Gravitational Force, the derivative of Gravitational Potential Energy?

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Discussion Overview

The discussion revolves around the relationship between gravitational force and gravitational potential energy, specifically questioning whether the gravitational force can be derived as the derivative of gravitational potential energy. The context includes considerations of the pioneer anomaly and its implications on gravitational calculations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents the equation for gravitational potential energy and calculates its derivative with respect to distance, proposing that this derivative represents gravitational force.
  • Another participant questions the relevance of the pioneer anomaly in this context, noting that it refers to an unexplained residual acceleration after accounting for known forces.
  • A participant reiterates the main question about the relationship between gravitational force and gravitational potential energy, suggesting that if this relationship holds, it could provide insights into the truth of the matter.
  • Another participant agrees that gravitational force is the derivative of gravitational potential energy but expresses confusion about the implications of adding the pioneer anomaly to gravitational calculations.
  • There is a suggestion that if the pioneer anomaly exists, it could invalidate previous estimates of the sun's mass, contingent on the assumption that the anomaly is due to gravitational attraction from the sun.
  • A later reply proposes a simplified model for the additional solar mass needed to account for the anomalous acceleration, indicating that the additional acceleration can be expressed in terms of an extra mass.

Areas of Agreement / Disagreement

Participants express differing views on the implications of the pioneer anomaly and its relationship to gravitational calculations. While there is some agreement on the derivative relationship between gravitational force and potential energy, the discussion remains unresolved regarding the effects of the pioneer anomaly.

Contextual Notes

There are limitations in the assumptions made regarding the pioneer anomaly and its impact on gravitational force calculations, as well as the dependence on the definitions of gravitational potential energy and force.

kmarinas86
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For the equation:

[itex]U=\frac{-GMm}{h}[/itex]

Where [itex]h[/itex] is the distance between the center of masses [itex]M[/itex] and [itex]m[/itex].

In Calculus, they teach you derivatives.
The derivative of [itex]U[/itex] with respect to [itex]h[/itex] is:

[itex]dU=d\left(\frac{-GMm}{h}\right)[/itex]
[itex]dU=\frac{GMm}{h^2}[/itex]

Which is the gravitational force.

Were I to apply this knowledge to the pioneer anomaly, I would deduce that the gravitational potential energy would be equal to the integral of the force with respect to [itex]h[/itex]:

[itex]g_{pioneer}=8.74*10^{-10}\frac{m}{s^2}[/itex]
[itex]dU=\frac{GMm}{h^2}+mg_{pioneer}[/itex]
[itex]dU=d\left(\frac{-GMm}{h}+mg_{pioneer}h\right)[/itex]
[itex]U=\frac{-GMm}{h}+mg_{pioneer}h[/itex]

Are my premises true?
 
Last edited:
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It's not clear what you are trying to do. For one thing, the pioneer anomaly refers to an unexplained residual acceleration after all known forces (like gravity from known masses) have been accounted for.
 
Main point:
Is the Gravitational Force the derivative of Gravitational Potential Energy?

A motive:
In case if this is correct, this would be provide information of its truth.

Secondary (following) point:
Is the main point still true if the pioneer anomalous acceleration is added into the acceleration due to gravity? Note that if the pioneer anomaly exists, it invalidates previous estimates of the sun's mass.

A motive:
In case if this is false, this would be provide information of its falsity.
 
Last edited:
kmarinas86 said:
Main point:
Is the Gravitational Force the derivative of Gravitational Potential Energy?
Yes.

Secondary (following) point:
Is the main point still true if the pioneer anomalous acceleration is added to the acceleration due to gravity?
Again, no idea what you are doing here.
Note that if the pioneer anomaly exists, it invalidates previous estimates of the sun's mass.
Only if you assume that the unexplained acceleration is due to the gravitational attraction of the sun.

Are you trying to model the additional solar mass needed to account for the anomalous acceleration? If so, no need to work so hard. If the sun had an extra mass [itex]\Delta M[/itex], the additional acceleration would be:
[tex]\Delta g = \frac{G \Delta M}{h^2}[/tex]

where h is the distance from the sun's center to the pioneer.
 

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