Final answer:
The distance x from the center of the planet where the net force on an object is zero can be found by equating the gravitational force exerted by both the planet and the moon and solving for x. When the masses used are the same, x is equal to the distance D divided by 2.
Step-by-step explanation:
To find the distance x from the center of the planet where the net force on an object due to gravity would be zero, we invoke the concept of gravitational forces equal and opposite at that point. The gravitational force exerted by the planet on the object is given by Newton's law of gravitation as Fp = G * M * m / (x^2), and the force exerted by the moon on the object is Fm = G * m * M / ((D-x)^2). We want the forces to be equal such that Fp = Fm, thus G * M * m / (x^2) = G * m * M / ((D-x)^2). Simplifying, we have M / (x^2) = M / ((D-x)^2). From here, we can solve for x, x^2 * (D-x)^2 = M / M, which simplifies to x = D/2 given that both masses of the planet and the moon are equal for this specific balance point. Therefore, if the values of mass for the moon and planet were to change, the expression for x would need to be adjusted accordingly.