The universal law of gravitation determines the amount of force exerted by a constant mass (M) on another constant mass (m), separated by a distance (r), is given by the expression F = ( -GmM )/r^2
a) What mathematical expression characterises the instant rate of change of the force F in respect to the distance (r) separating the two masses moving towards each other?
I got dF/dt = 2GmM/r^3 dr/dt
b) What's the rate of change?
No idea what to do here, they don't give me any numbers.
To find the rate of change of the force F with respect to the distance r, you need to take the derivative of the force equation with respect to r. So let's differentiate the expression F = (-GmM)/r^2 with respect to r.
a) Find the derivative of F with respect to r:
dF/dr = d/dt[(-GmM)/r^2]
To find dF/dr, we need to use the chain rule of differentiation. First, differentiate (-GmM)/r^2 with respect to r, then multiply it by the derivative of r with respect to time (dr/dt).
Let's start by differentiating (-GmM)/r^2 with respect to r:
d/dt[(-GmM)/r^2] = (GmM)(d/dr)[1/r^2]
Now we differentiate 1/r^2:
(d/dr)[1/r^2] = (-2)/r^3
Now we have:
dF/dr = (GmM) * (-2)/r^3
To find the rate of change of F with respect to time, we multiply dF/dr by dr/dt:
dF/dt = (GmM) * (-2)/r^3 * dr/dt
So, the mathematical expression characterizing the instant rate of change of the force F in respect to the distance r is:
dF/dt = (GmM) * (-2)/r^3 * dr/dt
b) To determine the rate of change, you would need to have information about the values of G (gravitational constant), m, M, r, and dr/dt (the rate at which the distance between the two masses is changing). Without these values, it is not possible to determine the exact rate of change.