#### Transcript Angular Momentum

Chapter 11 – Gravity Lecture 1 April 6, 2010 Kepler’s Laws Newton’s Universal Law of gravitation Gravitational and inertia mass Gravitational potential energy Kepler’s (empirical) Laws • Kepler’s First Law: – All planets move in elliptical orbits with the sun at one focus • Kepler’s Second Law: – The radius vector drawn from the Sun to a planet sweeps out equal areas in equal times, dA/dt = const. (the law of equal areas) • Kepler’s Third Law: T 2 Cr 3 – The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit. Today we can understand the physical reasons for these laws … Let’s remind us first of the geometry of the ellipse and then discuss the three laws. Ellipses e = c/a Planetary Orbits Kepler found that the orbit of Mars was an ellipse, not a circle. Kepler’s Third Law Kepler had access to very good data from the astronomer Tycho Brahe in Prague. See table for today’s data. After many years of work Kepler found an intriguing correlation between the orbital periods and the length of the semimajor axis of orbits. T Cr 2 3 Of the satellites shown revolving around Earth, the one with the greatest speed is • • • • • 1 2 3 4 5 Of the satellites shown revolving around Earth, the one with the greatest speed is • • • • • 1 2 3 4 5 The constant-area law. The orbits of two planets orbiting a star are shown. The semimajor axis of planet A is twice that of planet B. If the period of planet B is TB, the period of planet A is A. 2TB B. 2 2 TB C. 3TB D. E. 3 TB 4TB AÕ rA rB B A The orbits of two planets orbiting a star are shown. The semimajor axis of planet A is twice that of planet B. If the period of planet B is TB, the period of planet A is A. 2TB B. 2 2 TB C. 3TB D. E. 3 TB 4TB AÕ rA rB B A Newton’s Universal Law of Gravity m M F12 F21 r Newton’s Law of Gravity • Newton’s law of gravity will provide a physical theory of Kepler’s laws. m M F12 F21 r mM F 12 G 2 rˆ12 R Magnitude of force mM Fg G 2 r G 6.67 10 11 Nm 2 kg The Cavendish experiment Measuring G • G was first measured by Henry Cavendish in 1798 • The apparatus shown here allowed the attractive force between two spheres to cause the rod to rotate • The mirror amplifies the motion • It was repeated for various masses mM Magnitude G 2 r 7/20/2015 G 6.67 10 11 N m 2 / kg 2 Physics 201, UW-Madison 16 Gravitational and inertial mass • Gravitation is a force that acts on the gravitational mass Fg G mg M g r2 (the masses are the source) • Newton’s Law of motion acts on the inertial mass F mi a • In principle, they are not necessarily related, that the gravitational mass mg is not the same as mi (but they are, up to the current experimental accuracy) Kepler’s Third Law derived from Newton’s Law Easily derived for a circular orbit Centripetal force = gravitational force Fcent v2 m mR 2 R mM Fg G 2 R mM R2 4 2 M R 2 G 2 T R mR 2 G 2 T R 3 GM const. 2 2 T 4 Kepler’s Third Law 2 4 T2 R3 GM S Kepler’s Third Law derived from Newton’s Law 4 2 3 T R GM S Extension for elliptical orbits, (Without proof R a) Kepler’s Third Law 2 2 4 T2 a3 GM S Where a is the semimajor axis A woman whose weight on Earth is 500 N is lifted to a height of two Earth radii above the surface of Earth. Her weight • • • • • decreases to one-half of the original amount. decreases to one-quarter of the original amount. does not change. decreases to one-third of the original amount. decreases to one-ninth of the original amount. A woman whose weight on Earth is 500 N is lifted to a height of two Earth radii above the surface of Earth. Her weight A. B. C. D. E. decreases to one-half of the original amount. decreases to one-quarter of the original amount. does not change. decreases to one-third of the original amount. decreases to one-ninth of the original amount. From work to gravitational potential energy. In the example before, it does not matter on what path the person is elevated to 2 Earth radii above. Only the final height (or distance) matters for the total amount of work performed. Potential Energy m 2 r r r12 M1 Force (1,2) G mM rˆ1,2 2 r1,2 G 6.67 1011 N m2 / kg2 Work done to bring mass m from initial to final position. f f mM PE W F dr G 2 dr r i i 1 1 GmM r dr GmM rf ri i f 2 Zero point is arbitrary. Choose zero at infinity. Mm r PE(r ) 0 PE G Mm U(r) G r Demo