Physics 1501: Lecture 16, Pg 1 Physics 1501: Lecture 16 Todays Agenda l Announcements HW#6: Due Friday October 14 Includes 3 problems from Chap.8 l Topics Chap.8: Gravity and planetary motion Keplers laws Energy and escape velocity Physics 1501: Lecture 16, Pg 2 Gravitation according to Sir Isaac Newton l Newton found that a moon / g = l and noticed that R E 2 / R 2 = Universal Law of Gravitation: l This inspired him to propose the Universal Law of Gravitation: |F Mm |= GMm / R 2 RRERE a moon g G = 6.67 x m 3 kg -1 s -2 Physics 1501: Lecture 16, Pg 3 Gravity... F l The magnitude of the gravitational force F 12 exerted on an object having mass m 1 by another object having mass m 2 a distance R 12 away is: F l The direction of F 12 is attractive, and lies along the line connecting the centers of the masses. R 12 m1m1 m2m2 F F 12 F F 21 Physics 1501: Lecture 16, Pg 4 Lecture 16, Act 1 Force and acceleration l Suppose you are standing on a bathroom scale in Physics 203 and it says that your weight is W. What will the same scale say your weight is on the surface of the mysterious Planet X ? l You are told that R X ~ 20 R Earth and M X ~ 300 M Earth. (a) 0.75 (b) (c) (a) 0.75 W (b) 1.5 W (c) 2.25 W E X Physics 1501: Lecture 16, Pg 5 Keplers Laws l Much of Sir Isaacs motivation to deduce the laws of gravity was to explain Keplers laws of the motions of the planets about our sun. l Ptolemy, a Greek in Roman times, famously described a model that said all planets and stars orbit about the earth. This was believed for a long time. l Copernicus (1543) said no, the planets orbit in circles about the sun. l Brahe (~1600) measured the motions of all of the known planets and the position of 777 stars (ouch !) l Kepler, his student, tried to organize all of this. He came up with his famous three laws of planetary motion. Physics 1501: Lecture 16, Pg 6 Keplers Laws l After 20 years of work on Tycho Brahes data, Kepler formulated his three laws: 1. All planets move in elliptical orbits with the sun at one focal point. 2. The radius vector drawn from the sun to a planet sweeps out equal areas in equal times. 3. The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit. l It was later shown that all three of these laws are a result of Newtons laws of gravity and motion. Physics 1501: Lecture 16, Pg 7 Keplers Third Law Lets start with Newtons law of gravity and take the special case of a circular orbit. This is pretty good for most planets. Physics 1501: Lecture 16, Pg 8 Keplers Second Law This one is really a statement of conservation of angular momentum. Physics 1501: Lecture 16, Pg 9 Keplers Second Law 2. The radius vector drawn from the sun to a planet sweeps out equal areas in equal times. R dR dA Physics 1501: Lecture 16, Pg 10 Keplers Second Law R dR dA Physics 1501: Lecture 16, Pg 11 Example l The figure below shows a planet traveling in a clockwise direction on an elliptical path around a star located at one focus of the ellipse. When the planet is at point A, l a. its speed is constant. l b. its speed is increasing. l c. its speed is decreasing. l d. its speed is a maximum. l e. its speed is a minimum. Active Figure Physics 1501: Lecture 16, Pg 12 Lecture 16, Act 2 Satellite Energies Imagine a comet in an elliptical orbit around the sun such that at the perigee it is a distance r from the sun and at the apogee a distance 2r. What is the ratio of the comet s speed at perigee versus that at apogee, v p /v a. (a) 1/4 (b) (c) (a) 1/4 (b) 1/2 (c) 1 (d) 2 (e) 4 Hint: Use conservation of L Physics 1501: Lecture 16, Pg 13 Example l A satellite is in a circular orbit about the Earth at an altitude at which air resistance is negligible. Which of the following statements is true? l a. There is only one force acting on the satellite. l b. There are two forces acting on the satellite, and their resultant is zero. l c. There are two forces acting on the satellite, and their resultant is not zero. l d. There are three forces acting on the satellite. l e. None of the preceding statements are correct. Physics 1501: Lecture 16, Pg 14 Energy of Planetary Motion A planet, or a satellite, in orbit has some energy associated with that motion. Lets consider the potential energy due to gravity in general. Define r i as infinity U r RERE 0 Physics 1501: Lecture 16, Pg 15 Energy of a Satellite A planet, or a satellite, also has kinetic energy. We can solve for v using Newtons Laws, Plugging in and solving, Physics 1501: Lecture 16, Pg 16 Energy of a Satellite So, an orbiting satellite always has negative total energy. A satellite with more energy goes higher, so r gets larger, and E gets larger (less negative). Its interesting to go back to the solution for v. v is smaller for higher orbits (most of the energy goes into potential energy). Physics 1501: Lecture 16, Pg 17 Lecture 16, Act 3 Satellite Energies l A satellite is in orbit about the earth a distance of 0.5R above the earths surface. To change orbit it fires its booster rockets to double its height above the Earths surface. By what factor did its total energy change ? (a) 1/2 (b) (c) (a) 1/2 (b) 3/4 (c) 4/3 (d) 3/2 (e) 2 Physics 1501: Lecture 16, Pg 18 Escape Velocity l Normally, if I throw a ball up in the air it will eventually come back down and hit the ground. l What if I throw it REALLY hard ? l Two other options 1)I put it into orbit. 2)I throw it and it just moves away forever i.e. moves away to infinity Physics 1501: Lecture 16, Pg 19 Orbiting l How fast to make the ball orbit. l I throw the ball horizontal to the ground. l We had an expression for v above, Physics 1501: Lecture 16, Pg 20 Escape Velocity l What if I want to make the ball just go away from the earth and never come back ? l (This is something like sending a space ship out into space.) l We want to get to infinity, but dont need any velocity when we get there. l This means E TOT = 0 Why ?? Physics 1501: Lecture 16, Pg 21 Example l Which of the following quantities is conserved for a planet orbiting a star in a circular orbit? Only the planet itself is to be taken as the system; the star is not included. l a. Momentum and energy. l b. Energy and angular momentum. l c. Momentum and angular momentum. l d. Momentum, angular momentum and energy. l e. None of the above.