2. 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?

3. Gravity Research in the 21 st Century!

4. 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)/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 = 6.6726  0.0008  10 -11 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,,,,

5. 4 Fundamental Forces of Nature Sources 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!

6. 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!

7. 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!

8. 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 

9. 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!