How do I develop physical intuition for the tidal force?

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SUMMARY

The discussion focuses on developing physical intuition for tidal forces, specifically relating to the 1/r^3 term in the Taylor series expansion of gravitational force. The user emphasizes that traditional Taylor series do not provide the necessary physical understanding. Instead, they propose a conceptual approach that considers the 'stretching' effect on two radially-opposite ends of a smaller body of length 'l' when influenced by a much larger mass 'M' at a distance 'R', where R is significantly larger than l.

PREREQUISITES
  • Understanding of gravitational force and its mathematical representation
  • Familiarity with Taylor series expansions
  • Basic knowledge of physical concepts related to force and distance
  • Concept of mass and its influence on gravitational interactions
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  • Explore the mathematical derivation of tidal forces in gravitational physics
  • Study the implications of the 1/r^3 term in gravitational force calculations
  • Investigate the concept of stretching forces in physics
  • Learn about the effects of distance on gravitational interactions in astrophysics
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Students of physics, educators teaching gravitational concepts, and anyone interested in deepening their understanding of tidal forces and gravitational interactions.

Simfish
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I know that it corresponds to the 1/r^3 term in the Taylor series expansion of the gravitational force. But Taylor series expansions can't give me any physical intuition. By physical intuition, I mean that I want to know why the coefficients for the 1/r^3 term are the way they are.
 
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There is a better way (in my opinion) to think about it:
Consider the 'stretching' (force difference) on the two radially-opposite ends of a body (say, length 'l') near another far more massive body (mass 'M'). Make the assumption that the distance between the two objects ('R') is much larger than the size of the smaller body; e.g. l << R
 

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