2006: Assoc. Prof. R. J. Reeves Gravitation 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 Gravitation 1.2 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 ( ) determined to measure the positions of the planets sufficiently accurately to establish one viewpoint or the other. Kepler ( ) 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 Gravitation 1.3 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. T 2 ~ R 3. 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 Gravitation 1.4 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 Gravitation 1.5 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 N m 2 /kg 2 G remains the most difficult (and least accurate) fundamental constant. The 2002 CODATA value is G = ± x N m 2 /kg 2

2006: Assoc. Prof. R. J. Reeves Gravitation 1.6 Newton’s Universal Law of Gravitation Two point masses m 1 and m 2 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 Gravitation 1.7 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 Gravitation 1.8 Tests of Newton’s Law of Gravitation Is Newton’s Law exactly 1/r 2 ? 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: Observations on the rotation of binary stars although not as accurate concur with a 1/r 2 law. Currently there are many experiments testing gravity on mm length scales as this is the postulated size of new dimensions in string or brane theory. –The best results show 1/r 2 down to 0.14mm.