Velocity/orbital speed

[sheevz]

Homework Statement

find the speed of a satillite in orbit 1000km above the Earth's surface.
mass of Earth 5.98*10^24kg
mass of satellite = 50.0kg
radius of earth=6.38*10^6m

Homework Equations

so i used V= square root of G*mass of earth/r

The Attempt at a Solution

V= square root of (6.673*10^-11)(5.98*10^24)/7.38*10^6)
note***7.38*10^6 is the radius of the Earth plus the 1000km above the Earth's surface***

i get V= 7.353*10^3m/s

ok now my real question is if i DECREASE the distance above the Earth let's say by 100km i know that the velocity or my speed will increase but can someone tell me why this increases? i understand how in figuring out with numbers but why does the speed increase as the distance gets closer to the center?

 

[dynamicsolo]

You can look at this in terms of force or in terms of energy.

In terms of centripetal force, gravity supplies this force, so we have

m·(v^2)/r = GmM/(r^2) .

When you solve this for the speed along the (circular) orbit, we find

(v^2) = GM/r .

Because gravity is an inverse-square force, a stable orbit requires a higher speed along an orbit closer to the center of attraction. (This works the same way for electrostatic forces.)

In terms of energy, a bound orbit has negative mechanical energy. Since the gravitational force is attractive, the potential energy in a system is always negative (zero potential being assigned to infinite separation between masses in a system). As total mechanical energy is kinetic plus potential energy, and kinetic energy is always positive, a closer orbit has a more negative potential energy and a more negative total mechanical energy. The virial theorem (you can look up a proof) tells us that for gravity, U = -2K . So as the potential and total mechanical energy become more negative, the kinetic energy becomes more positive.

This increase in kinetic energy leads to what is called a "gravithermal catastrophe". In the limit as the radius of the orbit shrinks to zero, the kinetic energy would run away to infinity. In reality, this doesn't happen, since material particles would ultimately get in each other's way. What this does make possible, though, is "heating by gravitational collapse", which is basically why we have stars...