Question

The center of a moon of mass m is a distance D from the center of a planet of mass M. At some distance x from the center of the planet, along a line connecting the centers of planet and moon, the net force on an object will be zero. (a) Derive an expression for x in terms of m, M, and D. (b) If the net force is zero a distance 2/3D from the planet, what is the ratio R of the mass of the planet to the mass of the moon, M/m?

Answer 1

To find the point x where the net gravitational force is zero, gravitational forces due to the planet and moon are equated and the equation solved for x. When x is 2/3D from the planet, substituting into the equation allows for solving the mass ratio R = M/m.

Step-by-step explanation:

Derivation of Expression for x and Calculation of Mass Ratio

To derive an expression for x, the distance from the center of the planet at which the net force on an object is zero due to the gravitational pull of both a moon and the planet, we need to set the gravitational forces by the two bodies equal at point x. This results in the equation Gm/(D - x)^2 = GM/x^2, where G is the gravitational constant, m is the mass of the moon, M is the mass of the planet, and D is the distance between the centers of the moon and the planet.

The equation can be rearranged to find x in terms of m, M, and D. If the net force is zero a distance 2/3D from the planet (x = 2/3D), we substitute x into the equation and solve for the mass ratio R = M/m.