# Kepler, Newton, Gravity, error

• NCStarGazer
In summary: Thanks jmb2000 for the follow up and the kind words about my book!Ok, now summarize the conversation:In summary, the conversation discussed the relationship between the gravitational equation and centripetal force, and how they can be used to solve for G. The conversation also touched on Kepler's law and the center of mass in relation to circular orbits. The final conclusion was that a typo caused confusion in the equation and it was corrected.
NCStarGazer
Looking at the gravitational equation

F=G*(M*m)/r^2

and centripetal force

F = ((V^2)/r)*m

If you set the two equal and solve for G you get:

G = ((V^2)*r)/M

Substituting (4*pi^2*r^2)/T^2 for V^2 you now have

G = (4*pi^2*r^3)/(M*T^2)

With solution for G, look at Kepler's law with Newton's update,

(M+m)*P^2 = (4*pi^2*a^3)/G

Substituting the G solved for into Kepler's equation and consider working with a perfect circle, r will equal a and T will equal P.

Now, once you substitute in the solved G, simplify...

You Get

(M+m)*P^2 = P^2*m

This is obviously not true!

Please let me know where the error is in my observation.

Thanks!

NCStarGazer said:
look at Kepler's law with Newton's update,

(M+m)*P^2 = (4*pi^2*a^3)/G

No! The law is $MP^2 = 4\pi^2a^3/G$ (see here). In which case there is no contradiction (hardly surprising given the above form of Kepler's Law is derived from Newton's law of gravity).

Well... I am confused, as the link you reference has 4π2a3 = P2G(M + m) under "Conversions for Unknowns", which is the same as what I referenced (M+m)*P^2 = (4*pi^2*a^3)/G.

A bit out of order. jmb's response first.

jmb said:
NCStarGazer said:
(M+m)*P^2 = (4*pi^2*a^3)/G
No! The law is $MP^2 = 4\pi^2a^3/G$

Sorry, jmb, but NCStarGazer has the right expression.

$$\tau^2 = \frac{4\pi^2}{G(M+m)} \,a^3$$
The source of NCStarGazer's error:
NCStarGazer said:
Looking at the gravitational equation

F=G*(M*m)/r^2

and centripetal force

F = ((V^2)/r)*m
You are assuming that the r in Newton's law of gravity and the r in the centripetal force equation are one and the same. They aren't. Both objects are orbiting their center of mass. Given that the distance between a pair of objects of mass m and M is r, the distance from the object with mass m to the center of mass is

$$r_{cm} = \frac M{M+m}\,r$$

The centripetal force needed to sustain a circular orbit about the center of mass is thus

$$F=\omega^2 r_{cm} = \left(\frac{2\pi}{\tau}\right)^2\,\frac{M}{M+m}\,r$$

Combining this with the gravitational force yields

$$\frac {GMm}{r^2} = \left(\frac{2\pi}{\tau}\right)^2\,\frac{M}{M+m}\,r$$

Solving for the period,

$$\tau^2 = \frac{4\pi^2}{G(M+m)}\,r^3$$

D H

Thanks, I see how the center of mass can bring clarity. In working with your post I am still having trouble getting

$$\frac{GMm}{r^2}= \left(\frac{2\pi}{\tau}\right)^2\frac{M}{M+m}\,r$$

To yield your answer, I placed it in Maple and it always returns a version that has Gm(M+m); not G(M+m) in denominator ( I cannot find how to reduce the Gm to be only G), I did it by hand and I came up with the same. Did I miss something or was there a typo or such?

Appreciate the help!

A typo. ω2r has units of acceleration, not force. Restarting with the equation for the center of mass,

$$r_{cm} = \frac M{M+m}\,r$$

The centripetal force needed to sustain a circular orbit about the center of mass is thus

$$F=m\omega^2 r_{cm} = \left(\frac{2\pi}{\tau}\right)^2\,\frac{Mm}{M+m}\,r$$

Combining this with the gravitational force yields

$$\frac {GMm}{r^2} = \left(\frac{2\pi}{\tau}\right)^2\,\frac{Mm}{M+m}\,r$$

Solving for the period,

$$\tau^2 = \frac{4\pi^2}{G(M+m)}\,r^3$$

D H said:
Sorry, jmb, but NCStarGazer has the right expression.

Ughh, my bad. I read this when I was very tired and tried to answer the question too quicky (something I really shouldn't do!).

I saw NCStarGar's use of $r$ in the centripetal and gravitational force equations and assumed he was making the assumption that $M>>m$ and ignoring centre of mass issues. Of course had I taken on the significance of what he/she meant by "Newton's update" I would have realized that wasn't the intent...

Sorry NCStarGazer and thanks to D H for spotting this. Very embarrassed now!

## 1. Who is Johannes Kepler and what are his contributions to science?

Johannes Kepler was a German astronomer and mathematician who lived during the 16th and 17th centuries. He is most well-known for his three laws of planetary motion, which describe the movement of planets around the sun. These laws were a major advancement in understanding the mechanics of the solar system and laid the foundation for Isaac Newton's theory of gravity.

## 2. What is Newton's law of universal gravitation?

Newton's law of universal gravitation is a physical law that describes the force of gravity between two objects. It states that the force of gravity is directly proportional to the product of the masses of the two objects and inversely proportional to the square of the distance between them. This law explains the motion of objects in the solar system, as well as other phenomena such as tides.

## 3. How does gravity affect the motion of objects?

Gravity is a force that attracts objects with mass towards each other. This force affects the motion of objects by causing them to accelerate towards each other. The strength of the gravitational force depends on the masses of the objects and the distance between them. This is why objects with larger masses, like planets, have a stronger gravitational pull than smaller objects, like humans.

## 4. What is an error in scientific measurements?

Error in scientific measurements refers to the difference between the measured value and the true value of a physical quantity. This can be caused by a variety of factors, such as limitations of the measuring instrument, human error, or external influences. Scientists must always take into account and minimize errors in their measurements to ensure the accuracy of their results.

## 5. How do scientists account for errors in their experiments?

Scientists use a variety of methods to account for errors in their experiments. This includes using precise and accurate measuring instruments, conducting multiple trials, and applying statistical analysis to their data. They also take into consideration potential sources of error and make adjustments to their methods to reduce them. Additionally, peer review and replication of experiments by other scientists can help identify and address errors in scientific research.

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