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Newton's Laws of Motion Calculator

In this and other pages associated with the Newton calculator, the term satellite refers to any natural mass (m₂) such as a star, planet, moon, etc. freely orbiting a force-centre (m₁) such as a black-hole, star, planet, etc.

Orbital path of a satellite around a force-centre

Fig 1. Orbital System

The Orbital System

A typical orbital system is shown in Fig 1, where a satellite; e.g. a planet, moon or comet is orbiting a force-centre; e.g. a star or planet. The perigee (or perihelion) is the point in the orbit at which the satellite is closest to the force-centre and the apogee (or aphelion) is the point in the orbit at which the two are farthest apart. These two points are opposite each other in the orbital ellipse.

Laws of Motion

A detailed description and explanation for all the formulas used in this calculator can be found on our web-page: Laws of Motion.

Newton developed his laws without the knowledge of spin theory, which is understandable given that it wasn't known in the 17th century exactly how much spin existed in the sun or its orbiting planets. Spin theory defines the laws governing the spin (rotation) in force-centres and their orbiting bodies. This theory remained unsolved until 2017 but can now be found on our web-page; Planetary Spin

Without this theory, Newton's laws were incomplete and may be the reason why it was considered necessary to invent dark matter to account for a discrepancy between the calculated and the expected mass in our Milky Way galaxy. Spin theory has permitted the completion of Newton's laws, with which it is now possible to accurately calculate the properties of our Milky Way the results from which are as expected, thereby discounting the need for dark matter.

For example:
NASA suggests that our sun is 1E+21 metres from the centre of the galaxy and that it will take 230 million years to orbit the Milky Way, and Wikipedia suggests that the galaxy contains 250 billion ± 150 billion stars
Using this information, along with the sun's known properties, we can now calculate the sun's orbit in the Milky Way using Newton's laws of motion thus:
m₁ = 1.1229556625E+43 kg
m₂ = 1.9885E+30 kg
Tₒ = 7.258248E+15 s
= 1E+21 m (9.99900039E+20 gives the most accurate solution)
G = 6.67359232004332E-11 m³/s²/kg
The resultant calculation shows the following verification errors:
F̌ε = 0
F̂ε = 1.44074E-08
Řε = 4.13736E-13
i.e. the Milky Way's force centre is 1.1229556625E+43 kg rotating at 3.7E-10 radians per second

Moreover, the calculation will be equally applicable even if NASA and Wikipedia are incorrect, albeit the force-centre mass and angular speed will vary slightly. According to the spin-theory calculations, the average density of the force-centre is closer to that of iron than of a neutron star. It is, however, sufficiently large to trap light (i.e. it is a black hole).

Newton's laws of motion apply to all force-centres with orbiting bodies, including atoms. Therefore, this calculator may be used to calculate the performance of electrons in atoms assuming that gravity is the governing coupling force, which of course is not quite correct as the electrical attraction is much stronger.

Newton's laws also show that any satellite (star, planet, moon, etc.) can be extracted from its own orbit and slotted into any other. Try inserting Jupiter in earth's orbit, it works! However there is a reason why earth is where it is in relation to the sun and Jupiter is where it is. It is all due to the materials from which they are composed.

Newton's Gravitational Constant (G)

Newton never specified his gravitational constant (G) when he published his Principia, which is understandable because no doubt he did not realise that it is actually based upon elementary particles. This constant has continued to be the subject of estimation; until now.

'G' has now been solved and can be found on our web-page; Laws of Motion and its correct value is defaulted in the Newton calculator. Hence you will notice the difference between the 'units of convenience' (N.kg²/m²) that have been applied to this constant to date and its actual units (m³/s²/kg) as provided in the calculator.

Calculation Errors

The Newton calculator generates errors (F̌ε, F̂ε & Řε) in its calculations each of which provides an indication of the accuracy of your input data, i.e. errors do not indicate calculation errors.

These errors are defined as follows:

Řε represents the error between two alternative methods used to calculate the radial distance between the force-centre and its satellite at the apogee of its orbit. One radial distance (Řₒ) is generated from the geometric properties of the elliptical orbit and the other (Řₑ) is generated independently using the potential (gravitational) and kinetic energies in the satellite.
The error is calculated as follows: Řε = (Řₒ - Řₑ) / Řₒ

F̌ε represents the error between two alternative methods used to calculate the gravitational force between the force-centre and its satellite at the perigee of its orbit. One force () is generated from Newton's formula and the other (F̌c) is generated independently from the centripetal force in the satellite due to its orbital velocity.
The error is calculated as follows: F̌ε = ( - F̌c) / F̌ε

F̂ε represents the error between two alternative methods used to calculate the gravitational force between the force-centre and its satellite at the apogee of its orbit. One force () is generated from Newton's formula and the other (F̂c) is generated independently from the centripetal force in the satellite due to its orbital velocity.
The error is calculated as follows: F̂ε = ( - F̂c) / F̌ε

F̌ε and Řε should never be greater than ±1E-06 unless Newton's gravitational constant is incorrectly entered.

F̂ε may well be greater than this due to inaccurate input data. It should be understood that whilst Newton's laws are accurate, the complex nature of force-centres and their satellites is such that their properties are not accurately known.

You may alter the input data to improve (reduce) errors but bear in mind it will probably take the correction of more than one variable to minimise errors, and where more than one variable is altered, you would still be guessing.

Newton's Laws of Motion Calculator – Technical Help

Units

You may use any units you wish, but you must be consistent. Refer to Input Data and Output Data below for examples of the {units} you get out from those you enter. The {units} provided below are those for the defaulted data.

Units in [square brackets] are fixed; i.e. they cannot be altered

Input Data

m₁ = the mass of the force-centre {kg}
m₂ = the mass of the satellite {kg}
Tₒ = the period for the satellite to complete an orbit of its force-centre [seconds]
= the minimum radial distance between the centres of mass of the force-centre and the satellitee (@ the satellite's perigee) {m}
G = Newton's gravitational constant which is provided in the Newton calculator in m³/s²/kg, you will need to convert this value into your preferred units if different
(e.g. 1.19176793676718E-09 in³/lb/s²).

Output Data

Orbital Shape
a = the major semi-axis of the elliptical orbit {m}
b = the minor semi-axis of the elliptical orbit {m}
e = the eccentricity of elliptical orbit (0 ≤ e < 1) {0 means the orbit is circular}
p = the orbital [half] parameter
ƒ = the radial distance between the centres of mass of the force-centre and the satellite at the satellite's perigee {m}
x = the radial distance between the centre of mass of the force-centre and the geometric centre of the ellipse {m}
L = the total length (circumference) of a single orbit {m}
A = the total area enclosed by orbital ellipse {m²}

At its Perigee
= the gravitational force between the force-centre and the satellite at its orbital perigee calculated using Newton's formula {N}
F̌c = the centripetal force in the satellite at its orbital perigee {N}
= the gravitational acceleration exerted by the force-centre at the satellite's orbital perigee {m/s²}
= the satellite's orbital velocity (not its centripetal velocity) at its orbital perigee {m}
PEᴾ = the potential (gravitational) energy in the satellite at its orbital perigee {J}
KEᴾ = the kinetic energy in the satellite at its orbital perigee {J}
Eᴾ = the total energy in the satellite at its orbital perigee {J}

At its Apogee
Řₒ = the radial distance between the centres of mass of the force-centre and the satellite at its orbital apogee calculated using the geometry of the elliptical orbit {m}
Řₑ = the radial distance between the centres of mass of the force-centre and the satellite at its orbital apogee calculated using the energy in the satellite {m}
= the gravitational force between force-centre and the satellite at its orbital apogee calculated using Newton's formula {N}
F̂c = the centripetal force in the satellite at its orbital apogee {N}
= the gravitational acceleration exerted by the force-centre at the satellite's orbital apogee {m/s²}
= the satellite's orbital velocity (not its centripetal velocity) at its orbital apogee {m}
PEᴬ = the potential (gravitational) energy in the satellite at its orbital apogee {J}
KEᴬ = the kinetic energy in the satellite at its orbital apogee {J}
Eᴬ = the total energy in the satellite at its orbital apogee {J}

Constants
h = Newton's motion constant {m²/s} (Laws of Motion; Corollary 1)

Errors
F̌ε, F̂ε & Řε = refer to Calculation Errors above

Applicability

The Newton calculator can be used for any force-centre/satellite system in which gravity is the sole coupling force.

Accuracy

The calculations in the Newton calculator are completely accurate, more so than Newton's original calculations due to the incorporation of an accurate value for Newton's gravitational constant 'G'.

Further Reading

You will find further reading on this subject in reference publications(61, 62, 63 & 65)

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