What is the relationship between force and distance in planetary motion?

[rudransh verma]

https://www.feynmanlectures.caltech.edu/I_09.html
9-7
"From this figure we see that the horizontal component of the force is related to the complete force in the same manner as the horizontal distance x is to the complete hypotenuse r, because the two triangles are similar. Also, if x is positive, Fx is negative. That is, Fx/|F|=−x/r, or Fx= −|F|x/r= −GMmx/r3. Now we use the dynamical law to find that this force component is equal to the mass of the planet times the rate of change of its velocity in the x-direction".

I don't understand when the ratio of corresponding magnitudes are equal for similar triangles why is it taking -ve sign with x?

 

[kuruman]

I don't understand when the ratio of corresponding magnitudes are equal for similar triangles why is it taking -ve sign with x?

Look at the drawing. The force F_x points to the left (is -ve) whilst x is +ve. Now imagine the planet being on the other side of the y-axis at the mirror-image point. In this case F_x points to the right (is +ve) whilst x is -ve because it on the negative side. The -ve sign in front of x/r makes sure that the gravitational force is attractive and points in the right direction regardless of whether x is +ve or -ve.

 

[rudransh verma]

Look at the drawing. The force F_x points to the left (is -ve) whilst x is +ve. Now imagine the planet being on the other side of the y-axis at the mirror-image point. In this case F_x points to the right (is +ve) whilst x is -ve because it on the negative side. The -ve sign in front of x/r makes sure that the gravitational force is attractive and points in the right direction regardless of whether x is +ve or -ve.

But mathematically speaking we cannot put -ve sign. We are just using the property of similar triangles. One ratio is not equal to -ve of another ratio.

 

[kuruman]

But mathematically speaking we cannot put -ve sign. We are just using the property of similar triangles. One ratio is not equal to -ve of another ratio.

You asked and I replied. Similar triangles can be used to establish relations between magnitudes without reference to signs. This doesn't mean that we are prohibited to put a -ve sign where it belongs. Here, we are describing a physical situation using the language mathematics. Therefore, we are perfectly entitled to put -ve signs where they are needed in order to match the mathematical description to the observed behavior of the system.

 

[rudransh verma]

The -ve sign in front of x/r makes sure that the gravitational force is attractive and points in the right direction regardless of whether x is +ve or -ve.

we are describing a physical situation using the language mathematics.

Okay! Thanks.
Like hookes law it is also a law.