00:01
Okay, so we have this satellite at some radius that is orbiting the earth, and we're supposed to find the force equation in terms of the period and the masses.
00:14
So to do that, we've got to look at the centripetal force.
00:19
So that is what's keeping it in orbit, and that's going to be equal to the gravitational force, since that is what is acting on the satellite.
00:27
So if we set these equal to each other, the mass of the satellite will cancel at first.
00:33
The centripetal acceleration can be found by substituting omega squared over r.
00:39
And now we can solve this for the radius by multiplying through by the radius and dividing by omega squared.
00:47
Now omega squared is the rotational velocity, which we can put in terms of the period.
00:52
So 2 pi over the period.
00:53
And if we substitute that in for omega and then take the cube root of r, we get this expression.
01:01
Now we can substitute this back into our gravitational equation and simplify it.
01:09
So we're going to cube all of this stuff on the top so that it can go inside that cube root.
01:15
And so we'll have g cubed, m cubed, m cubed, and then we'll have this bottom term squared, and then everything under the cube root.
01:22
And so the g cubed over g squared will cancel down to g.
01:29
The mass of the earth will simplify as well.
01:34
Likewise with that m will not, the mass of the satellite will not simplify since it wasn't in the original radius equation.
01:53
And then we'll have simplified equation here...