Chapter 4. Motion and gravity
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1 Chapter 4. Motion and gravity Announcements Labs open this week to finish. You may go to any lab section this week (most people done). Lab exercise 2 starts Oct 2. It's the long one!! Midterm exam likely Oct 23 or 25, in class. Assignment 2 handed out Oct. 2, due Oct 10. Supplementary text problems (should be straightforward) 3-51, 3-54, 4-53, 4-57a,b,c Answers will be on web site
2 Objects in Motion : A review speed rate at which an object moves, i.e. the distance traveled per unit time [m/s; mi/hr] velocity an object s speed in a certain direction, e.g. 10 m/s moving east acceleration a change in an object s velocity, i.e. a change in either speed or direction is an acceleration [m/s2]
3 Newton s Laws of Motion Examples of the 3 laws of motion, paraphrased: 1. Move in straight lines unless outside force. 2. Forces cause accelerations. 3. Equal and opposite reactions.
4 Forces : A review Forces change the motion of objects. momentum the (mass x velocity) of an object force anything that can cause a change in an object s momentum As long as the object s mass does not change, the force causes a change in velocity, or an
5 Falling objects When you drop a ball: It starts with vo= 0 It accelerates down a = g = 10 m/s2 This value of g only true near the Earth's surface! After 1 sec, v=10 m/s After 2 sec, v=20 m/s. v = vo + a t The acceleration is of course caused by gravity We can calculate the mass of the Earth!
6 Universal Law of Gravitation Between every two objects there is an attractive force, the magnitude of which is directly proportional to the mass of each object and inversely proportional to the square of the distance between the centers of the objects.
7 Mass of the Earth A mass m feels a downward force and accelerates F = m a, but here a=g, so F = m g This downward force is of course gravity due to the Earth of mass M. Since we are the Earth's radius R above the Earth's center, gravity's force is GMm/R2 Setting these two equal : mg = GMm/R2 But the m's cancel, so for ANY object, g=gm/r2 So M = gr2/g = (10 m/s2)(6.4x106 m)2/(6.67x10-11nm2/kg2) = 6 x 1024 kg (confirm the units!)
8 Acceleration due to gravity Galileo experimentally showed that the acceleration an object experiences under gravity is independent of its mass. This means that 'inertial mass' (the tendency of objects to keep their motion) is the same as 'gravitational mass' (which appears in the law of gravity) Newton was puzzled by this... Einstein explained it. (see ASTR )
9 Central forces explain orbits The central (or centripital) force provided by gravity causes objects to orbit the origin of the force, continually changing the direction of the object's velocity If the force were to stop, linear motion resumes
10 In some sense, objects in orbit are always `falling' towards the gravitational center Newton performed this thought experiment with a cannon on a mountain top. A cannon ball shot fast enough will 'fall around the Earth' in an orbit. Something shot very fast can escape.
11 Orbital Paths : Conic sections Extending Kepler s First Law, Newton found that ellipses were not the only orbital paths. possible orbital paths circle (bound) e=0 ellipse (bound) 0<e<1 parabola (unbound) hyperbola (unbound)
12 Newton s Version of Kepler s 3rd Law Using calculus, Newton derived Kepler s three Laws from his own Law of Gravity. In the 3rd Law's most general form: P = 4 a / G (m1 + m2) SO : If you can measure the orbital period of two objects (P) and the distance between them (a), then you can calculate the sum of the masses of both objects (m1 + m2). TEST : Ensure you can recover Kepler's 3rd law for objects orbiting the Sun! (i.e., prove Earth's orbit takes 1 year given Msun, aearth in kg, km)
13 Celestial Mechanics......is the study of orbital motion Has been a driving force behind the development of mathematics and physics
14 An object's distance varies as it moves around its orbit (if e>0) Perihelion 'helios' = around Sun, so perihelion 'gee' = around Earth, so perigee Aphelion
15 Note period is independent of 'e'!?! Kepler's Third Law (and Newton's version of it also) depends ONLY on the semimajor axis 'a' While the speeds at various places around the orbit may differ, the period is the same if a is identical.
16 The next step in the scale So far in the course we have seen how to measure (in absolute units) The radius of the Earth The radius of the Moon The distance to the Moon But how does one figure out: The distance to the Sun? the scale of the planetary orbits (even in a relative sense)? Kepler managed the latter
17 Geometry establishes relative scale 1 AU defined to be semimajor axis of Earth; the astronomical unit becomes the yardstick Kepler used geometry of some planetary configurations to calculate the size of other planetary orbits. Interior planet (left) is easy: use greatest elongation. Exterior planet (right) needs more advanced trigonometry sin = r/(1au) so rau = sin. eg: Venus: =44 so r = 0.72 AU
18 But...but...but...what is an AU??? Astronomers didn't know the AU in terms of anything else (like kilometers). Estimates ranged from 8 million km (Copernicus) to 111 million km (Halley) Measuring the AU became a major goal of astronomy Halley realized that 'transits' could do the trick.
19 Transit: Mercury or Venus pass in front of the Sun's disk Doesn't happen every orbit. (Why?) Because Mercury is so small and close to the Sun, this is difficult to observe precisely. There was one in November 1999, and one in May 2003
20 Venus transits are better, but rare Halley realized that the transits of 1761 and 1769 could be used to measure the AU How? Page 218 of text. 1 Astronomical Unit = 1.50 x 108 km
21 Venus transits are better, but rare If observed from different latitudes and longitudes on Earth, the parallax of Venus on the solar disk and the timing of transit yields the AU. Worked in 1761, and 1% accuracy was obtained. Page 218 of text. 1 Astronomical Unit = 1.50 x 108 km
22 Some more orbital mechanics Objects circle around on their orbits forever until something `perturbs them' and changes their orbit. Examples: Passing close to another large object A spacecraft burning fuel Friction with a thin upper atmosphere The last example is why satellites in Earth orbit sometimes slowly spiral down to Earth. The first two examples are important for spacecraft mechanics.
23 How do I get to Mars??? Several spacecraft did this recently. Two arrived in January Surely the easiest spot to do the transfer is when the planets are closest, right?
24 How do I get to Mars??? Several spacecraft did this recently. Two arrived in January Surely the easiest spot to do the transfer is when the planets are closest, right? Wrong. Because things just don't `go straight' between the planets. You have to orbit the Sun.
25 If you orbit the Sun, how do you increase the size of your orbit? You DON'T accelerate towards Mars. You DO accelerate in the direction you are already moving. What does this do to the semimajor axis and eccentricity?
26 The Hohman transfer ellipse The thrust ('delta v') applied in the direction of motion increases the a and produces an e, so one is at perihelion P of an ellipse. With the right delta v you can get the aphelion A at Mars. This is an 'economical ' orbit; less fuel but not the fastest.
27 The Hohman transfer ellipse What is the Earth-Mars orbit's parameters? r2 = 1 AU = a(1-e) r1 = 1.52 AU = a(1+e) Add the equations: 2.52 AU=a(1+e)+a(1-e) 2.52 AU=2a, so a=1.26 AU The use 1AU=(1.26)(1-e) to calculate e=0.206 P = a3/2 = 1.41 yr for whole orbit, so 0.70 yr for half.
28 Getting the timing right Need to launch when Mars slightly ahead of Earth so that Mars is at the right place for arrival Exercise: Show this is when Mars is about 136 degrees before the opposition point of the launch date.
29 Orbit changes can also be natural Flying near a planet will cause a small object to 'change its orbit' The gravitation pull of the planet changes the orbit to a new one. Here, a comet --->> Pre-encouter: Large parabola Post-encounter Small ellipse Remains on ellipse Until another Jupiter encounter.
30 One can arrange that spacecraft `benefit' from the flybys. Free fuel!
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