How does newton's laws relate to kepplers law?

Answer 1

Newton's Law of Universal Gravitation is F = G M m / R 2 Kepler's Third Law is R 3/ T 2 = constant ( the value of this constant depends on the mass of the object that is being orbitted) Newton's law can also be written as F = m 4 (pi) 2 R / T 2 Combining this form of Newton's law with Kepler's law gives F = m 4 (pi) 2 R / T 2 = G M m / R 2 so R 3 / T 2 = 4 (pi) 2 G M so the constant in Kepler's law depends only on the mass of the object since 4, pi and G are constant

Answer 2

Kepler's Laws of Planetary Motion describe the motion of one object in orbit around another. Newton's Laws of Motion and the Law of Universal Gravitation describe how objects move in response to a force and how objects are attracted to each other. Kepler came first, and he laid some of the groundwork that Newton used in his laws. The two sets of laws are similar, and can actually be defined or derived in terms of each other.

Kepler's Laws:

The orbits of the planets form an ellipse, with the Sun as one of the two foci.

A line from the Sun to each of the planets sweeps out an area that is the same given any two equal amounts of time. (For each planet, that it.)

The square of the orbital period of a planet is proportional to the cube of the semi-major axis of the ellipse.

These laws need to be adjusted because they assume that the Sun's mass is many, many times that of the planets. In fact, the limit as the ratio of masses becomes infinity resolves to these laws. When you have two objects, such as a planet and a moon, things are not exactly the same, but you can still draw conclusions with the proper compensations.

Newton's Laws:

An object at rest in the absense of a force will stay at rest, and an object in motion in the absense of a force will stay in motion at the same velocity (which includes speed and direction).

The force required to accelerate an object is proportional to its mass and the acceleration, i.e. f = ma.

Any object with a force applied to it will exert an equal and opposite force against the force applied to it.

The force of gravity between any two objects is proportional to their mass and inversely proportional to the square of the distance between them, i.e. f = Gm1m2/r2.

Einstein:

Complicating all of this is relativity, because objects traveling at relatavistic speed, i.e. approaching the speed of light, experience deviations from these laws, deviations that I am going to ignore by saying that, in the general case, speeds of planets and moons are not that great. (Actually, the deviation is measurable even at our relatively low speed, but the error is small enough to ignore at this level.)