10.3 Motion In A Gravitational Field

Question 1

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?

Answer 1

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.

Question 2

What is the speed for circular motion in general?

Answer 2

v = 2pir/T where T is the period (the time for one full revolution).

Question 3

What is Kepler’s third law?

Answer 3

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

Question 4

How is the total energy of a moving satellite defined?

Answer 4

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.

Question 5

What is significant about the satellite being a potential well?

Answer 5

Energy must be supplied if it is to move away.

Question 6

What is escape speed?

Answer 6

The minimum launch sped of a projectile at the surface of a planet so that the projectile can move far away (to infinity).

Question 7

What is the total energy at launch at the surface of the planet?

Answer 7

Et = Ek + Ep = 1/2mv^2 - GMm/R where R is the radius of the earth.

Question 8

What is the formula for the speed needed by the satellite to just escape.

Answer 8

The total energy at infinite will be zero, by conservation of energy
1/2mv^2 - GMm/R = 0 —–> v^2 = 2GM/R