13.1 Newton's Law of Universal Gravitation
3 min read•june 24, 2024
revolutionized our understanding of the cosmos. It unified celestial and terrestrial mechanics, showing that the same laws govern both. This groundbreaking theory explained planetary motion and the behavior of objects on Earth.
The law states that every particle in the universe attracts every other particle with a force proportional to their masses and inversely proportional to the square of the distance between them. This simple yet powerful equation allows us to calculate gravitational forces between objects.
Historical Development and Fundamentals of Newton's Law of Universal Gravitation
Evolution of gravitational theory
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- Ancient times
- proposed idea of "natural place" for elements (earth, water, air, fire) with each seeking its natural place
- Heavenly bodies believed to follow different laws than objects on Earth, with perfect circular motion
- Middle Ages
- (Avicenna) introduced concept of momentum and inertia, laying groundwork for future developments
- and developed concept of impetus, a precursor to momentum, to explain motion
- Renaissance
- proposed heliocentric model of solar system, challenging Earth-centered view
- made key observations and experiments that laid foundation for Newton's work
- Discovered moons of Jupiter (, , , ), supporting Copernican model
- Studied motion of falling objects and projectiles, leading to concept of acceleration
- 's formulation
- Unified celestial and terrestrial mechanics under single law of gravitation, showing same laws govern both
- Developed inverse-square law of gravitation based on of planetary motion (elliptical orbits, equal areas, period-distance relation)
- Published groundbreaking work in 1687, which presented three laws of motion and Law of Universal Gravitation
Calculations with Newton's law
- Newton's Law of Universal Gravitation expressed as
- represents gravitational force between two objects, always attractive force
- is with value
- and are masses of the two interacting objects (kg)
- is distance between centers of the two objects (m)
- Force acts along line connecting centers of objects
- Magnitude of force directly proportional to product of masses and inversely proportional to square of distance ()
- To calculate force, identify masses and distance, then substitute values into equation
- Example: Calculate force between Earth (mass kg) and Moon (mass kg) with distance of m between centers
Complex gravitational interactions
- applies to gravitational forces
- Net gravitational force on object is vector sum of individual forces exerted by all other objects
- To find net force, calculate each individual force and add as vectors
- Example: Net force on Earth due to Sun and Moon is vector sum of individual forces
- considerations simplify calculations for symmetric mass distributions (spherical, cylindrical)
- Net gravitational force can be calculated as if all mass concentrated at center
- Applies to objects like or stars, treating them as point masses
- require integration over mass distribution
- Uses calculus to sum contributions from infinitesimal mass elements
- Example: Calculating gravitational force between two galaxies or within a star
- Gravitational potential energy given by
- Work done by gravitational force in moving object equals change in gravitational potential energy
- Symbols have same meaning as in force equation
- Example: Lifting object from Earth's surface to higher altitude increases gravitational potential energy
Mass and Gravitation
- : measure of object's resistance to acceleration when force is applied
- : measure of object's response to gravitational force
- : states that inertial mass and gravitational mass are equivalent
- This equivalence leads to being independent of an object's mass
- Newton's law implies , where objects can influence each other without physical contact
Key Terms to Review (43)
Action at a Distance: Action at a distance refers to the ability of an object to exert a force on another object without any physical contact or connection between them. This concept is central to Newton's Law of Universal Gravitation, which describes the gravitational force between any two objects in the universe, even those separated by vast distances.
Albert of Saxony: Albert of Saxony was a 14th century philosopher, theologian, and mathematician who made significant contributions to the development of physics, particularly in the context of Newton's Law of Universal Gravitation. He was a prominent figure in the scholastic tradition and his work helped pave the way for the scientific revolution that would follow.
Andromeda galaxy: The Andromeda Galaxy is a spiral galaxy approximately 2.537 million light-years from Earth. It is the nearest spiral galaxy to the Milky Way and on a collision course with it.
Aristotle: Aristotle was a renowned Greek philosopher who lived in the 4th century BCE. He made significant contributions to various fields, including physics, metaphysics, logic, ethics, politics, biology, and poetics, and his ideas have had a profound and lasting impact on Western thought.
In the context of 13.1 Newton's Law of Universal Gravitation, Aristotle's views on the nature of the universe and the motion of objects are particularly relevant. Aristotle believed that the Earth was the center of the universe and that all objects naturally moved towards their natural place, with heavier objects falling faster than lighter ones.
Brahe: Tycho Brahe was a Danish astronomer known for his accurate and comprehensive astronomical observations in the late 16th century. His data greatly influenced the development of later astronomical theories, including those by Kepler and Newton.
Callisto: Callisto is the second-largest moon of Jupiter and the third-largest moon in the Solar System. It is the fourth-largest moon overall and the largest of the Galilean moons. Callisto's composition is similar to that of the other Galilean moons, consisting of approximately equal amounts of rock and ice, making it one of the most heavily cratered objects in the Solar System.
Cavendish: Cavendish refers to Henry Cavendish, an 18th-century British scientist who is famous for his experiment to measure the gravitational constant (G). His work paved the way for a more accurate understanding of Newton's Law of Universal Gravitation.
Cavendish Experiment: The Cavendish experiment was a pioneering scientific investigation conducted in the late 18th century that provided the first experimental determination of the gravitational constant, a fundamental value in Newton's Law of Universal Gravitation. This landmark experiment demonstrated the existence of gravitational attraction between masses on a laboratory scale, laying the groundwork for our modern understanding of gravity.
Continuous Mass Distributions: Continuous mass distributions refer to the mathematical modeling of physical systems where the mass is distributed continuously throughout a volume or along a line, rather than being concentrated at discrete points. This concept is particularly relevant in the context of Newton's Law of Universal Gravitation, where the gravitational force between objects is calculated based on their continuous mass distributions.
Copernicus: Nicolaus Copernicus was a Renaissance-era polymath who formulated a model of the universe that placed the Sun rather than Earth at its center. This heliocentric model was revolutionary and laid the groundwork for modern astronomy.
Equivalence Principle: The equivalence principle is a fundamental concept in the theory of general relativity, which states that the effects of gravity are indistinguishable from the effects of acceleration. This principle establishes a deep connection between gravity and the geometry of spacetime, forming the foundation for Einstein's theory of general relativity.
Escape velocity: Escape velocity is the minimum speed an object must have to break free from a celestial body's gravitational influence without further propulsion. It depends on the mass and radius of the celestial body.
Escape Velocity: Escape velocity is the minimum speed required for an object to break free of a planet or moon's gravitational pull and enter into space without being pulled back down. This concept is crucial in understanding the motion of objects under the influence of gravity.
Europa: Europa is one of the four major moons of Jupiter, known for its icy surface and potential subsurface ocean. It is a prime target for exploration in the search for extraterrestrial life due to its unique characteristics and the possibility of harboring a habitable environment beneath its frozen exterior.
F = G(m1m2)/r^2: F = G(m1m2)/r^2 is the mathematical expression that represents Newton's Law of Universal Gravitation. It describes the force of gravitational attraction between two objects with masses m1 and m2, separated by a distance r, and governed by the universal gravitational constant G.
Galileo: Galileo Galilei was an Italian physicist and astronomer who made pioneering observations that laid the foundation for modern physics, particularly in mechanics. He is best known for his work on the motion of objects and the laws governing free fall.
Galileo Galilei: Galileo Galilei was an Italian astronomer, physicist, engineer, and philosopher who played a pivotal role in the scientific revolution of the 17th century. His groundbreaking contributions and discoveries had a profound impact on our understanding of motion, gravity, and the cosmos, laying the foundations for modern physics and astronomy.
Ganymede: Ganymede is the largest moon of Jupiter and the largest moon in the Solar System. It is a differentiated body with a rocky core and an icy shell, and is the only moon in the Solar System known to have its own magnetic field.
Gravitational Acceleration: Gravitational acceleration is the acceleration experienced by an object due to the Earth's gravitational pull. It is a fundamental concept that describes the rate of change in an object's velocity as it is acted upon by the force of gravity.
Gravitational Constant: The gravitational constant, denoted as 'G', is a fundamental physical constant that describes the strength of the gravitational force between two objects. It is a crucial parameter in understanding the laws of gravitation and the motion of objects under the influence of gravity.
Gravitational field: A gravitational field is a region of space surrounding a mass where another mass experiences a force due to gravity. It is represented by the gravitational field strength, denoted as $g$.
Gravitational Field: A gravitational field is a region of space surrounding a massive object, where the force of gravity is exerted on other objects. It describes the strength and direction of the gravitational force at every point in space, allowing the prediction of the motion of objects within that field.
Gravitational Mass: Gravitational mass is a fundamental property of an object that determines the strength of the gravitational force it exerts on other objects and the gravitational force it experiences from other objects. It is a measure of an object's gravitational attraction and is a key concept in both Newton's Law of Universal Gravitation and Einstein's Theory of Gravity.
Ibn Sina: Ibn Sina, also known as Avicenna, was a renowned Persian polymath who made significant contributions to various fields, including physics, mathematics, and medicine. He is particularly notable for his influential work on the concept of universal gravitation, which predated Isaac Newton's formulation of the law of universal gravitation.
Inertial Mass: Inertial mass is a fundamental property of an object that quantifies its resistance to changes in its state of motion. It is a measure of an object's inherent tendency to maintain its current state of rest or uniform motion, as described by Newton's first law of motion.
Inverse Square Law: The inverse square law is a fundamental principle that describes the relationship between a quantity and the distance from the source of that quantity. It states that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that quantity.
Io: Io is the innermost and most volcanically active of the four Galilean moons of Jupiter. It is the fifth-largest moon in the Solar System and the third-largest of the four Galilean moons, after Ganymede and Callisto.
Isaac Newton: Isaac Newton was an English mathematician, physicist, astronomer, and natural philosopher who is widely regarded as one of the most influential scientists of all time. His groundbreaking work in the fields of classical mechanics, optics, and mathematics laid the foundation for our understanding of the physical world and the laws that govern it.
Jean Buridan: Jean Buridan was a 14th-century French philosopher and logician who made significant contributions to the understanding of physics and mechanics. He is particularly known for his thought experiment involving a hypothetical donkey that is equally hungry and thirsty, unable to choose between two equidistant and equally attractive sources of food and water, and thus dies of both hunger and thirst.
Kepler's Laws: Kepler's laws are a set of three fundamental principles that describe the motion of planets around the Sun. These laws, formulated by the German astronomer Johannes Kepler in the early 17th century, provide a mathematical foundation for understanding the dynamics of celestial bodies and their orbits.
Meters per Second Squared: Meters per second squared (m/s²) is a unit of acceleration, which measures the rate of change in velocity over time. It represents the change in velocity, in meters per second, that occurs in one second. This unit is fundamental in understanding the concepts of motion, force, and gravity in physics.
Milky Way: The Milky Way is the galaxy that contains our Solar System, characterized by its spiral shape and vast number of stars, planets, and other celestial objects. It is part of the Local Group of galaxies within the universe.
Newton's Law of Universal Gravitation: Newton's Law of Universal Gravitation is a fundamental principle that describes the attractive force between any two objects with mass. It states that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Newtons: Newtons are the standard unit of force in the International System of Units (SI). They are named after Sir Isaac Newton, the renowned physicist who formulated the laws of motion and the theory of universal gravitation. Newtons are a fundamental concept in physics, as they quantify the amount of force acting on an object, which is crucial in understanding various physical phenomena.
Nicolaus Copernicus: Nicolaus Copernicus was a Renaissance-era astronomer who proposed a heliocentric model of the solar system, where the Earth and other planets revolve around the Sun, in contrast to the prevailing geocentric model. His groundbreaking work laid the foundation for modern astronomy and the scientific revolution.
Orbital Motion: Orbital motion refers to the circular or elliptical path that an object takes around another object due to the force of gravity between them. This term is central to understanding the dynamics of celestial bodies, such as planets orbiting the Sun, and the motion of satellites and other objects in space.
Planets: Planets are large, spherical celestial bodies that orbit around stars, such as the Sun, in the universe. They are distinguished from other objects in the solar system, like stars, moons, and asteroids, by their size, composition, and the way they move through space.
Principia Mathematica: The Principia Mathematica is a three-volume work on the foundations of mathematics, written by the philosophers Bertrand Russell and Alfred North Whitehead, and published between 1910 and 1913. It is considered a landmark in the history of mathematics and logic, as it aimed to derive all of mathematics from a small set of axioms and logical rules.
Satellites: Satellites are objects that orbit a larger body, such as a planet or a star, due to the force of gravity. They can be natural, like the Moon orbiting the Earth, or artificial, such as the satellites launched by humans to study the Earth, communicate, or explore space.
Superposition Principle: The superposition principle states that for linear systems, the net response caused by two or more stimuli is the sum of the individual responses that each stimulus would cause separately. This principle applies to various physical phenomena, including the behavior of waves, gravitational fields, and normal modes of vibration.
Symmetry: Symmetry is the quality of being made up of exactly similar parts facing each other or around an axis. It is a fundamental concept in physics that describes the invariance of a system under certain transformations, such as translations, rotations, or reflections.
Tides: Tides are the periodic rise and fall of sea levels caused by the gravitational pull of the moon and sun. They are a fundamental feature of the Earth's hydrosphere, influencing coastal regions, marine ecosystems, and various human activities.
Universal gravitational constant: The universal gravitational constant, denoted as $G$, is a fundamental physical constant that quantifies the strength of the gravitational force between two masses. Its value is approximately $6.674 \times 10^{-11} \text{Nm}^2\text{kg}^{-2}$.