WEBSITE NOTES: Gravitation. 1. Much of what we know about gravitation is due to the work of the following astronomers and mathematicians. (i) Nicolaus Copernicus (14731543), Poland, suggested that the Earth and all other planets revolve in circular orbits around the Sun, a heliocentric system, not the geocentric model that persisted for 1400 years. (ii) Tycho Brahe (15641601), Denmark, charted the positions of the planets and 777 stars for 20 years. (iii) Johannes Kepler (15711630), Germany, Brahe's assistant who studied the data from the charts for 16 years and finally formulated 3 laws of planetary motion. (iv) Galileo Galilei (15641642), Italy, who perfected the telescope and later was placed under house arrest and force to recant for supporting the heliocentric theory. (v) Isaac Newton (16421727), England, developed the Law of Universal Gravitation which states that all masses attract each other with a mutual force that varies with the inversesquare of the distance. (vi) Henry Cavendish (17311810), England, was the first experimenter to measure the force of gravity between masses in the laboratory, and the first to yield accurate values for Newton’s gravitational constant and the mass of the Earth. The Cavendish experiment, done in 179798, consisted of a torsion apparatus with metal spheres attached to rods. 2. Kepler's 3 laws of planetary motion state the following: Law (1): All planets revolve in elliptical, nearly circular, orbits around the Sun. Law (2): A straight line from a planet to the sun sweeps out equal areas in equal time intervals. Law (3): The cube of the orbital radius of any planet divided by the square of its period is constant. r^{3}/T^{2} = k 3. Newton's law of Universal Gravitation: "The force of attraction between two bodies is directly proportional to the product of their masses but varies inversely with the square of the distance between them." F = G·m_{1}·m_{2}/r^{2} . 4. The value of the Universal Gravitational constant, G, was also predicted by Newton. 5. In 1798 the value of G was carefully measured with a torsion apparatus by Henry Cavendish (17311810), England, confirming Newton's prediction. G = 6.67x10^{11} Nm^{2}/kg^{2} (known as "Big G") 6. The mass of the Earth can be found by using Newton's Gravitation Law. It is M_{E} = 5.98x10^{24} kg. The mass of the Sun can be found from the period and radius of a planet's orbit. The Sun's mass is computed to be M_{S} = 2.0x10^{30} kg. 7. The mass of a planet can be found only if it has a satellite orbiting it. 8. A satellite in a circular orbit, radius R, accelerates centripetally toward Earth at a rate equal to the acceleration of gravity at its orbital radius. The following properties of satellite motion can all be proven: (i) the velocity is given by the equation v = (2πR)/T (ii) acceleration due to gravity at the orbital radius, R, is g = (G·M_{E})/R^{2} (iii) the minimum or critical velocity for stable orbit is v = √(Rg) 9. All bodies have gravitational fields around them, which can be represented by a collection of vectors representing the force per unit mass at all locations. 10. The mass of an object can be determined in two ways, gravitationally and inertially. Both result in equivalent determinations of mass. 11. Albert Einstein (18791955), proposed that gravity is not a force, but a property of space itself. Mass curves space causing objects to be accelerated toward these massive bodies. 12. Einstein's theory, called the General Theory of Relativity, makes predictions slightly different from Newton's laws, but when tested, gives correct results. 13. Light has also shown to be deflected by massive celestial objects, and if a mass is large enough, light leaving it will be totally bent back to the object. This predicts that black holes in space exist. 14. And still, we need these steps to solve any problem in Physics: (i) read the problem and identify the given variables (ii) determine what you are asked to solve for (iii) find the correct motion formula to use (iv) use algebra to isolate the unknown (v) substitutein the given information and simplify. Kepler's Law Lab: Mars diagram.

