measured. Radioactive decay, a nuclear reaction, is a choice. In fact, Einstein tested his own equation with a lump of radium salt to see if it lost weight as it gave off radiation.

Today, the mass-energy equivalence relation has an important implication in nuclear industry.

``````From: "Black holes aren't black - After Hawking they shine!"
/C007571
Presented by Angie, Matthias and Thorsten
Team C007571, ThinkQuest Internet Challenge 2000 (http://www.thinkquest.org).