Two planets have the same mass, \(M .\) Each planet has a constant density, but the density of planet 2 is twice as high as that of planet \(1 .\) Identical objects of mass \(m\) are placed on the surfaces of the planets. What is the relationship of the gravitational potential energy, \(U_{1},\) on planet 1 to \(U_{2}\) on planet \(2 ?\) a) \(U_{1}=U_{2}\) b) \(U_{1}=\frac{1}{2} U_{2}\) c) \(U_{1}=2 U_{2}\) d) \(U_{1}=8 U_{2}\) e) \(U_{1}=0.794 U_{2}\)

The formula for gravitational potential energy (U) is: \(U = -\frac{GMm}{r}\) Where G is the Gravitational constant, M is the mass of the planet, m is the mass of the object, and r is the distance between the centers of the two masses (which is equal to the radius of the planet in this case).

The mass (M) of a planet can be expressed as a product of density (D) and volume (V): \(M = DV\) Since both planets are spherical, we can express their volumes in terms of their radii (R): \(V = \frac{4}{3}\pi R^3\)

Combining equations from steps 1 and 2, we can write the gravitational potential energy (U) in terms of density (D) and radius (R): \(U = -\frac{Gm}{R}(\frac{4}{3}\pi R^3D)\)

We are given that the density of planet 2 is twice the density of planet 1, and both planets have the same mass. Therefore: \(D_2 = 2D_1\) Since the mass of both planets is the same: \(M_1 = M_2\) And we know that \(M_1 = D_1V_1\) and \(M_2 = D_2V_2\), we get: \(D_1V_1 = D_2V_2\) Now we can substitute \(D_2\): \(D_1V_1 = (2D_1)V_2\) Which simplifies to: \(V_1 = 2V_2\) Since both planets are spheres, their volumes can be expressed in terms of their radii: \(V_1 = \frac{4}{3}\pi R_1^3\) and \(V_2 = \frac{4}{3}\pi R_2^3\) Substitute these expressions into the volume relationship equation: \(\frac{4}{3}\pi R_1^3 = 2(\frac{4}{3}\pi R_2^3)\) From this equation, we can determine the relationship between the radii of the two planets: \(R_1 = 2^{\frac{1}{3}}R_2\)

Now, using the expressions for gravitational potential energy, radii, and densities, we can find the relationship between \(U_1\) and \(U_2\): \(U_1 = -\frac{Gm}{R_1}\left(\frac{4}{3}\pi R_1^3D_1\right)\) \(U_2 = -\frac{Gm}{R_2}\left(\frac{4}{3}\pi R_2^3D_2\right)\) Now, let's substitute the density relationship, \(D_2 = 2D_1\), and the radius relationship, \(R_1 = 2^{\frac{1}{3}}R_2\): \(U_1 = -\frac{Gm}{(2^{\frac{1}{3}}R_2)}\left(\frac{4}{3}\pi (2^{\frac{1}{3}}R_2)^3D_1\right)\) \(U_2 = -\frac{Gm}{R_2}\left(\frac{4}{3}\pi R_2^3(2D_1)\right)\) Divide \(U_1\) by \(U_2\): \(\frac{U_1}{U_2} = \frac{-\frac{Gm}{(2^{\frac{1}{3}}R_2)}\left(\frac{4}{3}\pi (2^{\frac{1}{3}}R_2)^3D_1\right)}{-\frac{Gm}{R_2}\left(\frac{4}{3}\pi R_2^3(2D_1)\right)}\) This simplifies to: \(\frac{U_1}{U_2} = \frac{1}{2^{\frac{1}{3}}}\) So, the relationship between the gravitational potential energies of the two planets is: \(U_1 = 0.794U_2\) The correct answer is option e) \(U_{1}=0.794 U_{2}\).