Kepler's 3rd Law as a result of Newton's Laws

In summary, the conversation discusses showing that Kepler's 3rd Law holds when assuming a circular orbit with two gravitating bodies orbiting around the center of mass. The approach involves equating gravitational and centripetal forces and finding the constant of proportionality with the masses of the bodies. The individual has already shown this for one mass but is having difficulty differentiating between the two situations. They mention using Newton's 3rd law to establish a relationship between the distances from each body to the center of mass, r1 and r2, and the need to eliminate one of the variables. The conversation also touches on the ratio of r1 to r2 and how it changes with the ratio of m1 to m2, and the impact
  • #1
takbq2
32
0
thanks in advance for any and all help!

The question is to show that Kepler's 3rd Law holds when you assume circular orbit with TWO gravitating bodies both orbiting around the center of mass. (by equating gravitational and centripetal forces and finding the constant of proportionality which includes the masses of the gravitating bodies).

my attempt at a solution:

I already have shown it when assuming just one mass. I'm having a very hard time differentiating between the two situations.

I know I need to start by using Newton's 3rd law to get a relationship between r1 and a (where a is the distance between the two masses) and eliminate r2 or vice versa (r1 is the distance from m1 to the center of mass, and r2 is the distance from m2 to the center of mass)


Please help me if you can, I appreciate it so much.
 
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  • #2
please.. anyone?
 
  • #3
What do you think the ratio of r1 to r2 would be as compared to the ratio of m1 to m2?
 
  • #4
bigger m1 means smaller r1.. which would in turn mean smaller m2 and larger r2
 
  • #5
Alright, what is the ratio of r1 to r2 if m1=m2.

What happens to the ratio of r1:r2 if you now double the mass of m2?
 

1. What is Kepler's 3rd Law as a result of Newton's Laws?

Kepler's 3rd Law, also known as the Law of Harmonies, 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. This law was derived from Newton's Laws of Motion and Universal Gravitation.

2. How does Newton's First Law relate to Kepler's 3rd Law?

Newton's First Law of Motion, also known as the Law of Inertia, states that an object will remain at rest or in uniform motion unless acted upon by an external force. This law is reflected in Kepler's 3rd Law, as it explains how planets continue to orbit in a stable manner without any external force acting upon them.

3. What is the significance of Kepler's 3rd Law in understanding planetary motion?

Kepler's 3rd Law is significant because it helps us understand the relationship between a planet's orbital period and its distance from the sun. This law allows us to accurately predict the orbital periods of planets and other celestial bodies, and provides insight into the structure and dynamics of our solar system.

4. How does Newton's Universal Gravitation play a role in Kepler's 3rd Law?

Newton's Universal Gravitation states that every object in the universe attracts every other object with a force that is directly proportional to their masses and inversely proportional to the square of the distance between them. This law explains the gravitational force that allows planets to orbit around the sun, and ultimately leads to the formation of Kepler's 3rd Law.

5. What are some real-world applications of Kepler's 3rd Law as a result of Newton's Laws?

Kepler's 3rd Law has many real-world applications, including predicting the motion of satellites and other objects in orbit around the Earth, understanding the behavior of binary star systems, and even calculating the masses of distant planets and stars by observing their orbital periods and distances. It has also played a crucial role in the development of space exploration and our understanding of the universe.

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