10.3 Motion In A Gravitational Field Flashcards
What is the derivation for the gravitational force when a point mass m orbits a spherical mass M the orbital radius is r and the orbital speed is v?
The gravitational force provides the centripetal force on m and so
GMm/r^2 = m(v^2/r) ——-> v^2 = GM/r
The formula says that the loser m is to M the faster it moves.
What is the speed for circular motion in general?
v = 2pir/T where T is the period (the time for one full revolution).
What is Kepler’s third law?
It relates the period to the orbital redius. Taking M to be the mass of the sun and m the mass of a planet we deduce by coming the last two formulas that
(2pir/T)^2 = GM/r ——-> T^2 = (4pi^2/GM)r^3
How is the total energy of a moving satellite defined?
A satellite orbits a spherical mass M. The kinetic energy of the orbiting satellites is
Ek - 1/2mv^2 and since v^2 = GM/r we have that Ek = 1/2(GMm/r) The total energy is therefore
Et = Ep + Ek = 1/2(GMm/r) - GMm/r = -1/2(GMm/r)
the total energy is negative.
What is significant about the satellite being a potential well?
Energy must be supplied if it is to move away.
What is escape speed?
The minimum launch sped of a projectile at the surface of a planet so that the projectile can move far away (to infinity).
What is the total energy at launch at the surface of the planet?
Et = Ek + Ep = 1/2mv^2 - GMm/R where R is the radius of the earth.
What is the formula for the speed needed by the satellite to just escape.
The total energy at infinite will be zero, by conservation of energy
1/2mv^2 - GMm/R = 0 —–> v^2 = 2GM/R
