Question

With the usual assumption that the gravitational potential energy goes to zero at infinite distance, the gravitational potential energy due to the Earth at its center is a) positive. c) zero. b) negative. d) undetermined.

Short answer

Answer: b) Negative

Step-by-step solution

Recall the formula for gravitational potential energy

The gravitational potential energy (U) of an object near the Earth's surface can be calculated using the formula: U = -G * (m1 * m2) / r, where U is the gravitational potential energy, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between the objects.

Analyze the formula at the distance at the Earth's center

Considering that the Earth's center is the location at which the radius (distance between the object and the Earth's center) is the shortest possible value (r = 0), we can observe that the denominator of the formula (r) approaches zero. As r approaches zero while keeping the product of the other variables constant, the quotient (proportional to U) approaches infinity in a negative direction, given the negative sign in the formula.

Determine the sign of the gravitational potential energy

As we observed in the previous step, when the distance r approaches zero, the gravitational potential energy U becomes increasingly negative. This means that the gravitational potential energy due to the Earth at its center would be: - a) Positive: Incorrect, as we determined that U approaches negative infinity. - b) Negative: Correct, as the gravitational potential energy becomes increasingly negative. - c) Zero: Incorrect, as zero is reached only at infinite distance. - d) Undetermined: Incorrect, as our analysis determined the sign of the gravitational potential energy. So, the correct answer is b) negative.

Key concepts

Physics Education

Understanding the principles of physics is an essential part of education, especially for university students who are studying fields like engineering, natural sciences, or education itself. One of the fundamental topics in university-level physics is gravitational potential energy, a concept that helps students grasp interactions between massive objects.

It's critical to ensure that learners fully comprehend how gravitational potential energy is calculated and how it behaves under different conditions. For easier comprehension, educational resources can explain the concept by providing real-world examples or visual aids such as graphs to depict how potential energy changes with distance. By incrementally introducing the formula used to calculate gravitational potential energy and explaining each component thoroughly, educators can build a strong foundation for students to solve problems related to this concept.

Moreover, it's helpful to use step-by-step problem-solving techniques, as illustrated in the textbook solution, to guide students through logical reasoning and mathematical processes. This not only enhances mathematical skills but also encourages a deeper understanding of physical phenomena.

University Physics

Gravitational potential energy is a crucial part of the curriculum in 'University Physics' and relates to the broader understanding of gravitational forces. It is the energy an object possesses due to its position in a gravitational field. In a more advanced context, university physics students learn to apply calculus and differential equations to deal with the complexities of potential energy, especially as functions of distance or height.

In the context of the given exercise, the consideration of gravitational potential energy at the Earth's center provides a thought-provoking application of these principles. It challenges students to consider boundary conditions, such as what happens as the distance approaches zero. Such exercises are instrumental in enhancing problem-solving skills and prepare students for more complex scenarios involving gravity, such as orbital mechanics or astrophysics.

Understanding the relationship between distance and gravitational potential energy is a cornerstone in the study of physics at the university level and is an excellent example of how theoretical concepts have practical implications in scientific research and technological development.

Gravitational Constant

The gravitational constant (denoted as \(G\)) is one of the most fundamental constants in physics. It appears in Newton's universal law of gravitation and determines the strength of the gravitational force between two masses. Hence, it's paramount for students to understand the significance of \(G\) in equations dealing with gravitational phenomena.

With a value of approximately \(6.674 \times 10^{-11} \frac{Nm^2}{kg^2}\), the gravitational constant is remarkably small, which signifies the weak nature of gravity compared to other fundamental forces. It's interesting to note that despite its fundamental role, \(G\) is one of the least precisely measured physical constants.

In the solution provided in the textbook example, \(G\) is a critical factor in calculating the gravitational potential energy. Without knowing \(G\), it would be impossible to assess the potential energy's magnitude. Students must recognize that, although the exercise primarily focuses on identifying the sign of the potential energy value (negative), the gravitational constant is the backbone that allows us to build a quantitative understanding of gravitational forces in the universe.