Chapter 26: Law of Universal Gravitation

Table of Contents

26.1 Introduction to the Law of Universal Gravitation

In this chapter, we will explore the Law of Universal Gravitation which describes the attractive force between two masses. This law, formulated by Sir Isaac Newton, is a fundamental principle in classical physics and forms the basis for understanding planetary motion, celestial mechanics, and the behavior of massive objects in the universe.

The Earth-moon system obeys Newton’s Law of Universal Gravitation.

26.2 Newton’s Law of Universal Gravitation

The Universal Law of Gravitation states that every point mass attracts every other point mass with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. Mathematically, the gravitational force (F) between two masses (m1 and m2) is given by:

$F = \dfrac{Gm_1m_2}{R^2}$

where $G$ is the gravitational constant (approximately $6.674 \times 10^{-11} \dfrac{\text{N} \cdot \text{m}^2}{\text{kg}^2}$ ) and $R$ is the distance between the centers of the masses.

26.2.1 Gravitational Constant (G)

The gravitational constant, $G$ , is a fundamental constant in nature that determines the strength of the gravitational force between two masses. Its value is the same throughout the universe.

26.3 Gravitational Field

A gravitational field is a region in space where a mass experiences a gravitational force due to the presence of another mass. The strength of the gravitational field, represented by the symbol g, is defined as the gravitational force per unit mass experienced by an object in the field.

26.3.1 Gravitational Field near the Earth’s Surface

At the Earth’s surface, the strength of the gravitational field is approximately 9.81 N/kg. This value is often rounded to 9.8 N/kg or even 10 N/kg for simplicity in calculations.

26.4 Gravitational Potential Energy

The gravitational potential energy $(U)$ of an object is the work done against the gravitational force to move the object from a reference point to a specific position in a gravitational field. For objects near the Earth’s surface, the gravitational potential energy is given by:

$U = m g h$

where $m$ is the mass of the object, $g$ is the gravitational field strength, and $h$ is the height of the object above the reference point. However, we can more generally express the gravitational potential energy now as:

$U = -\dfrac{Gm_1m_2}{R}$

If you set both of these expressions for $U$ equal to each other, you can solve for $g$ and see why $g$ equals 9.81 N/kg on Earth. The exercise is left to the reader.

26.5 Planetary Motion and Kepler’s Laws

Johannes Kepler formulated three laws of planetary motion that describe the behavior of celestial bodies in the Solar System:

The Law of Orbits: Planets orbit the Sun in elliptical paths, with the Sun at one focus of the ellipse.
The Law of Areas: A line connecting a planet to the Sun sweeps out equal areas in equal intervals of time.
The Law of Periods: The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Chapter Summary

The Universal Law of Gravitation is a fundamental principle that describes the attractive force between any two masses. This law, combined with Newton’s laws of motion, provides a foundation for understanding celestial mechanics, planetary motion, and the behavior of massive objects in the universe. Gravitational fields, potential energy, and Kepler’s Laws are essential concepts for studying gravity and its effects on objects in space.

With this chapter complete, we have covered the key topics in our Physics 1 course. As you continue your studies, remember that physics is not only about memorizing equations and formulas but also about developing a deep understanding of the principles that govern the physical world. Always seek to build a strong conceptual foundation and practice problem-solving to hone your skills. Good luck as you continue to further explore the fascinating realm of physics!

Enroll on Canvas

This course uses Canvas for homework assignments, quizzes, and exams. These assignments are open to everyone. Anyone is allowed to enroll in the Canvas course. In fact, this is highly encouraged as it will help you track your progress as you go through the course. Graded feedback will help you get an idea for what your grade would actually be in a Physics 1 college course. Use this link to enroll in the Canvas course .

Congratualtions on completing TruPHY 101!

Continue to TruPHY 201: Physics II

Back to Chapter 25: Oscillators and Mechanical Waves

Do you prefer video lectures over reading a webpage? Follow us on YouTube to stay updated with the latest video content!

Chapter 26: Law of Universal Gravitation

26.1 Introduction to the Law of Universal Gravitation