A planet of mass 81M and its moon of mass M are separated by a distance 10d . A spaceship of mass m is parked on the line joining them at a point where the net gravitational force on it is zero. What is the gravitational potential energy of the spaceship at this position?

Step-by-step solution

Let the spaceship be at a distance x from the planet. The net gravitational force on it is zero, so the gravitational fields of the planet and the moon must be equal in magnitude at this point. G(81M)m x^2 = GMm (10d - x)^2 Taking the square root of both sides: 9 x = 1 10d - x 90d - 9x = x 10x = 90d x = 9d The spaceship is at a distance of 9d from the planet and 10d - 9d = d from the moon. The gravitational potential energy U of the spaceship is the scalar sum of the potential energies due to both bodies: U = - G(8