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.
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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.