Newtonprincipia _ the centripetal force of the Moving Object

Newtonprincipia _ the centripetal force of the Moving Object

 

An obvious difference between programming and mathematics is:
Each variable of the program must depend on a specific number,
We only need to know that a mathematical variable is a symbol;
The variables in the program must be discrete,
The variables in mathematics are continuous and can correspond to the vertices on the number axis one by one.

 

There are few formulas in the description below, and they are full of proportions.
I have heard that mathematics is a tool for Dummies,
And those smart people can find a solution in the most primitive way.
Let's take a look at how the centripetal force formula was derived from the apple-attacked Isaac in the 17th century.
In my opinion, many of the sub-components without symbols are involved.
The following content only describes the content of the centripetal force reversal formula, and others are skipped.

 

Mathematical Principles of Natural Philosophy
Volume 1: object motion
Part I: methods used to prove the initial ratio and final Ratio

Theorem I:

The ratio of all quantities and quantities tends to be zero in any limited time.
Before the end of the time, they are closer to each other than any difference, and eventually equal.
Explanation: If they do not want to wait, set the final difference to D.
Then, they cannot be closer to the given difference D, which is opposite to the assumption.

 

Theorem vi

If the string AB corresponding to any arc ACB given by the position,
And at a point A, it is in a continuous curvature space,

It is tangent to a line extending in two directions;
After that, point A and B are close to each other and overlap,
It is contained between the string and the tangent, reduced to infinity, and eventually disappears.
If the angle does not disappear, the angle contained in the arc acb and the tangent ad is equal to a straight line angle,
Therefore, the curvature is not continuous at point A, which is in conflict with assumptions.

Theorem VII

Under the same assumption, the final ratios of arc, string, and tangent are equal to each other.
When B is close to A, B and D are always considered to be extended to B and D in the distance, leading to BD parallel to the Section BD.
The angle dab disappears from the margin VI, so it is equal.
Therefore, the straight line AB, AD, and ACB in proportion to [AB, AD, and ACB disappear.
And they eventually have an equivalent ratio, which is evidenced by this.

 

Theorem XI

In all curves with finite curvature of the cut point, the right side of the cut point is calculated based on the quadratic ratio of the opposite side of the arc.

Situation:

Set AB as the arc, the tangent is AD, the right edge BD of the tangent is perpendicular to the tangent, And the right edge of the arc is AB
As a straight line Ag perpendicular to the opposite edge AB and the tangent ad, BG is handed over to G.
Then, point D, B, and g are near points D, B, and G,
Set J to the intersection of BD and Ag when vertex D and B finally reach,
Obviously, gj can be smaller than any assigned distance.
Because the triangle Abd is similar to the triangle gab, so AB is an equivalent item of BD and GA,
Therefore, square (AB): Square (AB) = (AG: Ag) * (BD: BD ).
Because gj can be less than any assigned distance,
Therefore, AG: Ag can be the same-value ratio. The difference is smaller than the difference given by any given difference.
Therefore, the final difference between square (AB): Square (AB) and BD: BD is smaller than any given difference.
According to the theorem I, the final ratio of Square (AB) and square (AB) is the same as that of BD and BD.
That is, evidence.

 

Part II: centripetal force
Proposition I, theorem I

Area, which is drawn by the radius of an object moving on the track to the center of the motionless force,
Stay on a non-moving plane and proportional to the time

Time is divided into equal segments, and in the first segment, the object draws a straight line AB due to its inherent motion.
In the second period, if the same object has no obstacle, it will reach the cpoint,
Draw the line BC that is equal to AB, and direct the, B, and C directions to the center of the same line as, BS, and CS
The area ASB and ASC are equal.
However, when an object reaches B, it is assumed that the centripetal force has a powerful impact,
As a result, the object is tilted from a straight line BC and moved forward on the straight line BC,
The second time period is completed, and the object is found in C, which is in the same plane as the triangle ASB.
Connection SC. Because Sb and CC are parallel, the area SBC is equal to the area SBC, so it is equal to the area SAB.
Similarly, if the centripetal force has successively played a role in C, D, E, and so on,
Allows objects to draw straight lines such as CD, de, and EF in their respective time segments on the same plane.
The area SBC, SCD, SDE, and SEF are equal to each other.
Therefore, an equal area is drawn on the same plane at the same time,
And through combination, any area between SADS and SAFS is like drawing their time.
Now, the number of triangles increases infinitely and the width is reduced to infinity,
In the end, their weekly-line ADF is a curve.
Therefore, centripetal force:
The object is pulled back from the cutting line of the curve continuously;
The SAFS is proportional to the total SAFS of any area drawn, which is evidenced by this.

=== System 2

If AB and BC are the same object in the no-resistance Space
Draw successive arc strings in equal time to supplement the parallelogram abcv,
Then this diagonal line BV is where the arc is reduced to infinity,
Extends along two directions through the center of the Force.

=== Department 3

If the string AB, BC, de, and EF make the object in a non-resistance Space
Draw the arc string at the same time and supplement the parallelogram abcv and defz,
In the ratio of the forces of B to E, when those arcs are reduced to infinity,
According to the final ratio of the diagonal BV and ez.

=== System 4

Any object is pulled from a straight line in a space without resistance
And the ratio of various forces bent to the curve orbit,
As shown in the ratio of the arc vector drawn in the image and other time.
When the arc is reduced to infinity, the vector of the arc gathers at the center of the force and splits the string equally.
The vector here is half of the diagonal line mentioned in system 3.

 

Proposition IV, theorem IV

Various Objects draw different circles with equal motion, and the centripetal force tends to the center of the circle;
And the square of the arc drawn in the same time is divided by the radius of the circle.
The system of proposition I 2 shows that these forces tend to the center of the circle;
From the system of proposition I 4, they are like the positive vector of the arc drawn in a very short period of time.
Let's say XI, as if the square of the string corresponding to the arc is divided by the diameter of the circle,
Let's use the theorem VII, as if the square of the arc is divided by the diameter of the circle.
Therefore, because these arcs are drawn at any equal time,
And the diameter is like its radius, so,
The force is like dividing the square of any arc drawn at the same time by the radius of the circle.
This is the evidence.

=== System 1

Because the arc is like the speed of an object,
The secondary Proportional Ratio of the centripetal force based on the given velocity is inversely proportional to the radius.

=== System 2

Also, because the cycle time is proportional to the radius and the speed is inversely proportional to the composite ratio,
The centripetal force is a combination of proportional ratios from the radius and quadratic inverse ratios of the cycle time.

=== Department 3

Therefore, if the cycle time is equal, the speed is like the radius, and the centripetal force is like the radius. And vice versa.

=== System 4

If both the cycle time and speed are based on the 1/2 ratio of the radius, the centripetal force is equal to each other, and vice versa.

According to system 1, we can know that when the ratio of speed is 1/2 times of the radius,
With the same centripetal force, the cycle time calculated at this time
Is the Combination ratio between the proportional radius and the inverse speed,
Therefore, the cycle time is based on the 1/2 ratio of the radius.

=== Department 5

If the cycle time is like a radius, and the velocity is equal, the centripetal force is inversely proportional to the radius, and vice versa.

 

 

... Centripetal force has been discussed here...

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Newtonprincipia _ the centripetal force of the Moving Object