#### Transcript 05_Circular_orbits_2012Sep27

1 Physics: Mechanics Dr. Bill Pezzaglia Circular Motion & Gravity Updated: 2012Sep27 2 Circular Motion & Gravity A. Angular Kinematics B. Central Force and Orbits C. Ficticious Forces 3 A. Angular Kinematics 1. Angles 2. Angular Velocity 3. Angular Acceleration 1. Angles Perhaps 5000 BC people changed from a nomadic culture to agrarian, settling in Sumer. Sumerians needed a calendar to tell them when to plant food. 4 Surviving Babylonian “Cuneiform” Clay Tablets of astronomical positions of sun & planets x 5 The Ecliptic Circle 6 The Babylonians determined the exact path of the sun through the zodiac constellations Hammurabi [1810-1750 BC] • school of scribes defines sexagessimal numbers (base 60). 7 (a) Angles in sexagesimal • Degrees: 360=circle = 24 hours • Arcminutes (minutes of arc): 60’=1 • Arcseconds (seconds of arc): 60”=1’ 8 (b) Angle in Radians • Circle: 360=2 radians • Or 1 radian =57.3 • Arc length formula is easy in radians: s=R • Its messier in degrees: sR 57.3 9 (c) Angular Displacement 10 • Analogous to linear displacement • Definition: how far its rotated: =f- I • There is a “wrap around” ambiguity, e.g. if you look at the clock and it says 3, and later it says 5, is it 2 hours later or 14 hours later? • Technically angular displacement is a pseudovector (not a “vector”), it represents a plane, not a line. 2. Angular Velocity (a) Definition: (analogous to average velocity) • Expressed in terms of period “T” for one complete revolution: • For constant angular velocity we have simple equation t 2 T 0 t 11 (b) Units of Angular Velocity • SI units: radians per second t Other systems used: • Degrees/second • Rpm: revolutions per minute (example, convert 1 rpm to rads/sec): 1 rev min 1min 2 rads 0.105 rads sec 60 sec 1rev 12 (c) Tangential Velocity • The (tangential) speed at a spot distance “r” from axis can be calculated as the total distance traveled (circumference of circle) in one period: 2 r vt T vt r 13 3. Angular Acceleration (a) Definition t • SI Units: rads/sec2 • Kinematic equations for rotation under constant angular acceleration are completely analogous to those for linear acceleration: t 12 t 2 2 2 14 15 B. Central Force & Orbits 1. Centripetal Force 2. Orbits 3. Newton’s Law of Gravity 1. Centripetal Force (a) Newton (1684) introduces concept of the force towards the center needed to make an object go in a circular path. 2 v 2 F m mR R 16 (b) Turning the Curve 17 •What is the maximum speed a car can go around a curve of radius “R” given friction coefficient . Does the result depend upon the mass of the car (should truck go slower than a car)? v Rg (c) Banked Curves •What banking angle will make a car turn a curve even if no friction? v Rg tan 18 2. Orbitals (a) Galileo deduces that if the cause of the centripetal acceleration is gravity (centripetal force) then we can calculate the orbital speed 2 v ac g R v gR 19 Tycho (1546-1601) 20 •Tycho Brahe measuring star positions (without a telescope) •Measurements of position of Mars showed deviations from circular orbits! •Asks Kepler to come and interpret data. b. Johannes Kepler (1571-1630) 21 •Tycho at first invited Kepler to help in analysis of his data, but then jealously wouldn’t let him have the information. •On his deathbed he gave Kepler the data. •Kepler used it (particular data on Mars), to develop three laws of planetary motion. 2c.1 Kepler’s 1st Law (1605) • Law No. 1. Each planet moves around the Sun in an orbit that is an ellipse, with the Sun at one focus. – This is contrary to the earlier belief that the orbits were perfect circles or combinations of circles. 22 23 • Ellipses, circles (parabolas and hyperbolas) are “conic sections”, studied first by the greeks. • But it would NEVER occur to the greeks that an orbit is an ellipse. (why?) Fig 2-3, p.45 24 Drawing an ellipse Focus Focus Fig 2-4, p.45 The Ellipse Do you remember any of this from high school geometry? 25 26 Highly eccentric Focus Focus Not very eccentric 27 Planet orbits tend to have low eccentricity (nearly circular). Comet orbits tend to be highly eccentric. Fig 2-10, p.53 28 2c.2 Kepler’s 2nd Law (1609) Kepler also noticed that when Mars is closest to the Sun in its elliptical orbit, it moves faster than when it is farther away. This led him to formulate his Second Law of Planetary Motion. 29 2c.2 Kepler’s 2nd Law (Equal areas in Equal Times) According to his second law, a planet moves fastest when closest to the Sun (at perihelion) and slowest when farthest from the Sun (at aphelion). As the planet moves, an imaginary line joining the planet and the Sun sweeps out equal amounts of area (shown as colored wedges in the animation) in equal intervals of time. 56 30 2c.2 Kepler’s 2nd Law Kepler’s 2nd law is actually a form of conservation of angular momentum A 1 L 1 mrv 2 rv 2 t m 2m 31 2c.3 Kepler’s 3rd Law: “Harmonic Law” 32 Planets closer to the sun move faster. This is consistent with his 2nd law, that showed a planet will move faster at perihelion. He searched for a relationship between orbital period and distance to the sun. 2c.3 Kepler’s 3rd Law (1618) • The square of the orbital period (P) is directly proportional to the cube of the semimajor axis of the orbit (a). P2 = a3 This law explains the proportions of the sizes of the orbits of the planets and the time that it takes them to make one complete circuit around the Sun. [Note: in physics, the symbol “a” is also used to represent “acceleration”. Confused?] Why is it called the “harmonic law”? Kepler thought the spacing between planets was related to musical intervals. 33 34 35 An example of Kepler’s third law: The orbit of Mars (Recall: P2 = a3) Mars’ orbit period (P) is 1.88 years. P2 = 3.53 Kepler’s law says that P2 = a3, so 3.53 = a3. So then a = (3.53)1/3 (the cube root of 3.53), or 1.52. Thus, the semimajor axis (average distance of Mars from the Sun) is 1.52 Astronomical Units. But how big is an Astronomical Unit? Kepler didn’t know. 2c Inverse Square Law Edmond Halley (1656-1742) deduced that Kepler’s 3rd law implies that gravity (of sun) must decrease with square of distance. 2R v P 2 v 2 R ac 4 2 R P 2 3 P R 1 a 2 R 36 3. Newton’s 4th Law: Gravity • In 1679 Hooke writes Newton, proposing inverse square law. • In 1684 Wren, Hooke and Halley discussed, at the Royal Society, whether the elliptical shape of planetary orbits was a consequence of an inverse square law of force depending on the distance from the Sun. • Later in the same year in August, Halley visited Newton in Cambridge and asked him what orbit a body would follow under an inverse square law of force • Sr Isaac replied immediately that it would be an Ellipsis, the Doctor struck with joy and amasement asked him how he knew it, why, said he I have calculated it, whereupon Dr Halley asked him for his calculation without any farther delay, Sr Isaac looked among his papers but could not find it, but he promised him to renew it, and then to send it him. • Actually, he had apparently derived it in 1680, after correspondence from Hooke. Later Hooke claims Newton stole the idea and did not give him credit! 37 (a) The Law of Gravity • Newton’s 4th law: The mutual force between two bodies is proportional to their masses, and inversely proportional to square of distance. • Gravitation Constant “G” measured 100 years later by Cavendish. 38 b. Newton’s Orbit • Combining Newton’s 2nd and 4th laws, we see that the mass of the test body cancels out! • Orbital speed depends only on size of orbit (and mass of planet) GmM ma F 2 R 2 v GM a 2 R R GM v R 39 c. Newton-Kepler Law 40 •Newton derived Kepler’s 3rd law from scratch. More important, he showed that the mass of the central body can be determined from the orbital period and radius of a moon: 2 v mM m G 2 R R 2R v P 2 2 3 v R 4 R M 2 G G P The Newton-Kepler Law • In the Principia Newton also deduced Kepler's third law, but in an important new form • Mass of central body: M = a3/P2 – Orbital Radius “a” (in astronomical units) – Period “P” (in years) – Mass “M” in units of “solar masses” • To measure mass of – – – – Earth, use moon’s orbit Jupiter, use Galilean moons Sun, use orbits of planets Galaxy, use orbits of stars around galaxy 41 42 C. Fictitious Forces Aka d’Alembert or “pseudo” force 1. Accelerated Frames 2. Centrifugal Force 3. Applications 4. (Coriolis & Euler force?) 1. Accelerated Frames (a) The elevator • • You feel heavier in an elevator accelerating upwards You feel lighter in an elevator accelerating downward Add graphic here 43 1b. The Einstein Equivalence Principle Reference at rest with Gravity is indistinguishable to a reference frame which is accelerating upward in gravity free environment. The apple accelerating downward due to gravity looks the same as an apple at rest in space, with the floor accelerating upward towards it. 44 1c. Sideways acceleration •In an accelerated reference frame (e.g. accelerating truck) you “feel” forces that mimic gravity. 45 2. Centrifugal Force •Huygen (1659?): If you are in a rotating frame of reference, you will “feel” a force outward •At equator of earth it will make you “feel” less heavy by 0.5% 2 v 2 F m mR R 46 3. Applications •Textbook Knight (page 179) makes the rather “poor” statement: “A centrifugal force will never appear on a freebody diagram and never be included in Newton’s Laws”. •I strongly disagree with this overly conservative statement. It implies that “physics” should only be done in inertial frames. This is silly. You live on a rotating earth which is revolving around the sun, so not a single experiment ever done has been in an inertial frame. •There are many problems that are more easily solved by allowing yourself to “sit” on the object that is accelerating. 47 Example: loop the loop •How fast must roller coaster go so it won’t fall off the track? 2 v m mg R v gR 48