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?
F = -GmM/r^2
dF/dt = 2GmM/r^3 dr/dt
Note that as the distance decreases, the force increases in magnitude.
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To find the mathematical expression for the instantaneous rate of change of the force (F) with respect to the distance (r), we need to take the derivative of the force equation with respect to r.
Given the force equation:
F = (-GmM) / r^2
To find dF/dr, we can differentiate both sides of the equation with respect to r:
dF/dr = d/dx [(-GmM) / r^2]
Using the quotient rule of differentiation, which states that the derivative of a quotient is the derivative of the numerator times the denominator minus the derivative of the denominator times the numerator, divided by the square of the denominator.
Applying the quotient rule in this case, we have:
dF/dr = [(-2GmM) / r^3]
Therefore, the mathematical expression characterizing the instant rate of change of the force F with respect to the distance r is:
dF/dr = [(-2GmM) / r^3]