newtons law of gravitation says that the magnitude F of the force exerted by a body of mass m on a bod of mass M is F=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 te minue sign mean?
b)suppose it is known that the earth attracts an object with a force that decreases at a rate 2N/km when r=20,000km?
Do you proof your typing? You should. Part b) makes no sense
Part a) requires some thinking. What is a gradient? What happens when one is going in the negative direction?
I will be happy to critique your thinking on this.
To find dF/dr, we need to differentiate the equation F = GmM/r^2 with respect to r. Before going into the differentiation, let's review what dF/dr represents and what its meaning is.
a) Understanding dF/dr:
dF/dr represents the rate of change of the force F with respect to the distance r. It is essentially the derivative of the force function with respect to the distance function.
b) Interpreting the minus sign:
The minus sign in dF/dr implies that the force exerted by the body of mass m on the body of mass M is inversely proportional to the square of the distance between them. This means that as the distance r increases, the force decreases. The minus sign indicates that the force is attractive, pulling the bodies towards each other.
Now let's calculate dF/dr using the given formula F = GmM/r^2.
Differentiating F = GmM/r^2 with respect to r:
dF/dr = d/dx(GmM/r^2)
= -2GmM/r^3
b) Calculating the given rate of change:
We are given that the force decreases at a rate of 2N/km when r = 20,000 km.
To calculate this, substitute the values into the derived formula:
-2GmM/r^3 = -2(6.674 × 10^(-11))(m)(M)/(20000^3) = -2N/km
Simplifying the equation:
-2(6.674 × 10^(-11))(m)(M)/(20000^3) = -2N/km
Now you can solve the equation to find the values of m and M.
Please note that this step eliminates the constant G from the equation as it is not given in the problem statement.