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

AI Thread Summary
The discussion highlights the relationship between force and distance in planetary motion, emphasizing how similar triangles help establish this relationship. The horizontal force component, Fx, is shown to be negative when x is positive, indicating the attractive nature of gravitational force. This negative sign ensures that the force direction aligns with the physical behavior of the system, regardless of the position of the planet. Participants clarify that while similar triangles relate magnitudes without signs, the negative sign is necessary for accurate physical representation. The conversation concludes with a comparison to Hooke's Law, reinforcing the idea that these principles govern physical interactions.
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.
 
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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