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Analytical Proof of Newton's Force Laws

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1 Introduction2 Summary of Analytical Proof of Newton's Force Laws2.1 Planet Position in Polar Coordinates, r and2.2 Planet Velocity in x and y Directions2.3 Planet Acceleration in x and y Directions2.4 Equate Gravitational Force to Planet Inertial Force2.5 Planet Radial and Transverse Accelerations2.6 Replace Time Dependent Termin aR2.7 Replace Time Dependent Term, , in aR2.8 Obtain r as a Function of  and Confirm Kepler's First Law3 Polar Equation of Conics4 Proof of Kepler's Second Law5 Proof of Kepler's Third Law6 Newton's Analytical Estimate of G7 Two Methods of Calculating Moon Radial Acceleration7.1 First Method of Calculating Moon Radial Acceleration.7.2 Second Method of Calculating Moon Radial Acceleration7.2.1 Conservation of Orbital Energy7.2.1a Development of First Orbital Energy Equation7.2.1b Development of Second Orbital Energy Equation7.3 Relation of Moon Tangential Velocity and Radial Acceleration8 Conclusions Shown by this Analysis of Newton’s LawsAnalytical Proof of Newton’s Force Laws Page 1 Analytical Proof of Newton's Force Laws1 Introduction Many students intuitively assume that Newton's inertial and gravitational force laws, F ma= and ( )F GMmdistance M m=−2, are true since they are clear and simple. However, there is an analysis that ties the two equations together and demonstrates that they must be true. The analysis provides answers to questions such as, "Is the inertial mass exactly the same as the gravitational mass? Why is the exponent of distance, 2, and not 1.99 or 2.01 or 1? Why is a constant required in one law and not in the other?"The ideal way to prove new theoretical laws is to forecast the outcome of an experiment using the laws, perform the experiment, and find that the result is as forecast. But nature had already performed the experiment with planets in the solar system, and Kepler had determined the results. So, Newton, in his 1669 paper, "Mathematical Principles of Philosophy", (now part of the Great Books Series in local libraries), applied his force laws to the solar system and obtained the same results that Kepler had stated. This confirmed Newton's ideas, put physics on a firm mathematical basis and answered the above questions.2 Summary of Analytical Proof of Newton's Force LawsIn the 8 step procedure that follows, Newton's force laws are applied to the planet−sun system, and the planet (earth) path around the sun is shown to be an ellipse. This procedure below uses the mathematics found in first year college texts and explains the mathematics within the derivations as they are being evolved.1. Observations show that the planets follow a smooth curve around the sun. Sketch the planet at position P, using polar coordinates, r and θ , within an orthogonal coordinate system.Page 2 Analytical Proof of Newton’s Force Laws2. Differentiate planet position functions to obtain planet velocities, v vx y and .3. Differentiate planet velocities to obtain planet accelerations, a ax y and .4. Equate Newton's inertial and gravitational force laws as applied to the planet. In this step we assume that inertial mass is identical to gravitational mass and that the force of gravity decreases as the square of distance. The acceleration required in the inertial law is also assumed to be the planet radial acceleration. G must also be assumed to make the equations consistent. The end result of this analytical procedure must show that all these assumptions are correct, or Newton’s equations would not be true.5. Convert accelerations, a ax y and , to assumed planet radial, aR, and transverse, aT, accelerations.6. Replace the time dependent term, ddtθ, in the expression of aR, with a function of r.7. Replace the time dependent term, d rdt22, in the expression of aR, with a function of r and θ.8. Solve the differential equation containing r as a function of θ. The solution is the polar equation of an ellipse. This result is the same as Kepler's determination from astronomical data and analytically proves Newton's force equations.2.1 Planet Position in Polar Coordinates, r and θThis analytical proof of Newton's force laws begins with a planet P, moving along a smooth curve in a polar coordinate system as shown in Figure 1. The planet is moving relative to the stationary sun.Analytical Proof of Newton’s Force Laws Page 3•θPrxy0 yposition of Px position of PFigure 1SunEarthRadius vector, r, is attached to planet, P, and varies in length as P moves. Also, angle θ and its rate of change vary as P moves. Therefore, the velocity and acceleration of P vary continuously as the planet moves along its path. Recall that acceleration, velocity and force have magnitude and direction. Newton had previously proved that, as far as the force of gravity was concerned, the entire mass of the planet and sun can be considered to be at the center of their spheres. The radius vector starts at the center of the sun and ends at the center of the planet. Determine the x and y positions of P, as a function of r and θ, by using the trigonometric functions that are indicated by Figure 1.The x distance of P from the origin; Px = r cos θ.The y distance of P from the origin; Py = r sin θ.As time passes, P moves along its curve, making r and θ dependent upon time, t. The positions of P as functions of time are indicated as;( ) ( ) ( )P t r t tx = cos θ,( ) ( ) ( )P t r ty = sin tθ.This completes step 1.Page 4 Analytical Proof of Newton’s Force Laws2.2 Planet Velocity in x and y DirectionsFigure 2 indicates that the change in x and y distances is a function of both r and θ as P moves in time along its path. •θPrxy0x velocity of P yposition of Py velocity of Px position of PFigure 2The velocity of the planet, P, is the change of distance along the curve per the change in time. Or,change in distancechange in time .st=∆∆The calculus expression for velocity in the x direction, as the change in time is made very small is; ( )ddtPxt .Velocity in the y direction is;( ) ddtP ty.Analytical Proof of Newton’s Force Laws Page 5Therefore, the velocity of P in the x direction is;( ) ( )( )vddtr t tx = cosθ.And the velocity of P in the y direction is; ( ) ( )( )vddtr ty = sin tθ.The calculus rule for obtaining the derivative of the product of two variables is to multiply the first term times the derivative of the second term plus the second times the


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