Physics 114A - Mechanics Lecture 30 (Walker: Ch. 12.4-5) Gravitational Energy March 11, 2014

1. Physics 114A - MechanicsLecture 30 (Walker: Ch. 12.4-5)Gravitational EnergyMarch 11, 2014 John G. Cramer Professor Emeritus, Department of Physics B451 PAB jcramer@uw.edu

2. Announcements • Homework #9 is due at 11:59 PM on Sunday, March 16. • Do the UW Clicker Survey at:https://catalyst.uw.edu/webq/survey/ltres/228318 • This is the week for Physics 117 Lab Makeups. If you missed any 117 labs, you should use this opportunity. • Exam 3 will be returned on Thursday. However, scores for Parts 1 & 2 are posted on WebAssign. If you took Exam 3 and have a 0 posted for either Part 1 or 2, see Susan Miller immediately. • Requests for Exam 3 regrades must be submitted to Susan Miller by 4 PM on Monday, March 17. • I will compute a “Preliminary Grade” and post it on Catalyst, probably tomorrow. It will be based on Exams 1-3 and HW#1-8 only. You need to take the Final only if you want to try to improve this preliminary grade. However, you mustcomplete HW#9. Physics 114A - Lecture 30

3. Exam 3 Statistics 1.8 2.8 3.8 High = 96 Median = 58 Low = 23 Average = 59.1 Std Dev = 15.2 Physics 114A - Lecture 30

4. The Physics 114A Final Exam • On Monday, March 17 at 2:30 PM we will have the Final Exam, covering Chapters 1 through 13 of Walker. It will be a 100 point exam similar in format to the previous Exams, except that it will be all multiple choice. There will be some emphasis on Chapters 12 and 13 and on material covered in Chapters 1-11 but not tested on previous exams. • If you have successfully taken Exams 1-3 and are satisfied with your grades, the Final is optional. I will post “Estimated Course Grades” tomorrow. • There will be new assigned seating for the Final. These assignments will be posted on Catalyst later this week. Physics 114A - Lecture 30

5. Lecture Schedule (Part 3) We are here. Physics 114A - Lecture 30

6. Gravitational Potential Energy Gravitational potential energy of an object of mass m a distance r from the Earth’s center: Physics 114A - Lecture 30

7. Gravitational Potential Energy Very close to the Earth’s surface, the gravitational potential increases linearly with altitude: Gravitational potential energy, just like all other forms of energy, is a scalar. It therefore has no components; just a magnitude and sign. Physics 114A - Lecture 30

8. Gravitational Potential Energy A plot of the gravitational potential energy Ug looks like this: Physics 114A - Lecture 30

9. Energy Conservation Total mechanical energy of an object of mass m a distance r from the center of the Earth is: This confirms what we already know – as an object approaches the Earth, it moves faster and faster. Physics 114A - Lecture 30

10. Energy Conservation An object falling into the Earth’s gravity well, (e.g., a comet), initially with zero mechanical energy. Physics 114A - Lecture 30

11. Energy Conservation Another way of visualizing the gravitational potential well: Physics 114A - Lecture 30

12. Escape Speed Escape speed: the initial upward (or sideways) speed a projectile must have in order to escape from the Earth’s gravity. How much speed is required to climb out of the well? Physics 114A - Lecture 30

13. Example: Escape Speed A 1000 kg rocket is fired straight away from the surface of the Earth. What speed does it need to “escape” from the gravitational pull of the Earth and never return? (Assume a non-rotating Earth.) This is also the speed at which (in the absence of atmosphere) a meteor, falling from very far away, would strike the surface of the Earth. It is also called “escape velocity.” Physics 114A - Lecture 30

14. Energy Conservation Speed of a projectile as it leaves the Earth, for various launch speeds Physics 114A - Lecture 30

15. Clicker Question 1 An astronaut is transported by a series of rockets from the surface of the Earth to the surface of the Moon and then returned to the surface of the Earth. How does the energy UEM required to transport him from Earth to Moon compare with UME required to transport him from Moon to Earth? (a) UEM>UME (b)UEM=UME (c)UEM<UME Physics 114A - Lecture 30

16. Example: Crashing into the Sun Suppose the Earth were suddenly to halt its motion in orbiting the Sun. The gravitational force would pull it directly into the Sun. What would be its speed as it crashed? Physics 114A - Lecture 30

17. Example:The Height of a Projectile A projectile is fired straight up from the Earth’s South Pole with an initial speed vi= 8.0 km/s. Find the maximum height it reaches, neglecting air drag. Physics 114A - Lecture 30

18. Satellite Orbits and Energies The tangential velocity v needed for acircular orbit depends on the gravitationalpotential energy Ug of the satellite at theradius of the orbit. The needed tangentialvelocity v is independent of the mass m ofthe satellite (provided m<<M). Notice that to make v larger, you need to go deeper into the gravity well, i.e., to a lower orbit where –Ug is larger and r is smaller. Physics 114A - Lecture 30

19. Orbital Energetics The equationK = -½Ugis called “The Virial Theorem”. In effect, it says that for a planet in orbit around the Sun, if you turned its velocity by 90o, so that it pointed straight out of the Solar System, you would have only half the kinetic energy needed to escape the Sun’s gravity well. Physics 114A - Lecture 30

20. Example:The Total Energy of a Satellite Show that the total energy of a satellite in a circular orbit around the Earth is half of its gravitational potential energy. E Although derived for this particular case, this is a general result, and is called the Virial Theorem. The factor of ½ is a consequence of the inverse square law. Physics 114A - Lecture 30

21. Black Holes Black holes: If an object is sufficiently massive and sufficiently small, the escape speed will equal or exceed the speed of light, so that light itself will not be able to escape the surface. This is a black hole. It is believed that there is a large black hole (thousands of solar masses) at the center of our galaxy. It would have a radius of about 1.33 x 1010 m, about 0.089 AU. Physics 114A - Lecture 30

22. Gravitational Lensing Light will be bent by any gravitational field; this can be seen when we view a distant galaxy beyond a closer galaxy cluster. This is called gravitational lensing, and many examples have been found. Physics 114A - Lecture 30

23. Gravitational Lensing Physics 114A - Lecture 30

24. Example:A Gravity Map of the Earth Twin satellites launched March 2002 are making detailed measurements of the Earth’s gravitational field. They are in identical orbits, with one satellite in front of the other by 220 km. The distance between the satellites is continuously monitored with micrometer accuracy using onboard microwave telemetry equipment. How does the distance between the satellites change as the satellites approach a region of increased mass? As the satellites approach a region of increased mass, they are pulled forward by it, but the leading satellite experiences a greater force that the trailing satellite. Therefore, the distance between them increases, providing an indication of a mass concentration. Physics 114A - Lecture 30

25. g for a Solid Sphere Gravitational field of a solid uniform sphere Physics 114A - Lecture 30

26. g for a Hollow Sphere Gravitational field of a uniform spherical shell Physics 114A - Lecture 30

27. Tides Usually we can treat planets, moons, and stars as though they were point objects, but in fact they are not. When two large objects exert gravitational forces on each other, the force on the near side is larger than the force on the far side, because the near side is closer to the other object. This difference in gravitational force across an object due to its size is called a tidal force. Physics 114A - Lecture 30

28. Tides Tidal forces can result in orbital locking, where the moon always has the same face towards the planet – as does Earth’s Moon. If a moon gets too close to a large planet, the tidal forces can be strong enough to tear the moon apart. This occurs inside the Roche limit; closer to the planet we have rings, not moons. Physics 114A - Lecture 30

29. Tides This figure illustrates a general tidal force on the left, and the result of lunar tidal forces on the Earth on the right. Physics 114A - Lecture 30

30. Tides Which has a larger effect on the Earth’s ocean tides, the Sun or the Moon? It turns out that there are two factors that control the tidal effect of a given celestial object:(1) The third power of the angular size of the object in the sky, and (2) The massdensity of the object. By a remarkable celestial accident, the Sun and Moon have almost exactly the same size in the sky (so solar eclipses are possible), but the Moon has 3 times the mass density of the Sun. Therefore, the Moon’s tidal effect on Earth’s oceans is 3 times that of the Sun. Physics 114A - Lecture 30

31. End of Lecture 30 • For Thursday, read Walker, Chapter 13.1-4. • Homework Assignment 9 is due at 11:59 PM on Sunday, March 16. • Do the Clicker Survey at:https://catalyst.uw.edu/webq/survey/ltres/228318 Physics 114A - Lecture 30