:: wikimiki.org ::

Two-body Problem

Two-body problem

In mechanics, the two-body problem is a special case of the n-body problem that admits a closed form solution. The most commonly encountered version of the problem, involving an inverse square law force, is encountered in celestial mechanics and the Bohr model of the hydrogen atom. This problem was first solved by Isaac Newton. This article deals with the general case where it is not assumed that one body has a much smaller mass than the other one.

Statement of problem

We restrict ourselves to the classical case, with forces that depend only on the positions of the bodies and obey the strong form of Newton's third law. Letting

\mathbf_

and

\mathbf_

be the positions of the two bodies, and

m_

and

m_

be their masses, we have (from Newton's second law): :

\mathbf_(\mathbf_,\mathbf_) = m_ \ddot\mathbf_

\mathbf_(\mathbf_,\mathbf_) = m_ \ddot\mathbf_

Note that these two vector equations comprise six scalar differential equations, each of second order. Therefore, we will need twelve (

6 \times 2

) constants of integration.

Sketch of solution

We start by taking advantage of Newton's third law to reduce the two-body problem to two equivalent one-body problems, one for the center of mass of the system, and one for the relative motion of the two bodies. We can identify linear combinations of the dependent variables to decouple the equations. Adding the differential equations, we get :

m_\ddot\mathbf_ + m_\ddot\mathbf_ = (m_ + m_)\ddot\mathbf_ = 0

where :

\mathbf_ \equiv \frac

is the position of the center of mass (barycenter) of the system. Integration shows:
- the total momentum is constant (conservation of momentum)
- the center of mass remains at rest, or moves in a straight line at a constant velocity (see also Motion of the center of mass); this provides six of the constants of integration. Next, we notice that because of conservation of angular momentum, the equivalent one-body problem is really a two dimensional problem. This provides two more constants. At this point it is convenient to switch to polar coordinates. This is as far as we can go for the general problem. We focus on the inverse square law force, as the most important case of the two-body problem.

Reduction to a single body problem

Using the strong form of Newton's third law, as well as the fact that the magnitude of the force depends only on the distance between the bodies, we have that :

\mathbf_(\mathbf_,\mathbf_)  = - \mathbf_(\mathbf_,\mathbf_) = -F(|\mathbf_ - \mathbf_|) \frac

We now multiply the first equation by

\frac

, the second by

\frac

, and subtract, giving :

\ddot\mathbf_ - \ddot\mathbf_ = -(\frac + \frac)F(|\mathbf_ - \mathbf_|) \frac

or :

\mu \ddot\mathbf = -F(|\mathbf|)\frac

where :

\mathbf \equiv \mathbf_ - \mathbf_

and :

\mu \equiv \frac = \frac

is the reduced mass of the system. The positions of the bodies are

\frac\mathbf

and

-\frac\mathbf

, respectively. Thus we have reduced the problem to a one-body problem.

Reduction to two dimensions

Starting with the one-body differential equation above, we take the cross product with the linear momentum :

\mathbf = \mu \dot\mathbf

to get :

\mu \ddot\mathbf \times \mathbf = -\frac(\mathbf \times \mathbf)

But the first term on the right is a scalar, and the second term is the angular momentum :

\mathbf = \mathbf \times \mathbf

which, by conservation of angular momentum is a constant. :

\ddot\mathbf \times \mathbf

always points in the same direction, which means that the plane determined by

\ddot\mathbf

and

\dot\mathbf

is always the same, and the motion is therefore in a plane. This provides two more constants of integration.

Change of variables

Having reduced the problem to two dimensions, at this point it is convenient to switch to polar coordinates. In polar coordinates, the vector differential equation reduces to a scalar equation, due to the fact that the force, and therefore the acceleration, is always toward the origin. It can be shown that r-component of acceleration is :

\ddot - r \dot^

Therefore, we have :

\mu(\ddot - r \dot^) = F(r)

Another change of variables is useful: let

u \equiv \frac

Newtonian Gravity

Applying the gravitational formula we get that the position of the first body with respect to the second is governed by the same differential equation as the position of a very small body orbiting a body with a mass equal to the sum of the two masses, because m1.m2/μ=m1+m2. Assume:
- the vector r is the position of one body relative to the other (above called x)
- r, v, the semi-major axis a, and the specific relative angular momentum h are defined accordingly (hence r is the distance)
-

\mu=(m_1+m_2)\,

the standard gravitational parameter (the sum of those for each mass) where:
-

m_1

and

m_2

are the masses of the two bodies. Then:
- the orbit equation applies; recalling that the positions of the bodies are m2/(m1+m2) and -m1/(m1+m2) times r, respectively, we see that the two bodies' orbits are similar conic sections; the same ratios apply for the velocities, and, without the minus, for the angular momentum with respect to the barycenter and for the kinetic energies
- for circular orbits

rv^2 = r^3 \omega^2 = 4 \pi^2 r^3/T^2 = \mu

- for elliptic orbits:

4 \pi^2 a^3/T^2 = \mu

(with a expressed in AU and T in years, and with M the total mass relative to that of the Sun, we get

a^3/T^2 = M

)
- for parabolic trajectories

r v^2

is constant and equal to

2 \mu

- h is the total angular momentum divided by the reduced mass
- the specific orbital energy formulas apply, with specific potential and kinetic energy and their sum taken as the totals for the system, divided by the reduced mass; the kinetic energy of the smaller body is larger; the potential energy of the whole system is equal to the potential energy of one body with respect to the other, i.e. minus the energy needed to escape the other if the other is kept in a fixed position; this should not be confused with the smaller amount of energy one body needs to escape, if the other body moves away also, in the opposite direction: in that case the total energy the two need to escape each other is the same as the aforementioned amount; the conservation of energy for each mass means that an increase of kinetic energy is accompanied by a decrease of potential energy, which is for each mass the inner product of the force and the change in position relative to the barycenter, not relative to the other mass
- for elliptic and hyperbolic orbits

\mu

is twice the semi-major axis times the absolute value of the specific orbital energy For example, consider two bodies like the Sun orbiting each other:
- the reduced mass is one half of the mass of one Sun (one quarter of the total mass)
- at a distance of 1 AU: the orbital period is

\sqrt

year, the same as the orbital period of the Earth would be if the Sun would have twice its actual mass; the total energy per kg reduced mass (90 MJ/kg) is twice that of the Earth-Sun system (45 MJ/kg); the total energy per kg total mass (22.5 MJ/kg) is one half of the total energy per kg Earth mass in the Earth-Sun system (45 MJ/kg)
- at a distance of 2 AU (each following an orbit like that of the Earth around the Sun): the orbital period is 2 years, the same as the orbital period of the Earth would be if the Sun would have one quarter of its actual mass
- at a distance of

\sqrt[3] \approx 1.26

AU: the orbital period is 1 year, the same as the orbital period of the Earth around the Sun Similarly, a second Earth at a distance from the Earth equal to

\sqrt[3]

times the usual distance of geosynchronous orbits would be geosynchronous.

General Relativistic Gravity

In the general theory of relativity gravity behaves somewhat differently, but, to a first approximation for weak fields, the effect is to slightly strengthen the gravity force at small separations. Kepler's First Law is modified so that the orbit is a precessing ellipse, its major and minor axes rotating slowly in the same sense as the oribital motion. The law of conservation of angular momentum still applies (Kepler's Second Law). Kepler's Third Law would in principle be altered slightly, but in practice, the only way to measure the sum of the masses is by applying that Law as it stands, so there is effectively no change. These results were first obtained approximately by Einstein, and the rigorous two body problem was later solved by Howard Percy Robertson.

Examples

- a binary star, e.g. Alpha Centauri (approx. the same mass)
- a double planet, e.g. Pluto with its moon Charon (mass ratio 0.147)
- a binary asteroid, e.g. 90 Antiope (approx. the same mass)

Other uses

The phrase two body problem is also used jokingly by scientists to refer to the difficulty of married graduate students or postdocs finding jobs at the same university.

Mechanics

Mechanics refers to: #a craft relating to machinery (from the Latin mechanicus, from the Greek mechanikos, meaning "one skilled in machines"), or #a range of disciplines in science and engineering.

Mechanics in science and engineering

Mechanics can be seen as the prime, and even as the original, discipline of physics. It is a huge body of knowledge about the natural world. It also constitutes a central part of technology. That is, how to apply this knowledge for humanly defined purposes. Briefly stated, mechanics is concerned with the motion of physical bodies, and with the forces that cause, or limit, these motions, as well as with forces which such bodies may, in turn, give rise to. Due to the wide scope of the subject, one may well find topics that would not fit easily into even this general characterization. Thus the term "body“ needs to stand for a wide assortment of objects, including particles, projectiles, spacecraft, stars, parts of machinery, parts of solids, parts of fluids (gases and liquids), etc. The major division of the mechanics discipline separates classical mechanics from quantum mechanics. Historically, classical mechanics came first, while quantum mechanics is a comparatively recent invention. Classical mechanics is older than written history, while quantum mechanics didn't appear until the year 1900. Both are commonly held to constitute the most certain knowledge that exists about physical nature. Especially classical mechanics has therefore often been viewed as a model for other so-called exact sciences. Essential in this respect is the relentless use of mathematics in theories, as well as the decisive role played by experiment in generating and testing them. Quantum mechanics is, formally at least, of the widest scope, and can be seen as encompassing classical mechanics, as a sub-discipline which applies under certain restricted circumstances. If properly interpreted, there is no contradiction, or conflict between the two subjects, each simply pertains to specific situations. While it is true that, historically, quantum mechanics has been seen as having superseded classical mechanics, this is only true on the abstract, or fundamental, level. In practice, classical mechanics remains as useful as ever. In a somewhat analogous way, Einsteinian relativity has expanded the scope of mechanics. This is true for classical as well as quantum mechanics. Again, there are no contradictions, or conflicts, so long as the specific circumstances are carefully kept in mind. Just as one could, in the loosest possible sense, characterize classical mechanics as dealing with "large" bodies (such as engine parts), and quantum mechanics with "small" ones (such as particles), it could be said that relativistic mechanics deals with "fast" bodies, and non-relativistic mechanics with "slow" ones. However, "fast" and "slow" are relative concepts, depending on the state of motion of the observer. This means that all mechanics, whether classical or quantum, potentially needs to be described relativistically. On the other hand, as an observer, one may frequently arrange the situation in such a way that this is not really required. Other distinctions between the various sub-disciplines of mechanics, concern the nature of the bodies being described. Particles are bodies with little (known) internal structure, treated as mathematical points in classical mechanics. Rigid bodies have extension, but retain a simplicity close to that of the particle, adding just a few so-called degrees of freedom, such as orientation in space. Otherwise, bodies may be semi-rigid, i.e. elastic, or non-rigid, i.e. fluid. These subjects have both classical and quantum divisions of study. For instance, the motion of a spacecraft is described by classical mechanics, regarding its orbit and attitude (i.e. by rotation with respect to the fixed stars). While an atomic nucleus is described by quantum mechanics in analogous situations.

Sub-disciplines in mechanics

- Classical mechanics
- Newtonian mechanics, the original theory of motion (kinematics) and forces (dynamics)
- Lagrangian mechanics, a theoretical formalism
- Hamiltonian mechanics, another theoretical formalism
- Celestial mechanics, the motion of stars, galaxies, etc.
- Astrodynamics, spacecraft navigation, etc.
- Solid mechanics, Elasticity, the properties of (semi-)rigid bodies
- Acoustics, sound in solids, fluids, etc.
- Statics, semi-rigid bodies in equilibrium
- Fluid mechanics, the motion of fluids
- Continuum mechanics, mechanics of continua (both solid and fluid)
- Hydraulics, fluids in equilibrium
- Biomechanics, solids, fluids, etc. in biology
- Statistical mechanics, large assemblies of particles
- Relativistic or Einsteinian mechanics, universal gravitation
- Quantum mechanics
- Particle physics, the motion, structure, and reactions of particles
- Nuclear physics, the motion, structure, and reactions of nuclei
- Condensed matter physics, quantum gases, solids, liquids, etc.
- Quantum statistical mechanics, large assemblies of particles Formally, "fields" constitute a separate discipline in physics, distinct from mechanics, whether classical fields or quantum fields. In actual practice, however, subjects belonging to mechanics and fields are closely interwoven. Thus, for instance, forces that act on particles are frequently derived from fields (electromagnetic or gravitational), and particles generate fields by acting as sources. In fact, in quantum mechanics, particles themselves are fields, as described theoretically by the wave function. ko:역학 ja:力学

N-body problem

The letter n is mathematical notation and should be italicized and set in lower-case. The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics, i.e. Newton's laws of motion and Newton's law of gravity.

Mathematical formulation

The general n-body problem can be stated in the following way. For each body i, with mass m_i, let c_i(t) be its trajectory in 3-dimensional space, where the parameter t is interpreted as time. Then the acceleration c'(t) of each mass mi satisfies by the law of gravity: : $c_$ (t) = \gamma \sum_ m_j \frac The solutions of this system of differential equations give the positions as a function of time. The force on each mass m_i is :

F_i = c_

(t) m_i
Two-body problem
Main article: two-body problem If the common center of mass of the two bodies is considered to be at rest, each body travels along a conic section which has a focus at the centre of mass of the system (in the case of a hyperbola: the branch at the side of that focus). If the two bodies are bound together, they will both trace out ellipses; the potential energy relative to being far apart (always a negative value) has an absolute value less than the total kinetic energy of the system; the sum of both energies is negative. (Energy of rotation of the bodies about their axes is not counted here). If they are moving apart, they will both follow parabolas or hyperbolas. In the case of a hyperbola, the potential energy has an absolute value smaller than the total kinetic energy of the system; the sum of both energies is positive. In the case of a parabola, the sum of both energies is zero. The velocities tend to zero when the bodies get far apart. Note: The fact that a parabolic orbit has zero energy arises from the assumption that the gravitational potential energy goes to zero as the bodies get infinitely far apart. One could assign any value (e.g. 23 joules) to the potential energy in the state of infinite separation. That state is assumed to have zero potential energy (i.e. 0 joules) by convention. See also Kepler's first law of planetary motion.
Three-body problem
The three-body problem is much more complicated; its solution can be chaotic. In general, the three-body problem cannot be solved analytically (i.e. in terms of a closed-form solution of known constants and elementary functions), although approximate solutions can be calculated by numerical methods or perturbation methods. The restricted three-body problem assumes that the mass of one of the bodies is negligible; the circular restricted three-body problem is the special case in which two of the bodies are in circular orbits (approximated by the Sun - Earth - Moon system). For a discussion of the case where the negligible body is a satellite of the body of lesser mass, see Hill sphere; for binary systems, see Roche lobe; for another stable system, see Lagrangian point. The restricted problem (both circular and elliptical) was worked on extensively by many famous mathematicians and physicists, notably Lagrange in the 18th century and Henri Poincaré in at the end of the 19th century. Poincaré's work on the restricted three-body problem was the foundation of deterministic chaos theory. In the circular problem, there exist five equilibrium points. Three are collinear with the masses (in the rotating frame) and are unstable. The remaining two are located 60 degrees ahead of and behind the smaller mass (e.g., Jupiter) in its orbit about the larger mass (e.g., Sun), forming two equilateral triangles. For sufficiently small mass ratio of the primaries, these triangular equilibrium points are stable, such that (nearly) massless particles will orbit about these points as they orbit around the larger primary (Sun). The five equilibrium points of the circular problem are known as the Lagrange points. In 1912, Finland-Swedish mathematician Karl Fritiof Sundman developed a convergent infinite series that provides a solution to the restricted three-body problem. Unfortunately, getting the value to any useful precision requires so many terms (on the order of 10^8,000,000) that his solution is of little practical use.
- Trivia: The three-body problem is figured prominently in the Criminal Minds television series episode "Compulsion."
See also

- Many-body problem
- Euler's three-body problem
- Virial theorem
- Few-body systems
External links

- [http://www.geom.umn.edu/~megraw/CR3BP_html/cr3bp.html More detailed information on the three-body problem]
- [http://www.ifmo.ru/butikov/ManyBody.pdf Regular Keplerian motions in classical many-body systems]
- [http://www.ifmo.ru/butikov/Projects/Collection.html Applets demonstrating many different three-body motions] Category:Celestial mechanics ja:多体問題

Closed form

In mathematics, closed form can mean:
- a finitary expression, rather than one involving (for example) an infinite series, or use of recursion - this meaning usually occurs in a phrase like solution in closed form and one also says closed formula;
- a closed differential form: see closed and exact differential forms. Category:Mathematical disambiguation

Force

:For other senses of this word, see force (disambiguation). In physics, a force is an external cause responsible for any change of a physical system. For instance, a person holding a dog by a rope is experiencing the force applied by the rope on their hand, and the cause for its pulling forward is the force exercised by the rope. The kinetic expression of this change is, according to Newton's second law, acceleration, non kinetic expressions such as deformation can also occur. The SI unit for force is the newton.

Elementary concepts

Force in its most primitive definition can be thought of as that which when acting alone causes an object to accelerate. In a practical sense forces can be divided into two groups: contact forces and field forces. Contact forces require the physical contact of one object with other such as a hammer striking a nail or the force exerted by a gas under pressure - gas produced by exploding gunpowder forces a heavy ball out of a cannon. Field forces on the other hand need no physical medium of contact. Gravity and magnetism are examples of such forces. It should be noted however, that fundamentally all forces are in fact field forces. The force of hammer striking the nail in the previous example turns out to be a clash of the electric forces in both hammer and nail. Nevertheless it is appropriate in some cases to maintain these two classifications for ease of understanding.

Quantitative definition

In physics models, the point-like system is used, where objects are represented as one-dimensional points at their centre of mass. The only change the system can experience is a change of its momentum (its speed). Since the rise of the atomic theory, any physical system has been considered in classical physics as composed of point-like systems called atoms or molecules. Therefore, all forces can be defined by their effect; that is, by the change of movement they induce on point-like systems. This change of movement can be quantified by the acceleration (the derivative of velocity). The discovery by Isaac Newton that a given force will induce an acceleration in inverse proportion to a quantity called the mass of inertia or inertial mass which is independent of the speed of the system is Newton's second law. This law allows us to predict the effect of a force on any point-like system whose mass is known. It is usually written as: :F = dp/dt = d(m·v)/dt = m·a (in the case where m does not depend on t) where :F is the force (a vector quantity), :p is the momentum, :t is the time, :v is the velocity, :m is the mass, and :a=d²x/dt² is the acceleration, the second derivative with respect to t of the position vector x. If the mass m is measured in kilograms and the acceleration a is measured in metres per second squared, then the unit of force is kilogram × metre/second squared. This unit is called the newton: 1 N = 1 kg x 1 m/s². This equation is a system of three second-order differential equations with respect to the three-dimensional position vector which is an unknown function of time. This equation can be solved if F is a known function of x and some of its derivatives and if the mass m is known. Morevover the boundary conditions are required; for example, the values of the position vector and x and the velocity v at the starting time, say t=0. Of course, this formula is only useful if one knows the numerical values of F and m. The definition above is an implicit definition, arrived at as follows. One defines a reference system (one litre of water) and a reference force (the gravitational force applied by the Earth on it at the altitude of Paris). One takes Newton's second law for granted (one postulates that it is true) and measures the acceleration induced by the reference force on the reference system. This gives us a mass unit (1 kg) and a force unit (the older unit of 1 kilogram-force = 9.81 N). Once this is done, one can measure any force by the acceleration it induces on the reference system and measure the inertial mass of any system by measuring the acceleration induced on this system by the reference force. Force is often considered a fundamental quantity in physics, but there are more fundamental quantities, such as momentum (p = mass m x velocity v). Energy, measured in joules, is still less fundamental than force and momentum, because it is defined as work, and work is defined in terms of force. The two most fundamental theories of nature - quantum electrodynamics and general relativity - do not contain the concept of force at all. Although not the most fundamental quantity in physics, force is an important basic mathematical concept from which other concepts, such work and pressure (measured in pascals), are derived. Force is sometimes confused with stress.

Types of force

There are four known fundamental forces in nature.
- Nuclear forces acting between subatomic particles
- Electromagnetic forces between electric charges
- Weak forces arising from radioactive decay
- Gravitational forces between masses Quantum field theory accurately models the first three fundamental forces, but does not model quantum gravity. Quantum gravity on a large scale can, however, be described by general relativity. The four fundamental forces describe every observable phenomenon including the many other forces observed such as: Coulomb's force (the force between electrical charges), gravitational force (force between masses), magnetic force, frictional forces, centripetal, centrifugal, impact force, and spring force, to name a few. Forces can also be classified into conservative forces and nonconservative forces. Conservative forces are equivalent to the gradient of a potential, and include gravity, electromagnetic force, and spring force. Nonconservative forces include friction and drag.

Properties of force

Because momentum is a vector, then force, being its time derivative, is also a vector - it has magnitude and direction. Forces can be added together using the parallelogram of force. When two forces act on an object, the resulting force, the resultant, is the vector sum of the original forces. This is called the principle of superposition. The magnitude of the resultant varies from zero to the sum of the magnitudes of the two forces, depending on the angle between their lines of action. If the two forces are equal, but opposite, the resultant is zero. This condition is called static equilibrium, with the result that the object remains at rest or moves with a constant velocity. As well as being added, forces can be can also be broken down (or 'resolved'). For example, a horizontal force pointing northeast can be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Force vectors can also be three-dimensional, with the third (vertical) component at right-angles to the two horizontal components.

Forces in theory

The total (Newtonian) force, in newtons, on an object at any given time is defined as the rate of change of the object's velocity multiplied by the object's mass: :

\mathbf = \lim_ \frac

where :m is the inertial mass of the particle (measured in kilograms) :v_o is its initial velocity (measured in metres per second) :v is its final velocity (measured in metres per second) :T is the time from the initial state to the final state (measured in seconds); :Lim T→0 is the limit as T tends towards zero. Force was so defined to explain the effects of superimposing situations: if in one situation, a force is experienced by a particle, and if in another situation another force is experienced by that particle, then in a third situation, which (according to standard physical practice) is taken to be a combination of the two individual situations, the force experienced by the particle will be the vector sum of the individual forces experienced in the first two situations. This superposition of forces, and the definition of inertial frames and inertial mass, are the empirical content of Newton's laws of motion. There are other ways to look at the above definition of force. First, the mass of a body multiplied by its velocity is called its momentum, p, so the above definition is equivalent to: :

\textbf=

If F is not constant over Δt, then this is the definition of average force over the time interval. To apply it at an instant we apply an idea from calculus. If we graph p as a function of time, the average force will be the slope of the line connecting the momentum at two times. Taking the limit as the two times get closer together gives the slope at an instant, which is called the derivative: :

\textbf=

Many forces are associated with a potential energy field. For instance, the gravitational force acting upon a body can be seen as the action of the gravitational field that is present at the body's location. The potential field is defined as that field whose gradient is equal and opposite to the force produced at every point: :

\textbf=-\nabla U

The derivative of force with respect to time is called yank. Higher order derivatives are sometimes used, but they lack names because of their rarity. In most explanations of mechanics, force is usually defined only implicitly, in terms of the equations that work with it. Some physicists, philosophers and mathematicians, such as Ernst Mach, Clifford Truesdell and Walter Noll, have found this problematic and sought a more explicit definition of force. According to the Special theory of relativity the mass of an object increases as it's velocity and therefore it's energy increases. The law of force must then be modified to the following: :

\textbf=

where :v is the mass's velocity :c is the speed of light. Note that the equation is undefined if the mass's speed is equal to c because one then has to divide by zero. This is one reason most physicists believe an object with nonzero rest mass can not be accelerated to the speed of light, as this would require an infinite force.

Units of measurement

The SI unit used to measure force is the newton (symbol N), which is equivalent to kg·m·s⁻².

Non-SI units of force and mass

The F=m·a relationship can be used with any consistent units (SI or CGS). If these units are not consistent, a more general form, F=k·m·a, can be used, where the constant k is a conversion factor dependent upon the units being used. For example, in imperial engineering units, F is measured in "pounds force" or "lbf", m in "pounds mass" or "lb", and a in feet per second squared. In this particular system, one needs to use the more general form above, usually written F=m·a/g_c with the constant normally used for this purpose g_c = 32.174 lb·ft/(lbf·s²) equal to the reciprocal of the k above. As with the kilogram, the pound is colloquially used as both a unit of mass and a unit of force. 1 lbf is the force required to accelerate 1 lb at 32.174 ft per second squared, since 32.174 ft per second squared is the standard acceleration due to terrestrial gravity. Another imperial unit of mass is the slug, defined as 32.174 lb. It is the mass that accelerates by one foot per second squared when a force of one lbf is exerted on it. When the standard gee (an acceleration of 9.80665 m/s²) is used to define pounds force, the mass in pounds is numerically equal to the weight in pounds force. However, even at sea level on Earth, the actual acceleration of free fall is quite variable, over 0.53% more at the poles than at the equator. Thus, a mass of 1.0000 lb at sea level at the equator exerts a force due to gravity of 0.9973 lbf, whereas a mass of 1.000 lb at sea level at the poles exerts a force due to gravity of 1.0026 lbf. The normal average sea level acceleration on Earth (World Gravity Formula 1980) is 9.79764 m/s², so on average at sea level on Earth, 1.0000 lb will exerts a force of 0.9991 lbf. The equivalence 1 lb = 0.453 592 37 kg is always true, by definition, anywhere in the universe. If you use the standard gee which is official for defining kilograms force to define pounds force as well, then the same relationship will hold between pounds-force and kilograms-force (an old non-SI unit is still used). If a different value is used to define pounds force, then the relationship to kilograms force will be slightly different—but in any case, that relationship is also a constant anywhere in the universe. What is not constant throughout the universe is the amount of force in terms of pounds-force (or any other force units) which 1 lb will exert due to gravity. By analogy with the slug, there is a rarely used unit of mass called the "metric slug". This is the mass that accelerates at one metre per second squared when pushed by a force of one kgf. An item with a mass of 10 kg has a mass of 1.01972661 metric slugs (= 10 kg divided by 9.80665 kg per metric slug). This unit is also known by various other names such as the hyl, TME (from a German acronym), and mug (from metric slug). Another unit of force called the poundal (pdl) is defined as the force that accelerates 1 lbm at 1 foot per second squared. Given that 1 lbf = 32.174 lb times one foot per second squared, we have 1 lbf = 32.174 pdl. The kilogram-force is a unit of force that was used in various fields of science and technology. In 1901, the CGPM improved the definition of the kilogram-force, adopting a standard acceleration of gravity for the purpose, and making the kilogram-force equal to the force exerted by a mass of 1 kg when accelerated by 9.80665 m/s². The kilogram-force is not a part of the modern SI system, but is still used in applications such as:
- Thrust of jet and rocket engines
- Spoke tension of bicycles
- Draw weight of bows
- Torque wrenches in units such as "meter kilograms" or "kilogram centimetres" (the kilograms are rarely identified as units of force)
- Engine torque output (kgf·m expressed in various word orders, spellings, and symbols)
- Pressure gauges in "kg/cm²" or "kgf/cm²" In colloquial, non-scientific usage, the "kilograms" used for "weight" are almost always the proper SI units for this purpose. They are units of mass, not units of force. The symbol "kgm" for kilograms is also sometimes encountered. This might occasionally be an attempt to disintinguish kilograms as units of mass from the "kgf" symbol for the units of force. It might also be used as a symbol for those obsolete torque units (kilogram-force metres) mentioned above, used without properly separating the units for kilogram and metre with either a space or a centered dot.

Conversions

Below are several coversion factors between various mesurements of force:
- 1 kgf (kilopond kp) = 9.80665 newtons
- 1 metric slug = 9.80665 kg
- 1 lbf = 32.174 poundals
- 1 slug = 32.174 lb
- 1 kgf = 2.2046 lbf

Forces in everyday life

Forces are part of everyday life, with examples such as:
- gravity: objects fall, even after being thrown upwards, or slide and roll down
- friction: floors and objects that are not extremely slippery
- spring force, objects resist tensile stress, compressive stress and/or shear stress, objects bounce back.
- electromagnetic force: attraction of magnets
- movement created by force: the movement of objects when force is applied.

Forces in the laboratory

Founding experiments

- Galileo Galilei used rolling balls to disprove the Aristotelian theory of motion (1602 - 1607)
- Henry Cavendish's torsion bar experiment measured the force of gravity between two masses (1798)

Instruments to measure forces

- spring balance
- pivot balance
- forcemeter

History

Force was first described by Archimedes.

References

-
-
-

External links

- [http://jumk.de/calc/force.shtml Calculation: force F - English and American units to metric units]
- [http://calc.skyrocket.de/en/ Online Unit Converter - Conversion of many different units]
- [http://www.patbelford.com/gallery/web3d/education/forceworkpower/index.html Interactive demonstration of Force-Work-Power Relationship] Category:Introductory physics ko:힘 ms:Daya (fizik) ja:力 simple:Force (physics)

Hydrogen

|- | Critical temperature || 32.19 K |- | Critical pressure || 1.315 MPa |- | Critical density || 30.12 g/L (Bohr radius) Hydrogen (Latin: hydrogenium, from Greek: hydro: water, genes: forming) is a chemical element in the periodic table that has the symbol H and atomic number 1. At standard temperature and pressure it is a colorless, odorless, nonmetallic, univalent, highly flammable diatomic gas. Hydrogen is the lightest and most abundant element in the universe. It is present in water, all organic compounds (rare exceptions exist, like buckminsterfullerene) and in all living organisms. Hydrogen is able to react chemically with most other elements. Stars in their main sequence are overwhelmingly composed of hydrogen in its plasma state. The element is used in ammonia production, as a lifting gas, as an alternative fuel, and more recently as a power source of fuel cells. Despite its ubiquity in the universe, hydrogen is surprisingly hard to produce in large quantities on the Earth. In the laboratory, the element is prepared by the reaction of acids on metals such as zinc. The electrolysis of water is a simple method of producing hydrogen, but is economically inefficient for mass production. Large-scale production is usually achieved by steam reforming natural gas. Scientists are now researching new methods for hydrogen production; if they succeed in developing a cost-efficient method of large-scale production, hydrogen may become a viable alternative to greenhouse-gas-producing fossil fuels. One of the methods under investigation involves use of green algae; another promising method involves the conversion of biomass derivatives such as glucose or sorbitol at low temperatures using a catalyst. Yet another method is the "steaming" of Carbon, whereby hydrocarbons are broken down with heat to release hydrogen.

Basic features

Hydrogen is the lightest chemical element; its most common isotope comprises just one negatively charged electron, distributed around a positively charged proton (the nucleus of the atom). The electron is bound to the proton by the Coulomb force, the electrical force that one stationary, electrically charged nanoparticle exerts on another. The hydrogen atom has special significance in quantum mechanics as a simple physical system for which there is an exact solution to the Schrödinger equation; from that equation, the experimentally observed frequencies and intensities of the hydrogen's spectral lines can be calculated. Spectral lines are dark or bright lines in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies. At standard temperature and pressure, hydrogen forms a diatomic gas, H₂, with a boiling point of only 20.27 K and a melting point of 14.02 K. Under extreme pressures, such as those at the center of gas giants, the molecules lose their identity and the hydrogen becomes a liquid metal. Under the extremely low pressure in space—virtually a vacuum—the element tends to exist as individual atoms, simply because there is no way for them to combine. However, clouds of H₂ and singular hydrogen atoms are said to form in H I and H II regions and are associated with star formation, however the existence of singular hydrogen atoms is disputed.. Hydrogen plays a vital role in powering stars through the proton–proton and carbon–nitrogen cycle. These are nuclear fusion processes, which release huge amounts of energy in stars and other hot celestial bodies as hydrogen atoms combine into helium atoms. H₂ is highly soluble in water, alcohol, and ether. It has a high capacity for adsorption, in which it is attached to and held to the surface of some substances. It is an odorless, tasteless, colorless, and highly flammable gas that burns at concentrations as low as 4% H₂ in air. It reacts violently with chlorine and fluorine, forming hydrohalic acids that can damage the lungs and other tissues. When mixed with oxygen, hydrogen explodes on ignition. A unique property of hydrogen is that its flame is completely invisible in air. This makes it difficult to tell if a leak is burning, and carries the added risk that it is easy to walk into a hydrogen fire inadvertently. See also: hydrogen atom.

Applications

Large quantities of hydrogen are needed in the chemical and petrolium industries, notably in the Haber process for the production of ammonia, which by mass ranks as the world's fifth most highly produced industrial compound. Hydrogen is used in the hydrogenation of fats and oils (into items such as margarine), and in the production of methanol. Hydrogen is used in hydrodealkylation, hydrodesulfurization, and hydrocracking. The element has several other important uses.
- The element is used in the manufacture of hydrochloric acid, in welding processes, and in the reduction of metallic ores.
- It is an ingredient in rocket fuels.
- It is used as the rotor coolant in electrical generators at power stations, because it has the highest thermal conductivity of any gas.
- Liquid hydrogen is used in cryogenic research, including superconductivity studies.
- Since hydrogen is 14.5 times lighter than air, it was once widely used as a lifting agent in balloons and airships. However, this use was curtailed when the Hindenburg disaster convinced the public that the gas was too dangerous for this purpose.
- Deuterium, an isotope of hydrogen (hydrogen-2), is used in nuclear fission applications as a moderator to slow neutrons, and in nuclear fusion reactions. Deuterium compounds have applications in chemistry and biology in studies of reaction isotope effects.
- Tritium (hydrogen-3), produced in nuclear reactors, is used in the production of hydrogen bombs, as an isotopic label in the biosciences, and as a radiation source in luminous paints. There are no "hydrogen wells" or "hydrogen mines" on Earth, so hydrogen cannot be considered a primary energy source like fossil fuels or uranium. Hydrogen can however be burned in internal combustion engines, an approach advocated by BMW's experimental hydrogen car. However, it is currently difficult and dangerous to store and handle in sufficient quantity for motor fuel use. Hydrogen fuel cells are being investigated as mobile power sources with lower emissions than hydrogen-burning internal combustion engines. The low emissions of hydrogen in internal combustion engines and fuel cells are currently offset by the pollution created by hydrogen production. This may change if the substantial amounts of electricity required for water electrolysis can be generated primarily from low pollution sources such as nuclear energy or wind. Research is being conducted on hydrogen as a replacement for fossil fuels. It could become the link between a range of energy sources, carriers and storage. Hydrogen can be converted to and from electricity (solving the electricity storage and transport issues), from bio-fuels, and from and into natural gas and diesel fuel. All of this can theoretically be achieved with zero emissions of CO₂ and toxic pollutants.

History

Hydrogen was first produced by Theophratus Bombastus von Hohenheim (1493–1541)—also known as Paracelsus—by mixing metals with acids. He was unaware that the explosive gas produced by this chemical reaction was hydrogen. In 1671, Robert Boyle described the reaction between two iron fillings and dilute acids, which results in the production of gaseous hydrogen. In 1766, Henry Cavendish was the first to recognize hydrogen as a discrete substance, by identifying the gas from this reaction as "inflammable" and finding that the gas produces water when burned in air. Cavendish stumbled on hydrogen when experimenting with acids and mercury. Although he wrongly assumed that hydrogen was a compound of mercury—and not of the acid—he was still able to accurately describe several key properties of hydrogen. Antoine Lavoisier gave the element its name and proved that water is composed of hydrogen and oxygen. One of the first uses of the element was for balloons. The hydrogen was obtained by mixing sulfuric acid and iron. Harold C. Urey discovered Deuterium, an isotope of hydrogen, by repeated distilling the same sample of water. For this discovery, Urey received the Nobel prize for in 1934. In the same year, the third isotope, tritium, was discovered. Because of its relatively simple structure, hydrogen has often been used in models of how an atom works.

Electron energy levels

The ground state energy level of the electron in a Hydrogen atom is 13.6 eV, which is equivalent to an ultraviolet photon of roughly 92 nm. With the Bohr Model the energy levels of Hydrogen can be calculated fairly accurately. This is done by modeling the electron as revolving around the proton, much like the earth revolving around the sun. Except the sun holds earth in orbit with the force of gravity, but the proton holds the electron in orbit with the force of electromagnetism. Another difference between the Earth-Sun system and the Electron-Proton system is that, in this model, due to quantum mechanics the electron is allowed to only be at very specific distances from the proton. Modeling the hydrogen atom in this fashion yields the correct energy levels and spectrum.

Occurrence

quantum mechanics.]] Hydrogen is the most abundant element in the universe, making up 75% of normal matter by mass and over 90% by number of atoms. This element is found in great abundance in stars and gas giant planets. It is very rare in the Earth's atmosphere (1 ppm by volume), because being the lightest gas causes it to escape Earth's gravity, though when compounds are considered, it is the tenth most abundant element on Earth. The most common source for this element on Earth is water, which is composed two parts hydrogen to one part oxygen (H₂O). Other sources include most forms of organic matter (currently all known life forms) including coal, natural gas, and other fossil fuels. Methane (CH₄) is an increasingly important source of hydrogen. Throughout the Universe, hydrogen is mostly found in the plasma state whose properties are quite different to molecular hydrogen. As a plasma, hydrogen's electron and proton are not bound together, resulting in very high electrical conductivity, even when the gas is only partially ionised. The charged particles are highly influenced by magnetic and electric fields, for example, in the Solar Wind they interact with the Earth's magnetosphere giving rise to Birkeland currents and the aurora. Hydrogen can be prepared in several different ways: steam on heated carbon, hydrocarbon decomposition with heat, reaction of a strong base in an aqueous solution with aluminium, water electrolysis, or displacement from acids with certain metals. Commercial bulk hydrogen is usually produced by the steam reforming of natural gas. At high temperatures (700–1100 °C), steam reacts with methane to yield carbon monoxide and hydrogen. :CH₄ + H₂O → CO + 3 H₂ Additional hydrogen can be recovered from the carbon monoxide through the water-gas shift reaction: :CO + H₂O → CO₂ + H₂

Compounds

The lightest of all gases, hydrogen combines with most other elements to form compounds. Hydrogen has an electronegativity of 2.2, so it forms compounds where it is the more nonmetallic and where it is the more metallic element. The former are called hydrides, where hydrogen either exists as H^- ions or just as a solute within the other element (as in palladium hydride). The latter tend to be covalent, since the H⁺ ion would be a bare nucleus and so has a strong tendency to pull electrons to itself. These both form acids. Thus even in an acidic solution one sees ions like hydronium (H₃O⁺) as the protons latch on to something. Although exotic on earth, one of the most common ions in the universe is the H₃⁺ ion. Hydrogen combines with oxygen to form water, H₂O, and releases a lot of energy in doing so, burning explosively in air. Deuterium oxide, or D₂O, is commonly referred to as heavy water. Hydrogen also forms a vast array of compounds with carbon. Because of their association with living things, these compounds are called organic compounds, and the study of the properties of these compounds is called organic chemistry. organic chemistry

Forms

Under normal conditions, hydrogen gas is a mix of two different kinds of molecules which differ from one another by the relative spin of the nuclei. These two forms are known as ortho- and para-hydrogen (this is different from isotopes, see below). In ortho-hydrogen the nuclear spins are parallel (form a triplet), while in para they are antiparallel (form a singlet). At standard conditions hydrogen is composed of about 25% of the para form and 75% of the ortho form (the so-called "normal" form). The equilibrium ratio of these two forms depends on temperature, but since the ortho form has higher energy (is an excited state), it cannot be stable in its pure form. In low temperatures (around boiling point), the equilibrium state is comprised almost entirely of the para form. The conversion process between the forms is slow, and if hydrogen is cooled down and condensed rapidly, it contains large quantities of the ortho form. It is important in preparation and storage of liquid hydrogen, since the ortho-para conversion produces more heat than the heat of its evaporation, and a lot of hydrogen can be lost by evaporation in this way during several days after liquefying. Therefore, some catalysts of the ortho-para conversion process are used during hydrogen cooling. The two forms have also slightly different physical properties. For example, the melting and boiling points of parahydrogen are about 0.1 K lower than of the "normal" form.

Isotopes

Hydrogen is the only element that has different names for its isotopes. (During the early study of radioactivity, various heavy radioactive isotopes were given names, but such names are no longer used, although one element, radon, has a name that originally applied to only one of its isotopes.) The symbols D and T (instead of ²H and ³H) are sometimes used for deuterium and tritium, although this is not officially sanctioned. (The symbol P is already in use for phosphorus and is not available for protium.) ;¹H The most common isotope of hydrogen, this stable isotope has a nucleus consisting of a single proton; hence the descriptive, although rarely used, name protium. The spin of a protium atom is 1/2+. ;²H The other stable isotope is deuterium, with an extra neutron in the nucleus. Deuterium comprises 0.0184%–0.0082% of all hydrogen (IUPAC); ratios of deuterium to protium are reported relative to the VSMOW standard reference water. The spin of a deuterium atom is 1+. ;³H The third naturally occurring hydrogen isotope is the radioactive tritium. The tritium nucleus contains two neutrons in addition to the proton. It decays through beta decay and has a half-life of 12.32 years. Tritium occurs naturally due to cosmic rays interacting with atmospheric gases. Like ordinary hydrogen, tritium reacts with the oxygen in the atmosphere to form T₂O. This radioactive "water" molecule constantly enters the Earth's seas and lakes in the form of slightly radioactive rain, but its half-life is short enough to prevent a buildup of hazardous radioactivity. The spin of a tritium atom is 1/2+. ;⁴H Hydrogen-4 was synthesized by bombarding tritium with fast-moving deuterium nuclei. It decays through neutron emission and has a half-life of 9.93696x10^-23 seconds. The spin of a hydrogen-4 atom is 2-. ;⁵H In 2001 scientists detected hydrogen-5 by bombarding a hydrogen target with heavy ions. It decays through neutron emission and has a half-life of 8.01930x10^-23 seconds. ;⁶H Hydrogen-6 decays through triple neutron emission and has a half-life of 3.26500^-22 seconds. ;⁷H In 2003 hydrogen-7 was created ([http://physicsweb.org/articles/news/7/3/3 article]) at the RIKEN laboratory in Japan by colliding a high-energy beam of helium-8 atoms with a cryogenic hydrogen target and detecting tritons—the nuclei of tritium atoms—and neutrons from the breakup of hydrogen-7, the same method used to produce and detect hydrogen-5.

References

# # # # # #
- [http://www.riken.go.jp/engn/r-world/research/lab/wako/ribeam/ RIKEN Beam Science Laboratory, Japan - Heavy hydrogen research]
- [http://chartofthenuclides.com/default.html Nuclides and Isotopes] Fourteenth Edition: Chart of the Nuclides, General Electric Company, 1989 ;Book references:
-
-
-
-

External links

- [http://www.hydropole.ch/Hydropole/Intro/Phasediag.gif Hydrogen phase diagram.]
- [http://www.compchemwiki.org/index.php?title=Hydrogen Computational Chemistry Wiki] Category:Nonmetals Category:Fuels Category:Chemical elements ko:수소 ms:Hidrogen ja:水素 simple:Hydrogen th:ไฮโดรเจน

Isaac Newton

Sir Isaac Newton, PRS ( – ) was an English physicist, mathematician, astronomer, alchemist and philosopher associated with the scientific revolution and the advancement of heliocentrism. He was one of the most influential scientists in history. Among his scientific accomplishments, Newton wrote the Philosophiae Naturalis Principia Mathematica, wherein he described universal gravitation and, via his laws of motion, laid the groundwork for classical mechanics. With Gottfried Leibniz he shares credit for the development of calculus. Newton was the first to promulgate a set of natural laws that could govern both terrestrial motion and celestial motion, and is credited with providing mathematical substantiation for Kepler's laws of planetary motion, which he expanded by arguing that orbits (such as those of comets) could include all conic sections (such as the ellipse, hyperbola, and parabola). Newton realised that the spectrum of colours observed when white light passed through a prism is inherent in the white light and not added by the prism (as Roger Bacon had claimed in the 13th century), and also notably argued that light is composed of particles. Newton additionally developed a law of cooling, proved the binomial theorem, and discovered the principles of conservation of momentum and angular momentum. Newton is regarded by many as having "unrivalled mathematical genius" [see Dampier & Dampier]. The mathematician Joseph Louis Lagrange (1736-1813), Director of the Berlin Academy of Sciences, said this about Newton: ::"Newton was the greatest genius that ever existed and the most fortunate, for we cannot find more than once a system of the world to establish." [See Shapley.]

Biography

Early years

Newton was born in Woolsthorpe-by-Colsterworth (at Woolsthorpe Manor), a hamlet in the county of Lincolnshire. Newton was prematurely born and no one expected him to live; indeed, his mother, Hannah Ayscough Newton, is reported to have said that his body at that time could have fit inside a quart mug (Bell, 1937). His father, Isaac, had died three months before Newton's birth. When Newton was two years old, his mother went to live with her new husband, leaving her son in the care of his grandmother. According to E.T. Bell (1937, Simon and Schuster) and H. Eves: :Newton began his schooling in the village schools and was later sent to Grantham Grammar School where he became the top boy in the school. At Grantham he lodged with the local apothecary, William Clarke and eventually became engaged to the apothecary's stepdaughter, Anne Storer, before he went off to Cambridge University at the age of 19. As Newton became engrossed in his studies, the romance cooled and Miss Storer married someone else. It is said he kept a warm memory of this love, but Newton had no other recorded 'sweethearts' and never married. Cambridge University From the age of twelve until he was seventeen, Newton was educated at Grantham Grammar School. His family then removed him from school and attempted to make a farmer of him. However he was thoroughly unhappy with the work and eventually with the help of his uncle and of his schoolteacher, he managed to persuade his mother to send him back to school so that he might complete his schooling. This he did at the age of eighteen, achieving an admirable final report. His teacher said: :His genius now begins to mount upwards apace and shine out with more strength. He excels particularly in making verses. In everything he undertakes, he discovers an application equal to the pregnancy of his parts and exceeds even the most sanguine expectations I have conceived of him. In 1661 he joined Trinity College, Cambridge, where his uncle William Ayscough had studied. At that time, the college's teachings were based on those of Aristotle, but Newton preferred to read the more advanced ideas of modern philosophers such as Descartes, Galileo, Copernicus and Kepler. In 1665 he discovered the binomial theorem and began to develop a mathematical theory that would later become calculus. Soon after Newton had obtained his degree in 1665, the University closed down as a precaution against the Great Plague. For the next two years Newton worked at home on calculus, optics and gravitation. He later continued his studies at Woolsthorpe Manor. The popular tradition has it that Newton was sitting under an apple tree when an apple fell on his head, and that this made him understand that earthly and celestial gravitation are the same. A contemporary writer, William Stukeley, recorded in his Memoirs of Sir Isaac Newton's Life a conversation with Newton in Kensington on 15 April 1726, in which Newton recalled "when formerly, the notion of gravitation came into his mind. It was occasioned by the fall of an apple, as he sat in contemplative mood. Why should that apple always descend perpendicularly to the ground, thought he to himself. Why should it not go sideways or upwards, but constantly to the earth's centre." In similar terms, Voltaire wrote in his Essay on Epic Poetry (1727), "Sir Isaac Newton walking in his gardens, had the first thought of his system of gravitation, upon seeing an apple falling from a tree." These accounts are exaggerations of Newton's own tale about sitting by a window in his home (Woolsthorpe Manor) and watching an apple fall from a tree. It is now generally considered probable that even this story was invented by Newton in later life, to illustrate how he drew inspiration from everyday events.

Middle years

Mathematical research

Woolsthorpe Manor.]] Newton became a fellow of Trinity College in 1669. In the same year he circulated his findings in De Analysi per Aequationes Numeri Terminorum Infinitas (On Analysis by Infinite Series), and later in De methodis serierum et fluxionum (On the Methods of Series and Fluxions), whose title gave the name to his "method of fluxions". Newton is generally credited as the discoverer of the binomial theorem, an essential step toward the development of modern analysis. Newton and Gottfried Leibniz developed the theory of calculus independently, using different notations. Although Newton had worked out his own method before Leibniz, the latter's notation and "Differential Method" were superior, and were generally adopted throughout the world. Though Newton belongs among the brightest scientists of his era, the last twenty-five years of his life were marred by a bitter dispute with Leibniz, whom he accused of plagiarism. The dispute created a divide between British and Continental mathematicians that persisted even after Newton's death. He was elected Lucasian professor of mathematics in 1669. Any fellow of Cambridge or Oxford had to be ordained at the time. However the terms of the Lucasian professorship required that the holder not be active in the church (presumably so as to have more time for science). Newton argued that this should exempt him from the normal ordination requirement, and Charles II, whose permission was needed, accepted this argument. This prevented the conflict that would have occurred between his religious views and the orthodoxy of the church.

Optics

From 1670 to 1672 he lectured on optics. During this period he investigated the refraction of light, demonstrating that a prism could decompose white light into a spectrum of colours, and that a lens and a second prism could recompose the multicoloured spectrum into white light. He also showed that the coloured light does not change its properties, by separating out a coloured beam and shining it on various objects. Newton noted that regardless of whether it was reflected or scattered or transmitted, it stayed the same colour. Thus the colours we observe are the result of how objects interact with the incident already-coloured light, not the result of objects generating the colour. For more details, see Newton's theory of colour. Many of his findings in this field were critized by later theorists, the most well-known being Johann Wolfgang von Goethe, who postulated his own colour theories. Johann Wolfgang von Goethe From this work he concluded that any refracting telescope would suffer from the dispersion of light into colours, and invented a reflecting telescope (today, known as a Newtonian telescope) to bypass that problem. By grinding his own mirrors, using Newton's rings to judge the quality of the optics for his telescopes, he was able to produce a superior instrument to the refracting telescope, due primarily to the wider diameter of the mirror. (Only later, as glasses with a variety of refractive properties became available, did achromatic lenses for refractors become feasible.) In 1671 the Royal Society asked for a demonstration of his reflecting telescope. Their interest encouraged him to publish his notes On Colour, which he later expanded into his Opticks. When Robert Hooke criticised some of Newton's ideas, Newton was so offended that he withdrew from public debate. The two men remained enemies until Hooke's death. In one experiment, to prove that colour perception is caused by pressure on the eye, Newton slid a darning needle around the side of his eye until he could poke at its rear side, dispassionately noting "white, darke & coloured circles" so long as he kept stirring with "y^e bodkin." Newton argued that light is composed of particles; thus he could not explain the diffraction of light. Later physicists instead favoured a wave explanation of light to account for diffraction. Today's quantum mechanics recognises a "wave-particle duality"; however photons bear very little semblance to Newton's corpuscles (e.g., corpuscles refracted by accelerating toward the denser medium). Newton is believed to have been the first to explain precisely the formation of the rainbow from water droplets dispersed in the atmosphere in a rain shower. Figure 15 of Part II of Book One of the Opticks shows a perfect illustration of how this occurs. In his Hypothesis of Light of 1675, Newton posited the existence of the ether to transmit forces between particles. Newton was in contact with Henry More, the Cambridge Platonist who was born in Grantham, on alchemy, and now his interest in the subject revived. He replaced the ether with occult forces based on Hermetic ideas of attraction and repulsion between particles. John Maynard Keynes, who acquired many of Newton's writings on alchemy, stated that "Newton was not the first of the age of reason: he was the last of the magicians." Newton's interest in alchemy cannot be isolated from his contributions to science.² (This was at a time when there was no clear distinction between alchemy and science.) Had he not relied on the occult idea of action at a distance, across a vacuum, he might not have developed his theory of gravity. (See also Isaac Newton's occult studies.) In 1704 Newton wrote Opticks, in which he expounded his corpuscular theory of light. The book is also known for the first exposure of the idea of the interchangeability of mass and energy: "Gross bodies and light are convertible into one another...". Newton also constructed a primitive form of a frictional electrostatic generator, using a glass globe (Optics, 8th Query).

Gravity and motion

glass In 1679, Newton returned to his work on mechanics, i.e., gravitation and its effect on the orbits of planets, with reference to Kepler's laws of motion, and consulting with Hooke and Flamsteed on the subject. He published his results in De Motu Corporum (1684). This contained the beginnings of the laws of motion that would inform the Principia. The Philosophiae Naturalis Principia Mathematica (now known as the Principia) was published on 5 July 1687¹) with encouragement and financial help from Edmond Halley. In this work Newton stated the three universal laws of motion that were not to be improved upon for more than two hundred years. He used the Latin word gravitas (weight) for the force that would become known as gravity, and defined the law of universal gravitation. In the same work he presented the first analytical determination, based on Boyle's law, of the speed of sound in air. With the Principia, Newton became internationally recognised. He acquired a circle of admirers, including the Swiss-born mathematician Nicolas Fatio de Duillier, with whom he formed an intense relationship that lasted until 1693. The end of this friendship led Newton to a nervous breakdown.

Later life

nervous breakdown In the 1690s Newton wrote a number of religious tracts dealing with the literal interpretation of the Bible. Henry More's belief in the infinity of the universe and rejection of Cartesian dualism may have influenced Newton's religious ideas. A manuscript he sent to John Locke in which he disputed the existence of the Trinity was never published. Later works — The Chronology of Ancient Kingdoms Amended (1728) and Observations Upon the Prophecies of Daniel and the Apocalypse of St. John (1733) — were published after his death. He also devoted a great deal of time to alchemy (see above)². Newton was also a member of the Parliament of England from 1689 to 1690 and in 1701, but his only recorded comments were to complain about a cold draft in the chamber and request that the window be closed. Newton moved to London to take up the post of warden of the Royal Mint in 1696, a position that he had obtained through the patronage of Charles Montagu, 1st Earl of Halifax, then Chancellor of the Exchequer. He took charge of England's great recoining, somewhat treading on the toes of Master Lucas (and finagling Edmond Halley into deputy comptroller of the temporary Chester branch). Newton became Master of the Mint upon Lucas' death in 1699. These appointments were intended as sinecures, but Newton took them seriously, exercising his power to reform the currency and punish clippers and counterfeiters. He retired from his Cambridge duties in 1701. Ironically, it was his work at the Mint, rather than his contributions to science, which earned him a knighthood. Newton was knighted by Queen Anne in 1705. Newton was made President of the Royal Society in 1703 and an associate of the French Académie des Sciences. In his position at the Royal Society, Newton made an enemy of John Flamsteed, the Astronomer Royal, by attempting to steal his catalogue of observations. Newton died in London and was buried in Westminster Abbey. It is believed Newton never had a romantic relationship, and he is said to have died a virgin. There is some speculation that Newton had Asperger syndrome, a form of autism. See People speculated to have been autistic. His niece, Catherine Barton Conduitt³, served as his hostess in social affairs at his house on Jermyn Street in London; he was her "very loving Uncle"⁴, according to his letter to her when she was recovering from smallpox.

Religious views

4 The law of gravity became Newton's best-known discovery. He warned against using it to view the universe as a mere machine, like a great clock. He said, "Gravity explains the motions of the planets, but it cannot explain who set the planets in motion. God governs all things and knows all that is or can be done." His scientific fame notwithstanding, the Bible was Newton's greatest passion. He devoted more time to the study of Scripture and Alchemy than to science, and said, "I have a fundamental belief in the Bible as the Word of God, written by those who were inspired. I study the Bible daily." Newton himself wrote works on textual criticism, most notably An Historical Account of Two Notable Corruptions of Scripture. He also attempted, unsuccessfully, to find hidden messages within the Bible (See Bible code). Despite his focus in theology and alchemy, Newton tested and investigated these myths with the scientific method, observing, hypothesizing, and testing his theories. To Newton, his scientific and mythical experiments were one in the same, observing and understanding how the world functioned. Newton is often accused of being a Unitarian and Arian, and not believing in the church's doctrine of divine trinity. However, T.C. Pfizenmaier argued that he more likely held the Eastern Orthodox view of the Trinity rather than the Western one held by Roman Catholics, Anglicans, and most Protestants.⁷ In his own day, he was also accused of being a Rosicrucian (as were many in the Royal Society and in the court of Charles II). In his own lifetime, Newton wrote more on religion than he did on natural science. He believed in a rationally emanent world, but he rejected the hylozoism implicit in Leibniz and Baruch Spinoza. Thus, the ordered and dynamically informed universe could be understood, and must be understood, by an active reason, but this universe, to be perfect and ordained, had to be regular.

Newton's effect on religious thought

Newton and Robert Boyle’s mechanical philosophy was promoted by rationalist pamphleteers as a viable alternative to the pantheists and enthusiasts, and was accepted hesitantly by orthodox preachers as well as dissident preachers like the latitudinarians. Thus, the clarity and simplicity of science was seen as a way to combat the emotional and metaphysical superlatives of both superstitious enthusiasm and the threat of atheism, and, at the same time, the second wave of English deists used Newton's discoveries to demonstrate the possibility of a "Natural Religion." The attacks made against pre-Enlightenment "magical thinking," and the mystical elements of Christianity, were given their foundation with Boyle’s mechanical conception of the universe. Newton gave Boyle’s ideas their completion through mathematical proofs, and more importantly was very successful in popularizing them. Newton refashioned the world governed by an interventionist God into a world crafted by a God that designs along rational and universal principles. These principles were available for all people to discover, allowed man to pursue his own aims fruitfully in this life, not the next, and to perfect himself with his own rational powers. The perceived ability of Newtonians to explain the world, both physical and social, through logical calculations alone is the crucial idea in the disenchantment of Christianity. Newton saw God as the masterful creator whose existence could not be denied in the face of the grandeur of all creation.⁵'^6' But the unforeseen theological consequence of his conception of God, as Leibniz pointed out, was that God was now entirely removed from the world’s affairs, since the need for intervention would only evidence some imperfection in God’s creation, something impossible for a perfect and omnipotent creator. Leibniz's theodicy cleared God from the responsibility for "l'origine du mal" by making God removed from participation in his creation. The understanding of the world was now brought down to the level of simple human reason, and humans, as Odo Marquard argued, became responsible for the correction and elimination of evil. On the other hand, latitudinarian and Newtonian ideas taken too far resulted in the millenarians, a religious faction dedicated to the concept of a mechanical universe, but finding in it the same enthusiasm and mysticism that the Enlightenment had fought so hard to extinguish.

Newton versus the counterfeiters

Newton estimated that 20% of the coins taken in during The Great Recoinage were counterfeit. Counterfeiting was treason, punishable by death by drawing and quartering. As gruesome as the penalties were, the courts were not arbitrary or capricious. The rights of free men had a long tradition in England and the crown had to prove its case to a jury. The law also allowed for plea bargaining. Convictions of the most flagrant criminals could be maddeningly impossible to achieve; however, Newton proved to be equal to the task. He assembled facts and proved his theories with the same brilliance in law that he had shown in science. He gathered much of that evidence himself, disguised, while he hung out at bars and taverns. For all the barriers placed to prosecution, and separating the branches of government, English law still had ancient and formidable customs of authority. Newton was made a justice of the peace and between June 1698 and Christmas 1699 conducted some 200 cross-examinations of witnesses, informers and suspects. During this time he obtained the confessions he needed and while he could not resort to open torture, whatever means he did use must have been fearsome because Newton himself later ordered all records of these interrogations to be destroyed. However he did it, Newton won his convictions and in February 1699, he had ten prisoners waiting to be executed. Newton's greatest triumph as the king's attorney was against William Chaloner. Chaloner was a rogue with a devious intelligence. He set up phony conspiracies of Catholics and then turned in the hapless conspirators whom he entrapped. Chaloner made himself rich enough to posture as a gentleman. Petitioning Parliament, Chaloner accused the Mint of providing tools to counterfeiters. (This charge was made also by others.) He proposed that he be allowed to inspect the Mint's processes in order to improve them. He petitioned Parliament to adopt his plans for a coinage that could not be counterfeited. All the time, he struck false coins, or so Newton eventually proved to a court of competent jurisdiction. On March 23, 1699, Chaloner was hanged, drawn and quartered.

Enlightenment philosophers

Enlightenment philosophers chose a short history of scientific predecessors—Galileo, Boyle, and Newton principally—as the guides and guarantors of their applications of the singular concept of Nature and Natural Law to every physical and social field of the day. In this respect, the lessons of history and the social structures built upon it could be discarded. It was Newton’s conception of the universe based upon Natural and rationally understandable laws that became the seed for Enlightenment ideology. Locke and Voltaire applied concepts of Natural Law to political systems advocating intrinsic rights; the physiocrats and Adam Smith applied Natural conceptions of psychology and self-interest to economic systems, and sociologists critiquing the current social order fit history into Natural models of progress.

Newton's legacy

progress]] Newton's laws of motion and gravity provided a basis for predicting a wide variety of different scientific or engineering situations, especially the motion of celestial bodies. His calculus proved vitally important to the development of further scientific theories. Finally, he unified many of the isolated physics facts that had been discovered earlier into a satisfying system of laws. Newton's conceptions of gravity and mechanics, though not entirely correct in light of Einstein's Theory of Relativity, still represent an enormous step in the evolution of human understanding of the universe. For this reason, he is generally considered one of history's greatest scientists, ranking alongside such figures as Einstein and Carl Friedrich Gauss. In 1717, the Kingdom of Great Britain went on to an unofficial gold standard when Newton, then Master of the Mint, established a fixed price of £3.17.10 ½d per standard (22 carat) troy ounce, equal to £4.4.11 ½d per fine ounce. Under the gold standard the value of the pound (measured in gold weight) remained largely constant until the beginning of the 20th century. Newton is reputed to have invented the cat flap. This was said to be done so that he would not have to disrupt his optical experiments, conducted in a darkened room, to let his cat in or out. Newtonmas is a holiday celebrated by some scientists as an alternative to Christmas, taking advantage of the fact that Newton's birthday falls on December 25. In July 1992, the Isaac Newton Institute for Mathematical Sciences was opened at Cambridge University - it is regarded as the