Relating Newton's universal law of gravitation with Kepler's 3rd law

In summary: So you would have to make sure that you changed the units of 'a' and 'P' to AU and years respectively, and also make sure that G was expressed in appropriate SI units.In summary, the conversation discusses rewriting Newton's law as P2 = a3 under specific conditions and converting G to appropriate units. The speaker sets up the problem as a ratio and compares it to Kepler's 3rd Law. They also mention the importance of considering the units for G in the calculation.
  • #1
Rumor
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I'm trying to rewrite Newton's law - GMP2 = 4(pi)2r3 as P2 = a3 under these conditions: that the mass M is the sun's mass, the radius is 1 AU, and G being converted to the appropriate units.

I get stuck on this problem, and I'm hoping you guys are willing to help me out.

To begin with, to make things easier, I set this problem up as a ratio:
P2/a3 = 1 and P2/r3 = 4(pi)2/GM

and then I compare the two. In an elliptical orbit, the radius r can be equaled to a, so I get:

P2/r3 = P2/a3 = 4(pi)2/GM.

Does this look right to you guys so far? I was told that if I did it correctly, my answer should come out to 1, but I'm not getting that once I work things out. I thought that I wouldn't have to worry about the units of G in this setup, but maybe I'm wrong? I'm not sure.

To clarify, I'm using 1.98 x 1030kg for M and 6.67 x 10-11 m3/kgs2 for G.

Any help would be very much appreciated.
 
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  • #2
Rumor said:
I'm trying to rewrite Newton's law - GMP2 = 4(pi)2r3 as P2 = a3 under these conditions: that the mass M is the sun's mass, the radius is 1 AU, and G being converted to the appropriate units.

I get stuck on this problem, and I'm hoping you guys are willing to help me out.

To begin with, to make things easier, I set this problem up as a ratio:
P2/a3 = 1 and P2/r3 = 4(pi)2/GM

and then I compare the two. In an elliptical orbit, the radius r can be equaled to a, so I get:

I think you mean that in a circular orbit, the semi-major axis, 'a' is equal to the radius, 'r'.
Rumor said:
P2/r3 = P2/a3 = 4(pi)2/GM.

Does this look right to you guys so far? I was told that if I did it correctly, my answer should come out to 1, but I'm not getting that once I work things out. I thought that I wouldn't have to worry about the units of G in this setup, but maybe I'm wrong? I'm not sure.

To clarify, I'm using 1.98 x 1030kg for M and 6.67 x 10-11 m3/kgs2 for G.

Any help would be very much appreciated.

1. Of *course* you have to worry about the units for G. It's not a dimensionless quantity, it has units.

2. The whole thing with Kepler's 3rd Law is that it can be expressed as P2 = a3 if and only if you express 'a' in AU and 'P' in years. The reason is because, using those units, the constant of proportionality between P and a, namely 4(pi)2/GM, is equal to 1. But if you don't use those units, and you use SI units instead, then this constant will not be equal to 1.
 

What is Newton's universal law of gravitation?

Newton's universal law of gravitation states that every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

What is Kepler's 3rd law?

Kepler's 3rd law, also known as the "harmonic law", states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

How are Newton's law of gravitation and Kepler's 3rd law related?

Newton's law of gravitation and Kepler's 3rd law are related because they both describe the motion of objects in space. Newton's law explains the force of gravity between two objects, while Kepler's law describes the relationship between a planet's orbital period and its distance from the sun.

Can Kepler's 3rd law be derived from Newton's law of gravitation?

Yes, Kepler's 3rd law can be derived from Newton's law of gravitation. By using the equations for gravitational force and centripetal force, Kepler's law can be derived by equating the two forces and solving for the orbital period.

What is the significance of relating Newton's law of gravitation with Kepler's 3rd law?

The relationship between these two laws helps us understand the fundamental principles of gravity and how it affects the motion of objects in space. It also allows us to make predictions and calculations about the orbits of planets and other celestial bodies.

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