Anaya Nolan
Newton's Law of Universal Gravitation states that every particle in the universe attracts every other particle with a force that is proportional to their masses and inversely proportional to the square of the distance between their centers of mass. This law applies to a satellite because it, too, is governed by the same principles of gravity and motion as any other object in the universe. Newton's laws of motion state that objects in motion will continue moving in a straight line unless acted upon by a force. In the case of satellites, this force is gravity, which bends their paths and keeps them in orbit.
| Characteristics | Values |
|---|---|
| Newton's Law of Gravitation | States that two objects with masses m1 and m2, with a distance r between their centers, attract each other with a force F given by: F = Gm1m2/r^2, where G is the Universal Gravitational Constant |
| Application to Satellites | Satellites, such as Swift, can orbit the Earth in nearly circular paths due to the balance between horizontal and vertical motion |
| The horizontal velocity of a satellite must be maintained at a high enough rate to counterbalance its vertical motion caused by gravity | |
| The force acting upon a satellite toward Earth is inversely proportional to the square of its distance from the center of Earth | |
| The gravitational force on a satellite decreases as the distance from the Earth increases | |
| Limitations | Newton's Law does not fully explain the precession of the perihelion of planetary orbits, especially Mercury's orbit |
| In spiral galaxies, the orbiting of stars around their centers appears to contradict Newton's Law and general relativity, possibly due to the presence of dark matter |
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What You'll Learn
Newton's insight: Earth's gravity extends to the Moon
Newton's Law of Universal Gravitation states that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centres of mass. In other words, the force acting upon an object toward Earth is inversely proportional to the square of its distance from the centre of Earth.
Before Newton's law of gravity, gravity was thought to be associated with Earth alone. Newton's insight was that Earth's gravity might extend as far as the Moon, producing the force required to curve the Moon's path from a straight line and keep it in its orbit. He hypothesised that gravity is not limited to Earth, but that there is a general force of attraction between all material bodies.
Newton's theory can be applied to the Moon. The Moon is 60 Earth radii away from the centre of Earth. The acceleration the Moon experiences is 3600 times less than that of an apple falling on Earth. This is precisely the observed acceleration of the Moon in its orbit.
Newton's theory can also be applied to satellites. For example, the Swift satellite orbits the Earth in a nearly circular path. The force of the Earth's gravity on Swift is "vertical" – pointed towards the centre of the Earth. Swift moves horizontally at just the right rate so that as it falls vertically, its motion creates a circular path around the Earth. This balance between "horizontal" and "vertical" motion is what is meant by "being in orbit".
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The 'first great unification'
Newton's Law of Universal Gravitation states that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centres of mass. This law describes gravity as a force and marked the unification of the previously described phenomena of gravity on Earth with known astronomical behaviours. It is considered the "first great unification" as it brought together the understanding of gravity on Earth and in space.
Before Newton's law of gravity, there were many theories explaining gravity. As early as Aristotle, philosophers made observations about objects falling and developed theories about why they fell. By the 17th century, the scientific method had started to take root, and thinkers like René Descartes and Galileo Galilei made significant contributions to the understanding of matter, motion, and gravity. Galileo, for instance, wrote about experimental measurements of falling and rolling objects. Johannes Kepler's laws of planetary motion summarised Tycho Brahe's astronomical observations.
In 1666, Isaac Newton developed the idea that Kepler's laws applied not only to the orbit of the Moon around the Earth but also to all objects on Earth. This analysis required assuming that the gravitation force acted as if all of the Earth's mass were concentrated at its centre, an unproven conjecture at the time. By 1680, new values for the diameter of the Earth improved his orbit time calculations to within 1.6% of the known value. In 1687, Newton published his Principia, which combined his laws of motion with new mathematical analysis to explain Kepler's empirical results.
Newton's insight was that Earth’s gravity might extend as far as the Moon and produce the force required to curve the Moon’s path from a straight line and keep it in its orbit. He further hypothesised that gravity is not limited to Earth but that there is a general force of attraction between all material bodies. Newton's laws of motion show that objects at rest will stay at rest, and objects in motion will continue moving in a straight line unless acted upon by a force. Since planets move in ellipses, not straight lines, Newton proposed that gravity must be the force bending their paths.
Newton's Law of Universal Gravitation can be applied to satellites, such as the Swift satellite. Swift can stay in orbit for many years as long as its horizontal velocity is maintained at a high enough rate. The balance between "horizontal" and "vertical" motion is what is meant by "being in orbit." The force of the Earth's gravity on Swift is "vertical" and pointed towards the centre of the Earth, while Swift moves horizontally at just the right rate so that as it falls vertically, its motion creates a circular path around the Earth.
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The role of calculus
Newton's Law of Gravitation states that two objects with masses m1 and m2, with a distance r between their centers, attract each other with a force F given by: F = Gm1m2/r^2, where G is the Universal Gravitational Constant. This law applies universally, including to satellites in orbit.
Calculus, a mathematical technique invented by Newton, played a crucial role in the formulation and application of Newton's Law of Gravitation. Using calculus, Newton proved that the planets orbit the Sun due to the gravitational pull they experience from the Sun. This proof involved a thought experiment, known as Newton's cannon, in which a cannonball is fired horizontally from a mountain, causing it to drop vertically towards the Earth while simultaneously moving horizontally away from the mountain. By increasing the force with which the cannonball is fired, it would be possible to make it travel far enough that it would always miss the Earth, demonstrating the curvature of the Earth's surface and the effect of gravity.
Additionally, with calculus, Newton was able to derive all of Kepler's laws from his own laws of gravitation. This allowed him to prove that the shape of an orbit caused by the force of gravity should be an ellipse. Furthermore, he showed that the velocity of an object in an orbit increases near perihelion and decreases near aphelion.
In conclusion, calculus was an essential tool that enabled Newton to develop his Law of Gravitation and apply it to a wide range of phenomena, including the motion of satellites.
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The inverse square law
Newton's law of universal gravitation describes gravity as a force stating that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centres of mass. This is known as the inverse-square law.
The inverse-square law proposed by Newton suggests that the force of gravity acting between any two objects is inversely proportional to the square of the separation distance between the objects' centres. This means that an increase in the separation distance results in a decrease in the force of gravity acting between the objects.
Newton's second law can be written as:
F = Gm1m2/r^2
Where:
- F is the force of gravity
- G is the universal gravitational constant
- M1 and m2 are the masses of the two objects
- R is the distance between the centres of the objects
Newton's law of gravitation can be applied to satellites. For example, the Swift satellite launched by NASA can orbit the Earth in a nearly circular path. As Swift enters its orbit, it has a velocity that is purely "horizontal", meaning it moves parallel to the curved surface of the Earth at each point. However, the force of Earth's gravity on Swift is "vertical" and pointed towards the centre of the Earth. This balance between "horizontal" and "vertical" motion is what is meant by "being in orbit".
Newton's law of gravitation was superseded by Albert Einstein's theory of general relativity, but the universality of the gravitational constant remains intact and the law is still used as an excellent approximation of the effects of gravity in most applications.
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Limitations of Newton's theory
Newton's theory of gravitation has several limitations. The theory is only applicable for point masses on extended bodies and inertial frames of reference, which is a major constraint. For objects travelling at speeds comparable to light, Newton's law of motion does not match experimental results, and Einstein's theory of relativity must be applied.
Newton's theory does not fully explain the precession of the perihelion of the orbits of the planets, especially Mercury, which shifts faster than Newton predicted. There is a 43 arcsecond per century discrepancy between the Newtonian calculation and the observed precession. The predicted angular deflection of light rays by gravity calculated using Newton's theory is only half of the deflection observed by astronomers.
Newton's theory also assumes that gravitation is an action-at-a-distance force, meaning that without physical contact, any change in the position of one mass is instantly communicated to all other masses. However, Einstein's theory of special relativity dismissed this assumption, stating that the speed limit in the universe is the speed of light.
While Newton's theory of gravitation has been superseded by Einstein's theory of general relativity, it is still used as an excellent approximation of the effects of gravity in most applications. Relativity is required only when extreme accuracy is needed or when dealing with very strong gravitational fields, such as those found near extremely massive and dense objects or at small distances, like Mercury's orbit around the Sun.
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Frequently asked questions
Newton's Law of Gravitation can apply to a satellite because it states that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers of mass. This means that the force acting upon a satellite (and therefore its acceleration) toward Earth is inversely proportional to the square of its distance from the center of Earth.
The force of the Earth's gravity on a satellite is "vertical" – pointed towards the center of the Earth. The satellite moves horizontally at a rate that creates a circular path around the Earth. This balance between "horizontal" and "vertical" motion is what is meant by "being in orbit."
Newton's theory does not fully explain the precession of the perihelion of the orbits of the planets, especially that of Mercury. There is also a discrepancy between the Newtonian calculation and the observed precession of Mercury's orbit. In spiral galaxies, the orbiting of stars around their centers seems to disobey Newton's law of universal gravitation and general relativity, which astrophysicists explain by assuming the presence of large amounts of dark matter.
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