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?
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.