two planets of masses m and respectively.have a centre to center separation of r.at what distance from the center of the pianet of mass m do the gravitational force of planets cancel each other???
Assuming you meant masses M and m, then we need at distance d from M,
M/d^2 + m/(r-d)^2 = 0
M(r-d)^2 + md^2 = 0
(M+m)d^2 - 2Mrd + Mr^2 = 0
d = r/2 * (M+√Mm)/(M+m)
just as a sanity check, it's easy to see that d=r/2 when M=m
To find the distance from the center of the planet of mass m where the gravitational forces of the two planets cancel each other, we can use the concept of gravitational force.
The gravitational force between two objects can be calculated using Newton's law of universal gravitation:
F = G * (m1 * m2) / r^2
Where:
F is the gravitational force between the two planets
G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
m1 and m2 are the masses of the two planets
r is the distance between the centers of the two planets
Since the two planets have the same mass m, we can rewrite the formula as:
F = G * (m * m) / r^2
Now the problem states that we want to find the distance from the center of the planet of mass m where the gravitational forces cancel each other. This means that the gravitational force between the two planets should be equal to zero:
0 = G * (m * m) / r^2
To solve for r, we rearrange the equation:
r^2 = G * (m * m) / 0
Since the denominator is zero, we conclude that the gravitational forces between the two planets cannot cancel each other out, and there is no specific distance from the center of the planet of mass m where this occurs.
Therefore, the gravitational forces of two planets can never cancel each other out at any specific distance from the center of the planet of mass m.