Introduction Section 5.1

• Newton’s Universal Law of Gravitation: Every mass particle attracts every other particle in the universe with a force that varies as the product of the masses and inversely as the square of the distance between them.

F = - [G(mM)/r2] er er : Points from m to M

r = distance between m & M

Point masses are assumed

- sign F is Attractive!Aren’t we glad its not REPULSIVE?

Gravity Research in the 21st Century!

• Newton formulated his Universal Law of Gravitation in 1666! He didn’t publish until 1687! Principia – See http://members.tripod.com/~gravitee/– Delay? Needed to invent calculus to justify calculations for extended bodies!

Also, was reluctant to publish in general.

F = - [G(mM)/r2] er (point masses only!)

G (Universal Gravitation Constant)

– G was first measured by Cavendish in 1798, using a torsion balance (see text). – Modern measurements give:

G = 6.6726 0.0008 10-11 N·m2/kg2

G is the oldest fundamental constant but the least precisely known. Some others are: e, c, ħ, kB, me, mp, ,,,

4 Fundamental Forces of NatureSources of forces: In order of decreasing strength

Gravity is, BY FAR, the weakest of the four!NOTE: 10-36 = (10-6)6!

36

orders of

magnitude!

Universal Law of GravitationF = - [G(mM)/r2] er

– Strictly valid only for point particles!

– If one or both masses are extended, we must make an additional assumption: That the Gravitational field is linear Then, we can use the Principle of Superposition to compute the gravitational force on a particle due to many other particles by adding the vector sum of each force.

– The mathematics of this & of much of this chapter

should remind you of electrostatic field calculations from E&M! Identical math!

• If you understand E&M (especially field & potential calculations) you should have no trouble with this chapter!

F = - [G(mM)/r2] er (Point particles!) (1)– Consider a body with a continuous distribution of matter with mass

density ρ(r)– Divide the distribution up into small masses dm (at r) of volume dv

dm = ρ(r)dv – The force between a (“test”) point mass m & dm a distance r away is

(from (1)):

dF = - G[m(dm)/r2] er = - G[m ρ(r)dv/r2] er (2)

– The total force between m & an extended body with volume V & mass M = ∫ρ(r)dv Integrate (2)!

F = - Gm∫[ρ(r)dv/r2]er (3)

The integral is over volume V! Note: The direction of the unit vector er varies with r & needs to be integrated over also! Also, r2 depends on r!

F = - Gm∫[ρ(r)dv/r2]er (I)

The integral is

over the volume

V! er & r2 both

depend on r! • In general, (I)

isn’t an easy

integral! It should

remind you of the electrostatic force between a point charge & a continuous charge distribution!

• If both masses are extended, we need also to integrate over the volume of the 2nd mass!

Arbitrary

Origin

Gravitational Field

F = - Gm∫[ρ(r)dv/r2]er Integral over volume V

• Gravitational Field Force per unit mass exerted on a test particle

in the field of mass M = ∫ρ(r)dv. g (F/m)

• For a point mass: g - [GM/r2] er

• For an extended body: g - G∫[ρ(r)dv/r2]er

Integral over volume V

Note: The direction of the unit vector er varies with r & needs to be integrated over also! Also, r2 depends on r!

• g: Units = force per unit mass = acceleration! Near the earth’s surface, |g| “Gravitational Acceleration Constant” (|g| 9.8 m/s2 = 9.8 N/kg)

Analogous to E = (F/q) in Electrostatics!