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

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Discussion Overview

The discussion revolves around the relationship between force and distance in the context of planetary motion, particularly focusing on the mathematical representation of gravitational force and the use of similar triangles in deriving relationships between force components and distances. The scope includes theoretical and mathematical reasoning.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants reference the relationship between the horizontal component of force and distance using similar triangles, noting that the force component is expressed as Fx = -GMmx/r³.
  • There is a question regarding the necessity of the negative sign in the expression for Fx, with one participant suggesting that it is not mathematically justified to include a negative sign when discussing ratios of similar triangles.
  • Another participant argues that the negative sign is essential to ensure that the gravitational force remains attractive, regardless of the sign of x, and that it aligns the mathematical description with the physical behavior of the system.
  • Some participants emphasize that while similar triangles can establish relationships between magnitudes, the inclusion of negative signs is necessary for accurately describing physical situations.

Areas of Agreement / Disagreement

Participants express differing views on the mathematical justification for the negative sign in the context of similar triangles and gravitational force. There is no consensus on whether the negative sign is necessary or appropriate in the mathematical formulation.

Contextual Notes

The discussion highlights limitations in the mathematical treatment of physical concepts, particularly regarding the interpretation of signs in equations and their implications for physical behavior.

rudransh verma
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https://www.feynmanlectures.caltech.edu/I_09.html
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"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?
 
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rudransh verma said:
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 Fx 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 Fx 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.
 
kuruman said:
Look at the drawing. The force Fx 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 Fx 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.
 
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rudransh verma said:
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.
 
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kuruman said:
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.
kuruman said:
we are describing a physical situation using the language mathematics.
Okay! Thanks.
Like hookes law it is also a law.
 
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