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 less dense planet will have the larger gravitational potential energy, because it is at a larger distance from the center of the planet compared to the object on the surface of the denser planet.

Step-by-step solution

Understand the relation between density and radius

Given that two planets have the same mass, \(M\), and one is denser than the other, it means that the denser planet has a smaller volume compared to the less dense planet. The radius, \(r\), of the denser planet is smaller than the radius of the less dense planet. Since density (\(p\)) is given by mass divided by volume, for a sphere, we have the relation \(p = \frac{M}{\frac{4}{3}\pi r^3}\), where \(r\) is the radius of the sphere.

Determine the object with a larger magnitude of gravitational potential energy

The gravitational potential energy is given by the formula \(U=-G\frac{mM}{r}\). To maximize the magnitude of gravitational potential energy, we need to minimize the distance, \(r\), of the object from the center of the planet. Since the object on the surface of the less dense planet is at a larger distance from the center of the planet compared to the object on the surface of the denser planet, the object on the surface of the less dense planet will have a larger magnitude of gravitational potential energy. The answer is: c) The object on the surface of the less dense planet will have the larger gravitational potential energy.

Key concepts

Gravitational Potential Energy Formula

Understanding how the gravitational potential energy formula is applied in various contexts is crucial to grasping the interactions between masses in the realm of physics. The key formula used to calculate gravitational potential energy (\(U\)) is \(U = -G\frac{mM}{r}\), where \(G\) represents the gravitational constant (\(6.674 \times 10^{-11} \, \text{N} \cdot (\text{m/kg})^2\)), \(m\) is the mass of the object being affected by gravity, \(M\) is the mass of the planet or celestial body generating the gravitational field, and \(r\) is the distance from the center of the mass \(M\) to the object.

When exploring problems like the one presented, this formula helps to clarify why the object on the surface of a less dense planet—having a larger radius—would have greater gravitational potential energy. It’s all due to the inverse relationship between gravitational potential energy and the distance \(r\). A larger \(r\) in the denominator of the formula means a less negative (larger) value of \(U\), indicating greater potential energy. It's important to note that while the potential energy is negative due to the attractive nature of gravity, we often compare its magnitude, which is always positive.

Density and Radius Relationship

The relationship between the density of a planet and its radius plays a pivotal role in understanding gravitational phenomena. Denser planets have a greater mass per unit volume, which typically correlates with a smaller radius if the mass is constant. Derived from the formula for density, \(\rho = \frac{M}{\frac{4}{3}\pi r^3}\), we can see that for a given mass \(M\), an increase in density \(\rho\) corresponds to a decrease in the volume, and hence, a smaller radius \(r\).

In the context of our exercise, the denser planet, despite having the same mass as the less dense one, will have a smaller radius because its matter is more compacted. This factor is essential when using the gravitational potential energy formula, as it reveals that the distance from the center of the planet—which affects the potential energy—varies inversely with the density of the planet.

Planetary Mass and Gravity

The mass of a planet is directly linked to the gravity it exerts on surrounding objects. Newton’s law of universal gravitation states that every point mass attracts every other point mass by a force acting along the line intersecting both points. The force is proportional to the product of their masses and inversely proportional to the square of the distance between their centers, as shown by the formula \(F = G\frac{mM}{r^2}\).

It’s important for students to recognize that planetary mass is a measure of the gravitational attraction's strength that a planet can exert on an object. Despite both planets in our exercise having equal mass, the gravitational force felt at the surface will not differ. However, the alteration in the gravitational potential energy comes from the differences in radius due to the denser planet having a compacted distribution of its mass. This compactness results in a stronger gravitational field near the planet's surface, even though the mass—and therefore the total gravitational force—is the same for both planets.