Kepler's Law Program With Planets

In summary, at one moment, planets X and Y are aligned, making a straight line with the star, but during the next five years, planet X's angular displacement is 90.0°.
  • #1
hardygirl989
21
0

Homework Statement



Two planets X and Y travel counterclockwise in circular orbits about a star as shown in the figure below. The radii of their orbits are in the ratio 4:3. At one moment, they are aligned as shown in figure (a), making a straight line with the star. During the next five years, the angular displacement of planet X is 90.0° as shown in figure (b). What is the angular displacement of planet Y at this moment?
Answer = ____ revolutions


Homework Equations



ƩF=ma

The Attempt at a Solution



ƩF=ma
(GMplanetMstar)/r^2 = ma = Mplanet V^2 / r
GMstar = r^3ω^3=Rx^3ωx^3 = Ry^3ωy^3
ωy = (90 degrees/5 years)(4/3)^(3/2) = 16√3 degrees/year ≈ 27.712812921102 degrees/year

I think I have figured out most of this problem, but I can not figure out how to convert into revolutions. Can anyone help? Thanks.
 
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  • #2
One revolution of a planet corresponds to completing a full circular orbit. The number of degrees in a circle is 360, so there are that many degrees in a full revolution. Then the fraction of a revolution you've traveled is just the fraction of 360 degrees that you've traveled.
 
  • #3
If I divide by 360, I get 2√3/45, but this would make the number in revolutions per year then. How do I get the number into just revolutions?
 
  • #4
Well, if you have an angular speed, what you've got is essentially a number of revolutions per year that Planet Y traverses. You are told that the situation runs its course for five years. So how many revolutions does Planet Y traverse in that time?

While I'm not exactly sure what your work is saying, it does seem that there's an algebraic error in the second line going to the third line, since it would seem there should only be [itex]\omega^2[/itex], not cubed.
 
  • #5
I noticed that I forgot to post the image
http://www.webassign.net/serpse8/13-p-020.gif

You are right about the W^3 thing. It should be W^2, but I have it written that way on my paper...sorry. The R should be cubed tho because you multiply by the other R.

So I think in 5 yrs it goes 1/4 of a revolution? So, I do... (1/4)*360*27.7128 = 2494.15? That number seems too big...
 
  • #6
You have already accounted for the quarter of a revolution when solving for the angular speed of planet Y to begin with. That is, the correct form of your last line is

[tex]\omega_y^2 = \omega_x^2 \frac{R_x^3}{R_y^3},[/tex]

and to calculate [itex]\omega_x[/itex] you would say that it completes 1/4 revolutions per 5 years, instead of 90 degrees per 5 years. Or you could do what you did, and solve for a number of degrees per year, and convert it to revolutions per year. The point is, once you have [itex]\omega_y[/itex], you've found the number of revolutions Planet Y will complete in one year. It is simple from there to figure out how many it will complete in five years.
 
  • #7
Oh! I see now.

ωy = (1/4 revolutions in 5 years)(4/3)^(3/2)
ωy = (1/4)(4/3)^(3/2) = 0.385 rev

This is the correct answer! Thank you! :)
 
  • #8
How do I mark this thread as solved?
 

Related to Kepler's Law Program With Planets

1. What are Kepler's Laws of Planetary Motion?

Kepler's Laws of Planetary Motion are three laws that describe the motion of planets around the sun. The first law, also known as the law of ellipses, states that planets orbit the sun in elliptical paths with the sun at one of the foci. The second law, or the law of equal areas, states that a line connecting the planet and the sun will sweep out equal areas in equal times. The third law, also known as the law of harmonies, states that the square of a planet's orbital period is proportional to the cube of its semi-major axis.

2. How do Kepler's Laws impact our understanding of the solar system?

Kepler's Laws have greatly contributed to our understanding of the solar system and the laws of planetary motion. They provided a mathematical framework for understanding the motion of planets around the sun, and helped to disprove the previous geocentric model of the universe. These laws also allowed for the prediction of planetary positions and the discovery of new planets.

3. What is the significance of Kepler's Law Program With Planets?

The Kepler's Law Program With Planets is a computer program that simulates the motion of planets around the sun based on Kepler's Laws. It allows scientists to study and visualize the complex movements of planets in a simplified and interactive manner. This program has been used to further our understanding of planetary motion and has also been used to discover new exoplanets.

4. How was Kepler able to develop these laws?

Johannes Kepler was a German mathematician and astronomer who used the precise observations of his mentor, Tycho Brahe, to develop his laws of planetary motion. He spent years analyzing these observations and eventually discovered the patterns and relationships that led to his three laws. Kepler's mathematical background also played a crucial role in his ability to formulate these laws.

5. Are Kepler's Laws still relevant today?

Yes, Kepler's Laws are still relevant and widely used in modern astronomy and physics. They have been proven accurate and are a fundamental part of our understanding of the solar system and the universe. They have also been generalized and extended to other systems, such as binary stars and galaxies, demonstrating their continued relevance in our study of the universe.

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