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)/r 2 ] e r e r : 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 21 st Century!

Newton formulated his Universal Law of Gravitation in 1666! He didn’t publish until 1687! Principia –See –Delay? Needed to invent calculus to justify calculations for extended bodies! Also, was reluctant to publish in general. F = - [G(mM)/r 2 ] e r (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 =   N·m 2 /kg 2 G is the oldest fundamental constant but the least precisely known. Some others are: e, c, ħ, k B, m e, m p,,,,

4 Fundamental Forces of Nature Sources of forces: In order of decreasing strength Gravity is, BY FAR, the weakest of the four! NOTE: = (10 -6 ) 6 !   36  orders of   magnitude!

Universal Law of Gravitation F = - [G(mM)/r 2 ] e r –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)/r 2 ] e r (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)/r 2 ] e r = - G[m ρ(r)dv/r 2 ] e r (2) –The total force between m & an extended body with volume V & mass M = ∫ ρ(r)dv  Integrate (2)!  F = - Gm ∫ [ρ(r)dv/r 2 ]e r (3) The integral is over volume V! Note: The direction of the unit vector e r varies with r & needs to be integrated over also! Also, r 2 depends on r!

F = - Gm ∫ [ρ(r)dv/r 2 ]e r (I) The integral is over the volume V! e r & r 2 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 2 nd mass! Arbitrary Origin 

Gravitational Field F = - Gm ∫ [ρ(r)dv/r 2 ]e r 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/r 2 ] e r For an extended body: g  - G ∫ [ρ(r)dv/r 2 ]e r Integral over volume V Note: The direction of the unit vector e r varies with r & needs to be integrated over also! Also, r 2 depends on r! g: Units = force per unit mass = acceleration! Near the earth’s surface, |g|  “Gravitational Acceleration Constant” (|g|  9.8 m/s 2 = 9.8 N/kg) Analogous to E = (F/q) in Electrostatics!