Derivative of Newton's Law of Gravitation

[Bear_B]

Homework Statement

Newton's Law of Gravitation says that the magnitude (F) of the force exerted by a body of mass (m) on a body of mass (M) is

F = (GmM)/(r^2)

where G is the gravitational constant and r is the distance between the bodies.

Find dF/dr and explain it's meaning. What does the minus sign indicate?

Homework Equations

none

The Attempt at a Solution

using implicit differentiation:
dF/dr = d/dr [(GmM)/r^2]
= [(r^2)d/dr(GmM) - (GmM)d/dr(r^2)] / (r^4)
= [(r^2)G(d/dr(mM)) - (2rGmM)] / (r^4)
= [(r^2)G(m * dM/dr + M * dm/dr) - 2rGmM] / (r^4)

Ok, I know this can be simplified, but that seemed even more convoluted when I did that. I'm really not sure what to do from here. I expected the derivative to be something a little more intuitive and to result in a negative expression based on the question. I would expect dF/dr to equal a negative expression and the significance of the minus sign to be indicative of a force of attraction since the derivative is with respect to distance and the distance is decreasing. Please help me with the derivative, because If I can get that right, then I just need to express what's going on in words.

 

[tiny-tim]

Hi Bear_B!

(try using the X² tag just above the Reply box )

Newton's Law of Gravitation says that the magnitude (F) of the force exerted by a body of mass (m) on a body of mass (M) is

F = (GmM)/(r^2)

where G is the gravitational constant and r is the distance between the bodies.

Find dF/dr and explain it's meaning. What does the minus sign indicate?
…
dF/dr = d/dr [(GmM)/r^2]
= [(r^2)d/dr(GmM) - (GmM)d/dr(r^2)] / (r^4)

ooh, so complicated! no, m and M are constants.

Try again. ​

 

[Bear_B]

Ok, I had seen that result worked out elsewhere when I couldn't get on here. Can you please tell me why m and M are constants? I thought the equation to determine the gravitational force was for an unknown mass m and an unknown mass M at a unknown distance r from each other...knowing that m and M are constants makes the differentiation simple, I just want to understand why I am treating m and M as constants. Thanks

 

[Bear_B]

F = GmM/r²
dF/dr = GmM [ d/dr (r^-2)]
= GmM (-2r^-3)
= GmM (-2/r³)
dF/dr = -2GmM / r³

ok, so here is the answer, but why are m and M constants?

 

[D H]

Because they have nothing to do with r. They are masses.

 

[tiny-tim]

Can you please tell me why m and M are constants? I thought the equation to determine the gravitational force was for an unknown mass m and an unknown mass M at a unknown distance r from each other...knowing that m and M are constants makes the differentiation simple, I just want to understand why I am treating m and M as constants. Thanks

Hi Bear_B!

(just got up :zzz: …)

Yes, m and M are variables just like r,

but they are independent variables (they can be varied separately, without affecting the others).

So when the question asks "Find dF/dr", it means differentiate with respect to r, keeping all other variables constant.

 

[Bear_B]

"Find dF/dr", it means differentiate with respect to r, keeping all other variables constant.

Thanks. Now I understand the concept that the Force (F) is being taken with respect to distance (r) determining the rate of change of F as r changes for two objects with mass m and M. Since dF/dr is asking how F changes when r changes it has to assume that the m and M are constants unless otherwise indicated. I think I get it now. I just wanted a conceptual understanding of the problem beyond how to do the problem mechanically (because that is the easy part).

^^ This is my (basic) understanding of why the derivative assumes m and M as constants when differentiating with respect to r.