Analytical Proof of Newton's Force Laws

By George J. Spix

1 Introduction

Many students intuitively assume that Newton's inertial and gravitational force laws, and , are true since they are clear and simp le. 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 di stance, 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, a nd 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 Laws

In 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 q, within an orthogonal coordinate system

      2.  

       

    2. Differentiate planet position functions to obtain planet velocities, .
    3. Differentiate planet velocities to obtain planet accelerations, .
    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 i n 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, , to assumed planet radial, , and transverse, , accelerations.
    6. Replace the time dependent term, , in the expression of , with a function of r.
    7. Replace the time dependent term, , in the expression of , with a function of r and q.< /LI>
    8. Solve the differential equation containing r as a function of q. 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 q

This 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.

Radius vector, r, is attached to planet, P, and varies in length as P moves. Also, angle q 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 t he planet.

Determine the x and y positions of P, as a function of r and q, by using the trigonometric functions that are indicated by Figure 1.

The x distance of P from the origin; Px = < I>r cos q.

The y distance of P from the origin; Py = r sin q.

As time passes, P moves along its curve, making r and q dependent upon time, t. The positions of P as functions of time are indicated as;

This completes step 1.

 

 

 

2.2 Planet Velocity in x and y Directions

Figure 2 indicates that the change in x and y distances is a function of both r and q as P moves in time along its path.

 

The velocity of the planet, P, is the change of distance along the curve per the change in time. Or,

The calculus expression for velocity in the x direction, as thechange in time is made very small is;

Velocity in the y direction is;

 

 

Therefore, the velocity of P in the x direction is;

And the velocity of P in the y direction is;

 

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 derivative of the first.

Also, the derivative of the sine of an angle is the cosine of the angle times its derivative, and the derivative of the cosine of an angle is minus the sine of the angle times its derivative.

Using these differentiation rules;

The expression for the velocity of P in the x direction is,

 

The expression for the velocity of P in the y direction, following the same rules, is:

This completes step 2.

 

2.3 Planet Acceleration in x and y Directions

The next step is to obtain expressions for the planet accelerations in the x and y directions indicated in Figure 3.

 

Recall that acceleration is the rate of change of velocity.

Let the acceleration of the planet in the x direction be ax.

Let the acceleration of the planet in the y direction be ay.

Then;

and

Finding acceleration causes us to take the derivative, with respect to time, of velocity. Velocity, in turn, is the derivative of distance with respect to time. Therefore, acceleration is the second derivative of distance with respect to time. The deri vative of a derivative is called the second derivative. The symbol for the second derivative, in this case is;

 

Replace vx and vy with their derived expressions listed in Section 2.2. Follow the same differentiation rules as given above and obtain:

This completes step 3.

Accelerations ax and ay must be used to obtain expressions for the radial and transverse accelerations of the planet in Step 5.

 

2.4 Equate Gravitational Force to Planet Inertial Force

Newton's force of gravity law as applied to earth mass, m, and sun mass, M, is;

Where r is the radius vector, the varying distance from earth to sun, and G is the gravitational cons tant.

FGravitational is the amount of force that acts in a straight line between the planet and sun. This force would place the planet in free fall toward the sun if it were not for the counteracting planet inertial force.

What is the inertial force on the planet?

Newton’s inertial force law states that the inertial force is equal to the acceleration of the planet times the mass of the planet.

 

 

 

The inertial and gravitational forces must be equal to each other in magnitude but opposite in direction, or else the planet would leave its orbit. With unequal forces, the planet would fall into the sun, or attain a different orbit in a new equilibriu m path, or go spinning off into space. Since the planet does maintain its orbit, the sum of the two forces must be zero.

 

Divide through by m and obtain;

Then;

This is an important place in the proof where the inertial mass is assumed to be identical to gravity mass and the radial acceleration is shown opposite to the attraction of gravity. We must continue to be skeptical of these assumptions, including dist ance to the second power, until we derive the elliptical path of the planet around the sun.

This equation of the radial acceleration shows that aRadial is proportional to the inverse of distance squared.

By applying some mathematics we will modify the equation to obtain aRadial as a function of r and q. This radial acceleration equation is the basic equation that will evolve into the equation showing that the earth orbit is an ellipse.

 

Notice also that Newton's inertial force law can be considered simply as the definition of the unit of force. Once the standards of kilogram, meter and second are agreed upon, the unit of inertial force is established. We need a constant, (G), to make the gravitational units of force have the same dimensions and the same magnitude as inertial force units. But we have no reason (as yet), to believe that the inertial force, based on random but agreed upon standards, is directly proportional to the gravit ational force. We just assumed the equivalence when we canceled "m" in the above derivation. If the path of the earth around the sun is analytically determined to be an ellipse, then the assumption is correct.

This completes step 4.

The next part of this proof to find expressions for radial and transverse planet accelerations in step 5 of the procedure.