Question

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) \(U_{1}=0.794 U_{2}\)

Short answer

Answer: e) \(U_{1}=0.794 U_{2}\)

Step-by-step solution

Gravitational Potential Energy Formula

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).

Mass and Volume Formulas for Spheres

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): \(V = \frac{4}{3}\pi R^3\)

Express Gravitational Potential Energy in terms of Density and Radius

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

Determine the relationship between the densities and radii of the planets

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: \(R_1 = 2^{\frac{1}{3}}R_2\)

Determine the relationship between the gravitational potential energies of the two planets

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) \(U_{1}=0.794 U_{2}\).

Key concepts

Constant Density of Planets

Understanding the concept of constant density in planetary bodies is crucial to grasp the broader aspects of gravitational potential energy. Constant density refers to the uniform distribution of mass throughout the volume of an object. For planets, which are often approximated as spheres in physics problems, this implies that every equal volume segment of the planet carries the same amount of mass.

When two planets have the same mass but different densities, their volumes must differ to accommodate the mass in each case. If one planet has twice the density of the other, it means that the same mass is packed into a smaller volume, resulting in a smaller radius for the denser planet. Density is a critical factor in calculating gravitational potential energy because the energy not only depends on mass but also on how that mass is distributed within the volume of the planet.

Mass-Volume Relationship for Spheres

In the context of spherical planets, comprehending the mass-volume relationship is pivotal. The volume of a sphere is proportional to the cube of its radius, as described by the formula:
\[V = \frac{4}{3}\pi R^3\]
For a given mass, a sphere with a larger radius will have a lower density, because the mass is spread over a larger volume. This relationship showcases the intricate balance between mass, volume, and density. When handling problems involving gravitational potential energy, students should pay careful attention to how a change in radius affects the overall energy, especially since gravitational potential energy is inversely related to the radius when the mass remains constant.

Comparing Gravitational Forces

Comparing gravitational forces between two planets involves analyzing how the gravitational potential energy varies on their surfaces. The earlier exercise illustrates this by comparing the energy needed to bring an object from infinity to the surface of each planet.

Given that gravitational force dictates the potential energy, understanding that the force itself is contingent on the mass of the planets and the distance to their centers (i.e., their radii) is important. Being mindful that when comparing gravitational forces, or the potential energy resulting from these forces, the relationship between the mass of the planets and the distance to their centers—dictated by their respective densities—is the key factor in understanding why the gravitational potential energies, in this case, differ and by how much they do so.