A mass M is split into two parts, m and M - m, which are then separated by a certain distance. What ratio m/M maximizes the magnitude of the gravitational force between the parts?
F = G m (M -m)/r^2
maximize
m(M-m) where M is a constant total mass
y = m(M-m) = M m - m^2
dy/dm = M - 2 m
that = 0 when
M = 2 m
m = M/2
so split it in two
2m
To determine the ratio m/M that maximizes the magnitude of the gravitational force between two parts, we need to find the point where the force is the greatest.
The gravitational force between two masses is given by the equation:
F = (G * m1 * m2) / r^2
Where:
- F is the gravitational force
- G is the gravitational constant
- m1 and m2 are the masses
- r is the distance between the two masses
In this case, the two parts of mass M are separated by a certain distance, let's call it d. So, we have:
- m1 = m
- m2 = M - m
- r = d
Substituting these values into the gravitational force equation, we get:
F = (G * m * (M - m)) / d^2
To find the ratio m/M that maximizes the force, we can differentiate the equation with respect to m and set it equal to zero:
dF/dm = (G * (M - 2m)) / d^2 = 0
Solving for m:
M - 2m = 0
2m = M
m = M / 2
Therefore, the ratio m/M that maximizes the magnitude of the gravitational force between the parts is 1/2.