Question

Two planets have the same mass, \(M,\) but one of them is much denser than the other. Identical objects of mass \(m\) are placed on the surfaces of the planets. Which object will have the gravitational potential energy of larger magnitude? a) Both objects will have the same gravitational potential energy. b) The object on the surface of the denser planet will have the larger gravitational potential energy. c) The object on the surface of the less dense planet will have the larger gravitational potential energy. d) It is impossible to tell.

Short answer

Answer: The object on the surface of the denser planet will have the larger gravitational potential energy.

Step-by-step solution

Calculate the radii of the planets

To calculate the potential energy, we need to find the radii of the planets. Let's denote the density of the denser planet as \(\rho_1\) and the less dense planet as \(\rho_2\). The mass of each planet can be expressed as \(M = \rho_1 V_1 = \rho_2 V_2,\) where \(V_1\) and \(V_2\) are the volumes of the planets. For a spherical planet, we have \(V_1 = \frac{4}{3}\pi r_1^3\) and \(V_2 = \frac{4}{3}\pi r_2^3,\) where \(r_1\) and \(r_2\) are the radii of the denser and less dense planets respectively. Thus, using the mass and density, we can determine the radii of the planets: \(r_1 = \sqrt[3]{\frac{3M}{4\pi\rho_1}}\) and \(r_2 = \sqrt[3]{\frac{3M}{4\pi\rho_2}}.\)

Calculate the gravitational potential energy for the objects on each planet

Now that we have the radii for both planets, we can calculate the gravitational potential energy for each object using the formula \(U = -\frac{G\M\times m}{r}.\) For the denser planet, we have \(U_1 = -\frac{G(M)(m)}{r_1}\) and for the less dense planet, we have \(U_2 = -\frac{G(M)(m)}{r_2}.\)

Compare the gravitational potential energies

To compare the gravitational potential energies \(U_1\) and \(U_2,\) note that the magnitudes of the energies are inversely proportional to the radii of the planets (since the other factors are constants). As the denser planet has a smaller radius (\(r_1 < r_2\)), its gravitational potential energy magnitude will be larger, and the less dense planet's gravitational potential energy magnitude will be smaller. Therefore, the correct answer is: b) The object on the surface of the denser planet will have the larger gravitational potential energy.