Assumption on central forces between two particles

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SUMMARY

The discussion centers on Newton's force law for two particles interacting through a central force, specifically examining the implications of Newton's third law, F12 = -F21. Participants explore the requirements for Newtonian mechanics to be invariant under Galilean transformations, emphasizing the need for specific assumptions about the nature of the forces involved. Key assumptions include the equality of forces and their opposite directions, as well as the restriction of motion to a single axis due to the two-particle system. The conversation highlights the necessity of identifying these assumptions to fully address the problem presented.

PREREQUISITES
  • Understanding of Newton's laws of motion
  • Familiarity with Galilean transformations
  • Basic knowledge of vector forces and their properties
  • Ability to analyze motion in a two-particle system
NEXT STEPS
  • Study the implications of Newton's third law in multi-particle systems
  • Research the mathematical formulation of Galilean transformations
  • Explore the concept of force invariance in different reference frames
  • Examine case studies involving central forces in classical mechanics
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Students of classical mechanics, physics educators, and anyone interested in the foundational principles of Newtonian physics and their applications in particle dynamics.

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Homework Statement



Consider Newton’s force law for two particles interact through a central force F12(r1',r2',u1,u2), where by Newton’s third law F12 = -F21.

m1(d^2r1/dt^2) = F12(r1,r2,u1,u2)

m2(d^2r2/dt^2) = F21(r1,r2,u1,u2)

A. Show that Newtonian mechanics is form invariant with respect to a Galilean transformation?
B. What assumptions about the force must be made?

Homework Equations


Newton's third law:
F12 = -F21

Galilean Transform:
t'=t
m'=m
r'=r-vt

The Attempt at a Solution



I did the A. without a problem. It's the B. that troubles me a bit. I'm sure I'm missing what my prof wants me to find...

I wrote that the forces are equal and have opposite directions. The problem must lie on a single axis since there is only two particles. I'm not sure what other assumption we can make..

Thanks for the help!
 
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When you did part a, you had to make some assumptions.
What were they?

If you did not notice any assumptions, you did not complete part a.
Try going back over your working in more detail... check each step for its reasoning.
Though you may just be expected to list the assumptions for Galilean relativity.
 
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