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 the following.

F=(GmM)/r^2
Here G is the gravitational constant and r is the distance between the bodies.
(a) Find dF/dr

(b) Suppose it is known that Earth attracts an object with a force that decreases at the rate of 2 N/km when r = 30,000 km. How fast does this force change when r = 15,000 km?

dF/dr=-2GMm/r^3=-2F/r

b) dF/dr=-2 when r=30,000
but the relationship is inverse, so halving the distance will double the rate.

16

To find dF/dr, we need to differentiate the equation F = (GmM)/r^2 with respect to r.

(a) Find dF/dr:
To differentiate F = (GmM)/r^2 with respect to r, we can use the power rule of differentiation.

Differentiating (GmM)/r^2 with respect to r:
dF/dr = d/dx((GmM)/x^2)
= -2(GmM)/r^3

Therefore, dF/dr = -2(GmM)/r^3.

(b) To find how fast the force changes when r = 15,000 km, we can substitute the given data into dF/dr.

Given:
dF/dr = -2 N/km
r = 30,000 km

Substituting these values in the equation:
-2 N/km = -2(GmM)/(30,000 km)^3

Now we can solve for GmM:

-2(GmM)/(30,000 km)^3 = -2 N/km

Dividing both sides by -2:
(GmM)/(30,000 km)^3 = 1 N/km

Multiplying both sides by (30,000 km)^3:
GmM = (30,000 km)^3 N

Now we can use this value to find how fast the force changes when r = 15,000 km:

Given:
r = 15,000 km

Substituting the new value of r into the equation:
-2(GmM)/(15,000 km)^3 = dF/dr

Simplifying further:
-2(GmM)/(15,000 km)^3 = dF/dr

Therefore, the force changes at the same rate of -2 N/km when r = 15,000 km.

To find dF/dr, we need to calculate the derivative of the gravitational force equation with respect to r.

(a) First, let's differentiate the equation F = (GmM)/r^2 with respect to r.

Using the power rule, the derivative of r^2 is 2r.

So, dF/dr = -2(GmM)/r^3.

(b) To find how fast the force changes when r = 15,000 km, we substitute this value into the expression for dF/dr.

Let r = 15,000 km.

Using the given information that the force decreases at a rate of 2 N/km when r = 30,000 km, we have:

dF/dr = -2 N/km.

Substituting r = 15,000 km into dF/dr, we get:

dF/dr = -2(GmM)/(15,000)^3 N/km.

Now we can calculate the value for dF/dr by evaluating the expression.