Need help on syllabus dot point (1 Viewer)

[mAtboisLim]

'define the quantitative and qualitative relationship between orbital velocity, the gravitational constant, mass of the central body, mass of the satellite, and the radius using Kepler's third law of periods'


I have to make a presentation on this dot point and am unsure of what your meant to do exactly; I am especially confussed on where to find out about the qualitative part.

 

[Tommy_Lamp]

Qualitative: Keplers third law: i.e. Orbiting objects will map out equal areas in equal time regardless of the distance from the point of gravitation.

 

[zeropoint]

'define the quantitative and qualitative relationship between orbital velocity, the gravitational constant, mass of the central body, mass of the satellite, and the radius using Kepler's third law of periods'


I have to make a presentation on this dot point and am unsure of what your meant to do exactly; I am especially confussed on where to find out about the qualitative part.

An object, mass m, executing an orbit of uniform radius R about a body of mass M experiences a gravitational force F = GMm/r^2. If you equate this with the centripetal force, the m's cancel, and you arrive at an expression for the orbital speed v = Sqrt[GM/R]. Now you have an expression that answers all three points. Namely, the orbital speed is directly proportional to the square root of the mass of the central body and inversely proportional to the square root of the radius. The factor Sqrt[G] is a proportionality constant equal to the square root of the universal gravitational constant. Qualitatively, satellites orbit faster about more massive planets and the velocity of the satellite approaches infinity when the radius is very small and decreases asymptotically with increasing radius so the satellite is nearly motionless when R --> oo.

 

[mAtboisLim]

Thank you so much zeropoint -- that was immensely helpful.

Does the mass of the satellite come into the equation though?

 

[ashtor]

...but how does what you said use Kepler's Third Law??

 

[Constip8edSkunk]

Thank you so much zeropoint -- that was immensely helpful.

Does the mass of the satellite come into the equation though?

the mass cancels out when u equate the centripetal force eqn with the newtons


to get to the keplers 3rd law, its just substituting and rearranging the above
say let M be mass of earth and m be that of the satellite

ie. GMm/r^2 = mv^2/r
GM/r = v^2

now v = rw (w =omega), and w= 2pi/T => v=2rpi/T

so GM/r = 4r^2pi^2/T^2
or GM/4pi^2 = r^3/T^2

 

[Dash]

Refer to the following link for more info:
http://www.boredofstudies.org/community/showthread.php?t=14033

 

[wind]

This is what I've got from a website:

To achieve and maintain a stable orbit around a planet, a satellite must have a certain velocity. In general , we define the term orbital velocity (ω) to be the velocity required by a satellite to enter and maintain a particular orbit around a celestial object.

If we assume the orbit of the satellite around the celestial object is circular, we can use Kepler's Third Law (Law of Periods) to obtain an equation for the orbital velocity of the satellite. Starting with the Law of Periods Equation:

Substituting:

T = (2πr)/v

We obtain the orbital velocity:

v = Sqrt([GM]/r)