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Problem 20 Hard Difficulty

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 = \frac {GmM}{r^2} $
where $ G $ is the gravitational constant and $ r $ is the distance between the bodies.
(a) Find $ dF/dr $ and explain its meaning. What does the minus sign indicate?
(b) Suppose it is known that the earth attacks an objects an object with a force that decreases at the rate of 2 N/km when

$ r = 20,000 km. $ How fast does this force change when $ r = 10,000 km? $

Answer

a) $\frac{d F}{d r}=\frac{-2 G m M}{r^{3}}$
b) 16 $\mathrm{N} / \mathrm{km}$

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Video Transcript

here we have an equation that represents the force as a function of the distance between the bodies and we want to find D FDR. So the FDR would be the rate of change of the force as a function of the distance. So that would tell us how the force is changing as a result of the change in the distance. So let's rewrite the function as G m M times are to the negative second power. And now we confined its derivative by bringing down the negative, too. G m and M are constants, and we have our to the negative. Third, we can write rewrite this as negative to G m m over r cubed. All right, So for Part B, we're interested in figuring out the rate of change for a 10,000 kilometer distance, given some information that we know about a 20,000 kilometer distance. So here's what we know that if r equals 20,000 kilometers, dfd are is negative two. It says it is decreasing at a rate of two. I believe it would be Newtons per kilometre. If that's the unit I'm interpreting correctly. Okay, so here's what we can do we can substitute these numbers into our derivative and that will give us the value of the constant. And then we can use that value as their constant and find the unknown value. So we have negative two for Dft R is equal to negative two g m m. Over 20,000 cubed. Okay, now we can divide both sides by negative too. And that will just be a one. And then we can multiply both sides by 20,000 cubed. So we have 20,000 cubed equals GMM. Now let's use that to solve for the rate of change of the force when the distance is 10,000. So now we have rate of change of force with respect to radius is negative two times that value. We just got 20,000 cubed. That's our GMM over are cute. We can substitute are 10,000 in for our we can reduce 20,000 cubed over 10,000 cubed, and that's going to leave us with two. Cute. We have negative two times, two cubed and that's negative 16. Now the problem doesn't really require negative because it just asks How fast is the force changing? So the forces changing at 16 Newtons per kilometre