Real gravitational potential energy to kinetic energy

Click For Summary

Discussion Overview

The discussion revolves around the relationship between gravitational potential energy and kinetic energy, particularly in the context of varying gravitational forces at different distances from massive bodies. Participants explore the implications of gravitational equations for both small and large bodies, as well as the complexities introduced by finite sizes and non-spherical shapes.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes a gravitational force equation that accounts for the distance from the surface of a spherical body, suggesting that the force changes during free fall.
  • Another participant challenges the correctness of the integral used by the first participant, asserting that the integral of the proposed function is incorrect.
  • A later reply acknowledges a mistake in the initial formulation but maintains that the integral remains incorrect, emphasizing the need for proper definitions when dealing with finite-sized objects.
  • Some participants discuss the necessity of using the distance between the central points of objects rather than their surfaces when calculating gravitational forces.
  • There is a question about the implications of taking limits as distances approach zero, with a participant noting that the behavior of the limits will depend on the manner in which they approach zero.
  • One participant suggests that numerical simulations or multipole expansions may be required to analyze non-spherical bodies like Earth.

Areas of Agreement / Disagreement

Participants express disagreement regarding the correctness of the integral and the definitions used in gravitational equations. The discussion remains unresolved with multiple competing views on how to approach the problem.

Contextual Notes

Limitations include the dependence on definitions of distance and the need for adjustments in equations when considering finite-sized objects. The implications of non-spherical shapes and the behavior of limits are also not fully resolved.

omiros
Messages
29
Reaction score
0
This isn't a homework problem, but something that has being buzzing in my head.

I've been thinking about it cause at huge distances from Earth g changes and also in what happens for really dense objected and small things could change too.

The real equation for gravity no matter which body we are talking about with a spherical symmetry, is Fg=GmM1M2/(r+s)2,
with r = the radius of the main body and s = the distance of the body from the surface of the other one.

Let's consider M1the main one and M2the one that can be viewed as a particle. We know that ΔK = -∫s0Fds , however the force is chaning all through the 'free fall' and makes things more complicated.

The solution that I find is Fg=GmM1M2s/(r2+rs)

Where do I need the help?
I don't know if this is the right solution and I also don't know how to interprete the limiting cases, like limr→0 and lims→0 at the same time(in the center).
What are all the 3 different equations that describe the variance of the force?
Also could someone transform the energy equations if both bodies are 'big' but with different radius(not same r or mass)?
What energy do they have before they collide?
What would happen if they were like earth? (not spherical, with equations)
 
Physics news on Phys.org
Your integral is incorrect. The integral of -1/(r+s)^2 is 1/(r+s)
 
I am sorry. That is my mistake. I meant ΔK not Fgwith limits s and 0
 
Last edited:
This does not change the fact that your integral is wrong.

with r = the radius of the main body and s = the distance of the body from the surface of the other one.
That does not work if both objects have a finite size. The denominator has to be the distance between the central points of the objects. It is convenient to use r for this distance, if you use other definitions you have to adjust the denominator to get the same result.
As long as the spherical objects do not overlap, this formula is true independent of the sizes and distances of the objects.

What would happen if they were like earth? (not spherical, with equations)
You need a multipole expansion or numerical simulations.
 
mfb said:
This does not change the fact that your integral is wrong.

Did you try to solve it before you say it was wrong? If you simplify the two fractions that you get, this is the result.
 
Oh sorry, I was confused as you used s for integration limits and as variable at the same time.
Okay, the result looks good.
Where is the point in taking the limit s->0 (i.e. not lifting the body at all) then? The limit for r,s->0 will depend on the way they go to zero.
 

Similar threads

Undergrad Gravitational potential energy
  • · Replies 46 ·
2
Replies
46
Views
5K
High School Conceptually understanding change in potential energy with 0 net work
  • · Replies 9 ·
Replies
9
Views
2K
Graduate Gravitational potential energy -- Why is it always negative?
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Change of coordinates in a potential energy field
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad How to interpret this definition of potential energy?
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Question About Potential Energy
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad System, potential energy, and nonconservative forces
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Two separate "gravitational potential energy" equations?
  • · Replies 11 ·
Replies
11
Views
3K
Graduate A non-spherical Earth's gravitational potential?
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Gravitational Potential Reference Point
  • · Replies 6 ·
Replies
6
Views
3K