P113 Gravitation: Lecture 1

1. P113 Gravitation: Lecture 1 • Background to Newton’s Gravitation Law • Background to Newton’s Discovery • Copernicus, Brahe, Kepler, Einstein • Newton’s Law of Gravitation • Principle of superposition • Tests of Newton’s Gravitation Law 2006: Assoc. Prof. R. J. Reeves

2. Background to Newton’s Law of Gravitation • We know Newton’s Universal Law of Gravitation as • The motion of the planets had been observed for many centuries but there were debates as to whether they orbited the sun or earth. • Copernicus in 1543 proposed the heliocentric model - went against the religious doctrine of the time that the earth was the center of the universe. (Aristotle and Ptolemy) • Brahe (1546 -1601) determined to measure the positions of the planets sufficiently accurately to establish one viewpoint or the other. • Kepler (1572 - 1630) used the very accurate table of Brahe to derive 3 Laws of Planetary Motion. • Einstein’s General Theory of Relativity established the equivalence between gravity and acceleration. 2006: Assoc. Prof. R. J. Reeves

3. Background • Kepler’s Laws of Planetary Motion • Each planet moves around the sun in an ellipse, with the sun at one focus • The radius vector from the sun to the planet sweeps out equal areas in equal times • The square of the period of any planet is proportional to the cube of the semimajor axis of the orbit, i.e. T2 ~ R3. • Newton’s ~1666 idea was that the earth’s gravity on the moon exactly counterbalanced its centrifugal force. • He used his laws of mechanics and Kepler’s 3rd law to derive the inverse quadratic form. 2006: Assoc. Prof. R. J. Reeves

4. Derivation of the Law - 1 • The “centrifugal” force on the moon was • From the period of the circular motion of the moon • Then using Kepler’s 3rd Law • Therefore, the balancing gravitational force is 2006: Assoc. Prof. R. J. Reeves

5. Derivation of the Law - 2 • Because of symmetry Newton introduced the earth’s mass and finally the proportionality constant. • The constant G cannot be derived from theory and has a value determined by experiment. In 1798 Cavendish used a torsion balance to obtain the first value of G = 6.67 x 10-11 N m2/kg2 • G remains the most difficult (and least accurate) fundamental constant. The 2002 CODATA value is G = 6.6742 ± 0.0010 x10-11 N m2/kg2 2006: Assoc. Prof. R. J. Reeves

6. Newton’s Universal Law of Gravitation • Two point masses m1 and m2 separated by distance r act on each other with equivalent force given by • The Law is called universal because it applies to any mass throughout the universe acting on all other mass. • Question: Which direction should r point? 2006: Assoc. Prof. R. J. Reeves

7. Principle of Superposition • Newton’s force law gives us the relationship between two masses: • If there several masses, then the principle of superposition states that the total effect is the sum of the individual effects. • Checkpoint 3 2006: Assoc. Prof. R. J. Reeves

8. Tests of Newton’s Law of Gravitation • Is Newton’s Law exactly 1/r2? • The best experimental tests come from laser ranging of the earth-moon system. • Apollo astronauts left reflectors on the moon surface that bounce back laser pulses. • To date these results agree with an exact inverse square law to an accuracy of 1:1012. • Observations on the rotation of binary stars although not as accurate concur with a 1/r2 law. • Currently there are many experiments testing gravity on 0.1-0.3mm length scales as this is the postulated size of new dimensions in string or brane theory. • The best results show 1/r2 down to 0.14mm. 2006: Assoc. Prof. R. J. Reeves