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Our equation for work follows from the conservation of energy. If we consider some object then we expect that if we do work $W$ on it then its kinetic energy must increase by $W$. So the requirement for the equation for work is that it must be equal to the change in kinetic energy. Proving this is usually done using integral calculus, but since you give the ...

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The notion of work in physics was first formulated by the French mathematician Gustave Coriolis in Calculation of the Effect of Machines, or Considerations on the Use of Engines and their Evaluation published in 1829. Coriolis defined work as "weight lifted through a height". He was concerned with developing a term that could measure the units of work ...

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There are some physical quantities that are usefull (and this is under statement), like energy. It is conserved, it is a function of some other very important quantities that can help you describe the motion of the body etc. If you can justify energy, there should be no problem in justifying work, which is energy transfered to a body by some force. Quantity ...

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The rotation group, its representations, and their carrier spaces are fundamental parts of quantum mechanics. Every object in the universe is either a spin=0, 1/2, 1, 3/2, 2,… object. For the integer spin objects, the rotation group is O(3), and the rotation matrices contain only real numbers. However, there are half integer spin particles in the world, ...

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I suspect that they were lucky that their predictions agreed with reality so closely, but any prediction was going to have Neptune roughly (perhaps very roughly) in the same direction as Uranus, during the times when it affects Uranus the most. So I suspect their calculations meaningfully ruled out large swathes of sky, which improved odds of finding it.

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Although this may not be what you're looking for... They weren't "simply lucky." In fact, they didn't use Bode's law at all- they used calculations based on Neptune's supposed gravitational effect on Uranus. In fact, had the two used Bode's law, they would never have found Neptune, as the Bode "law" would predict a completely different location. (This is ...

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To avoid to decide if his derivative $\dot u(x)$ is a covariant or a contravariant object (or perhaps to go for the contravariant one). Seriously. Of course not rigorously, nor even formally. Duality will enter scene in the XIXth-XXth centuries. We got used to integrate a density across a path, or to multiply vector and covectors from the tangent and ...

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Because atoms do not show the linear Stark effect. As atoms have no permament electric dipole moment the leading order effect is the quadratic Stark effect, which is supressed by another factor of the field strength (which is already small compared to the atomic field strengths). Early precision spectrometry was based entirely on atoms in electric discharge ...

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The kilogram is the base unit of mass because electrical engineers in the late 19th century chose a particular set of practical electrical units. These practical units were a success, and we are still using them today: ohm, volt, ampere, and the joule. In 1874 the mechanical units cm, g, s ('CGS') were adopted as the coherent system of units for science. ...

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You seem to have a lot of questions, and other responses don't really answer the core so well. Why is Fermat's principle true? How did Fermat know it? Assume that you have any medium satisfying the wave equation, $v^2 \nabla^2 f = \ddot f$. This holds for taut strings, for light in the Maxwell equations, for vibrations on a drum, etc. Then it turns out ...

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Historically, conservation of energy may have been discovered by Julius Robert Mayer in 1842. When he was a youngster he tried to build a water wheel that drove an Archimedean screw to lift water back up to the top of the wheel and keep it turning. He found this to be impossible. The lesson stayed with him 'til later in life he became a doctor and studied ...

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Can I attribute the conservation of mechanical energy to the power of maths, especially vector calculus? No. Is this the true story behind the process during which people discover conservation of mechanical energy? No, energy conservation was historically a long debate over centuries. Is there a better way of arriving at the conclusion? For ...

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