Newton's take on Gravitation - the inverse square law
Newton's take on gravity and orbits - which is the genesis of our modern conception of it, is based on:
Universal Gravitation (1687, Principia)
Newton's take on orbits was quite different. For him, Kepler's laws were a manifestation of the bigger "truth" of universal gravitation. That is:
All bodies have gravity unto them. Not just the Earth and Sun and planets, but ALL bodies (including YOU). Of course, the gravity for all of these is not equal. Far from it. The force of gravity can be summarized in an equation:
or.... the force of gravitation is equal to a constant ("big G") times the product of the masses, divided by the distance between them (between their centers, to be precise) squared.
Big G = 6.67 x 10^-11, which is a tiny number - therefore, you need BIG masses to see appreciable gravitational forces.
This is an INVERSE SQUARE law, meaning that:
- if the distance between the bodies is doubled, the force becomes 1/4 of its original value
- if the distance is tripled, the force becomes 1/9 the original amount
- etc.
If we're talking about an object on Earth, the force due to gravity is "weight", discussed below.
Weight
Weight is a result of local gravitation. Since F = G m1 m2 / d^2, and the force of gravity (weight) is equal to m g, we can come up with a simple expression for local gravity (g):
g = G m(planet) / d^2
Likewise, this is an inverse square law. The further you are from the surface of the Earth, the weaker the gravitational acceleration. With normal altitudes, the value for g goes down only slightly, but it's enough for the air to become thinner (and for you to notice it immediately!).
Note that d is the distance from the CENTER of the Earth - this is the Earth's radius, if you're standing on the surface.
If you were above the surface of the earth an amount equal to the radius of the Earth, thereby doubling your distance from the center of the Earth, the value of g would be 1/4 of 9.8 m/s/s. If you were 2 Earth radii above the surface, the value of g would be 1/9 of 9.8 m/s/s.
The value of g also depends on the mass of the planet. The Moon is 1/4 the diameter of the Earth and about 1/81 its mass. You can check this but, this gives the Moon a g value of around 1.7 m/s/s. For Jupiter, it's around 2.5 m/s/s.
Orbits
We've discussed orbits before (with Kepler). Recall that orbits are elliptical, with the Sun at one focus - or with the central gravitating body at one of the two focal points. If it is a 2 (or more) star system, it becomes a bit more complex, but the idea is largely the same. Objects orbit if they have the right orbital (tangential) velocity for their distance away from the body. Needless to say, most bodies don't orbit, but rather crash into the planet or drift off into space.
General Relativity
The current view of gravitation is much more subtle, and is usually credited to Einstein, though others had an influence on the theory. Gravity is a function of the geometry of space - a curvature caused by the presence of mass, affecting space and time around it. If that seems complex, well, it is. The theory is known as the General Theory of Relativity. The time effects are often dealt with in the Special Theory of Relativity - this is where the "Twin Paradox" and other interesting phenomena come up. That is, what a person sees/experiences depends on their reference frame. It gets complex and weird pretty quickly. I'm happy to chat about this with you, if you like.
Universal Gravitation (1687, Principia)
Newton's take on orbits was quite different. For him, Kepler's laws were a manifestation of the bigger "truth" of universal gravitation. That is:
All bodies have gravity unto them. Not just the Earth and Sun and planets, but ALL bodies (including YOU). Of course, the gravity for all of these is not equal. Far from it. The force of gravity can be summarized in an equation:
or.... the force of gravitation is equal to a constant ("big G") times the product of the masses, divided by the distance between them (between their centers, to be precise) squared.
Big G = 6.67 x 10^-11, which is a tiny number - therefore, you need BIG masses to see appreciable gravitational forces.
This is an INVERSE SQUARE law, meaning that:
- if the distance between the bodies is doubled, the force becomes 1/4 of its original value
- if the distance is tripled, the force becomes 1/9 the original amount
- etc.
If we're talking about an object on Earth, the force due to gravity is "weight", discussed below.
Weight
Weight is a result of local gravitation. Since F = G m1 m2 / d^2, and the force of gravity (weight) is equal to m g, we can come up with a simple expression for local gravity (g):
g = G m(planet) / d^2
Likewise, this is an inverse square law. The further you are from the surface of the Earth, the weaker the gravitational acceleration. With normal altitudes, the value for g goes down only slightly, but it's enough for the air to become thinner (and for you to notice it immediately!).
Note that d is the distance from the CENTER of the Earth - this is the Earth's radius, if you're standing on the surface.
If you were above the surface of the earth an amount equal to the radius of the Earth, thereby doubling your distance from the center of the Earth, the value of g would be 1/4 of 9.8 m/s/s. If you were 2 Earth radii above the surface, the value of g would be 1/9 of 9.8 m/s/s.
The value of g also depends on the mass of the planet. The Moon is 1/4 the diameter of the Earth and about 1/81 its mass. You can check this but, this gives the Moon a g value of around 1.7 m/s/s. For Jupiter, it's around 2.5 m/s/s.
Orbits
We've discussed orbits before (with Kepler). Recall that orbits are elliptical, with the Sun at one focus - or with the central gravitating body at one of the two focal points. If it is a 2 (or more) star system, it becomes a bit more complex, but the idea is largely the same. Objects orbit if they have the right orbital (tangential) velocity for their distance away from the body. Needless to say, most bodies don't orbit, but rather crash into the planet or drift off into space.
General Relativity
The current view of gravitation is much more subtle, and is usually credited to Einstein, though others had an influence on the theory. Gravity is a function of the geometry of space - a curvature caused by the presence of mass, affecting space and time around it. If that seems complex, well, it is. The theory is known as the General Theory of Relativity. The time effects are often dealt with in the Special Theory of Relativity - this is where the "Twin Paradox" and other interesting phenomena come up. That is, what a person sees/experiences depends on their reference frame. It gets complex and weird pretty quickly. I'm happy to chat about this with you, if you like.
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