Gravitational Force, the derivative of Gravitational Potential Energy?

AI Thread Summary
The discussion centers around the relationship between gravitational force and gravitational potential energy, specifically whether the gravitational force is the derivative of gravitational potential energy. The equation for gravitational potential energy is presented, and its derivative with respect to distance shows that it equates to gravitational force. The conversation shifts to the pioneer anomaly, which refers to an unexplained acceleration that could affect calculations of gravitational forces and the sun's mass. The participants question the implications of incorporating the pioneer anomaly into gravitational calculations, noting that if it exists, it may challenge previous estimates of solar mass. Ultimately, the primary assertion that gravitational force is the derivative of gravitational potential energy is affirmed, while the implications of the pioneer anomaly remain uncertain.
kmarinas86
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For the equation:

U=\frac{-GMm}{h}

Where h is the distance between the center of masses M and m.

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

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

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 h:

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

Are my premises true?
 
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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.
 
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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 \Delta M, the additional acceleration would be:
\Delta g = \frac{G \Delta M}{h^2}

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