Solving Planetary Motion: Mass, Velocity & Energy

In summary: It is good to have more than one way of solving a problem, but this is not a different way, it's just equivalent algebra with some different symbols.
  • #1
Psyguy22
62
0

Homework Statement


A planet with a mass of 8.99·1021 kg is in a circular orbit around a star with a mass of 1.33·1030 kg. The planet has an orbital radius of 1.21·1010 m.

a) What is the linear orbital velocity of the planet?

b) What is the period of the planets orbit?

c) What is the total mechanical energy?

Homework Equations


Keplers 3rd law
[itex] (T^2/R^3)=(4π^2/G M) [/itex]


The Attempt at a Solution


I figured it would be easier to solve b first, So I solved for T
[itex] T=sqrt((4π^2*R^3)/(G M)) [/itex]

and came up with 6.2784...*10^5. But I am confused on what the units would be. Both seconds and years are wrong.
 
Physics news on Phys.org
  • #2
When you use this version of Kepler's third law, everything is in SI units. So, your value of T would be in seconds.

It's only the case when T is in years and R is in AU that the simpler version of Kepler's 3rd Law (T^2 = R^3) works. This is because you happen to have chosen a unit system in which the constants (4pi^2/GM) are equal to 1.

If you are getting the wrong answer in seconds, you need to double check your arithmetic.
 
  • #3
You don't need Kepler's 3rd law here. You know the centripetal acceleration of the planet from equation ##a=G\frac{M}{r^{2}}##, where ##M## is the mass of the star and ##r## is the orbital radius. You also know that for motion in a circular orbit, ##a=\frac{v^{2}}{r}##. From these eqs you can solve the orbital velocity ##v##, and knowing the velocity the calculation of orbital period and mechanical energy are trivial.
 
  • #4
hilbert2 said:
You don't need Kepler's 3rd law here. You know the centripetal acceleration of the planet from equation ##a=G\frac{M}{r^{2}}##, where ##M## is the mass of the star and ##r## is the orbital radius. You also know that for motion in a circular orbit, ##a=\frac{v^{2}}{r}##. From these eqs you can solve the orbital velocity ##v##, and knowing the velocity the calculation of orbital period and mechanical energy are trivial.

Kepler's 3rd Law in the form above can be derived easily by equating the centripetal and gravitational accelerations exactly as you have done above. So I don't see what is the harm in using a mathematically-equivalent thing. (You'll get the period first, and knowing the period, the calculation of orbital velocity and mechanical energy are trivial).
 
  • #5
^ Actually, Keplers 3rd law holds for more general elliptical orbits, not only circular, and it can't be derived simply from the formulae for circular orbits. Of course it doesn't matter whether one uses Kepler's laws or more detailed mechanical considerations in solving the problem.
 
  • #6
Thanks both of you. @ Cephid, you were right. I must of typed numbers wrong the first time.
 
  • #7
hilbert2 said:
^ Actually, Keplers 3rd law holds for more general elliptical orbits, not only circular, and it can't be derived simply from the formulae for circular orbits. Of course it doesn't matter whether one uses Kepler's laws or more detailed mechanical considerations in solving the problem.

I know that. The point I was making was that if you assume a circular orbit, and equate centripetal and gravitational accelerations, you will obtain the *exact* equation for Kepler's 3rd Law that is given in the OP, which is a special case of Kepler's 3rd for circular orbits. I know that if you derive it for the most general case, you will find that r is replaced by the semi-major axis of the ellipse, but that's not relevant to this problem.

In any case, you seem to have missed the actual point I was making, which is that I see no utility in advising the OP to use a second method that is mathematically-equivalent to the one he/she is already using.
 

1. What is planetary motion?

Planetary motion refers to the movement of planets in our solar system around the sun. It is governed by the laws of motion and gravity.

2. How is mass related to planetary motion?

Mass is a measure of the amount of matter in an object. In planetary motion, the mass of a planet affects its gravitational pull and influences its orbit around the sun.

3. What is the role of velocity in planetary motion?

Velocity is the measure of the speed and direction of an object's movement. In planetary motion, the velocity of a planet determines the shape and size of its orbit around the sun.

4. How does energy play a part in planetary motion?

Energy is required for any object to move. In planetary motion, the potential energy of a planet is converted into kinetic energy as it moves in its orbit around the sun.

5. What factors affect planetary motion?

Several factors can affect planetary motion, including the mass and velocity of the planet, the gravitational pull of other celestial bodies, and any external forces acting on the planet.

Similar threads

  • Introductory Physics Homework Help
Replies
6
Views
636
  • Introductory Physics Homework Help
Replies
7
Views
969
  • Introductory Physics Homework Help
Replies
1
Views
760
  • Introductory Physics Homework Help
Replies
2
Views
2K
  • Introductory Physics Homework Help
Replies
10
Views
2K
  • Introductory Physics Homework Help
Replies
7
Views
2K
  • Introductory Physics Homework Help
Replies
16
Views
319
  • Introductory Physics Homework Help
Replies
2
Views
436
  • Introductory Physics Homework Help
Replies
5
Views
2K
  • Introductory Physics Homework Help
Replies
11
Views
1K
Back
Top