Unraveling Gravity: Newton's Universal Law and 21st Century Research

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)/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?

3. Gravity Research in the 21st 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)/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, ,,,

5. 4 Fundamental Forces of NatureSourcesof forces: In order of decreasing strength  36  orders of  magnitude! Gravity is, BY FAR, the weakest of the four! NOTE: 10-36 = (10-6)6!

6. Universal Law of Gravitation F = - [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 Superpositionto 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)/r2] er(Point particles!)(1) • Consider a body with a continuous distribution of matter with mass densityρ(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, r2depends on r!

8. 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 

9. 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, r2depends 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!