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Transcript
Chapter 7: Rotational Motion and the Law of Gravity Objectives • Be able to distinguish between a rotation and a revolution. • Be able to distinguish between frequency and period. • Be able to calculate tangential speed. • Understand the concept of a centripetal acceleration. Circular Motion revolution: object moving in a circular (or elliptical) path around an axis point rotation: object spinning around its axis period (T): time required for one complete cycle frequency (f): number of cycles per unit time T 1 f hertz (hz): cycles/second Uniform Circular Motion v tangential speed d 2 r v 2 r f t T r What is the tangential speed (in m/s) of a palm tree on the equator? What is it for a Ponderosa pine in Polson? Rearth = 6380 km Centripetal Acceleration Dv = vf – vi = vf + (– vi) vi r q vf d vf Dv q – vi q r centripetal acceleration (ac): a center-seeking change in velocity Dv d v r Dv v t v r Dv v 2 t r v2 ac r Objectives • Understand the concept of centripetal force. • Be able to identify or give examples of forces acting as centripetal forces. • Be able to solve centripetal force problems. Centripetal Force centripetal force: any center-seeking force that results in circular motion v Fc v v Fc is unbalanced: it causes a change in velocity. Fc and v are perpendicular: no net work is done by Fc so the KE (and speed) remains constant. F m a Fc m a c m v2 Fc r m ( 2 r T ) 2 Fc r m 4 2 r Fc T2 Centripetal Forces Forces acting as centripetal forces: hammer throw (tension) motorcycle cage (normal force) car turning on road (friction) moon orbiting earth (gravity) e- orbiting nucleus (electromagnetic) Centripetal Force At what maximum speed that a car make a turn of radius 12.3 meters if the coefficient of friction between the tires and the road is 1.94? What is the magnitude of the Fc if the mass of the car is 1383 kg? Twirl-O Problem On the popular Twirl-O, a passenger is held inside a large spinning cylinder. If the radius of the ride is 4.0 m, with what rotational period must the ride rotate in order for the passenger to not fall? The ms between the wall and the passenger is 0.60. Objectives • Understand how Newton’s third law relates to the concept of a “centrifugal” force. • Explain how simulated gravity could be achieved on a spacecraft. • Be able to solve simulated gravity problems. “Centrifugal Force” The force equal-and-opposite to a centripetal force is known as a centrifugal force. can on bug (FC is FN) bug on can (~ FW) From the bug’s point of view, it feels like the normal force exerted upward by the ground. Simulated Gravity FN = FC FW simulated weight (FW) = FC = m · aC = m · g 4 r aC 2 T 2 A simulated gravity can be produced by adjusting r and T. If r = 95 m, what does T need to be ? Centripetal Force Extra-Credit At what minimum height will a Hot Wheels car make it around the loop-the-loop without falling? Hint: at the top of the loop the only force acting is Fw (= Fc) h=? find the equation r Objectives • Explain the factors that affect the force of gravity between two objects. • Understand the concept of the universal gravitational constant, G. • Be able to solve gravitation problems. Universal Gravitation 1660s: Isaac Newton first realized that gravity keeps the moon in orbit around the earth (FG = Fc) gravity: an attractive force between two masses What factors affect the strength of the force? FG ~ m1· m2 FG ~ 1 / r2 m1 m2 FG ~ r2 m1 m2 FG G r2 Universal Gravitational Constant • “Big G” was first measured by Cavendish in 1797 • G = 6.67 x 10-11 Nm2/kg2 Mass of the Earth The earth has a radius of 6380 km. If a 1.0 kg mass weighs 9.81 N, what is the mass of the earth? Universal Gravitation Problem How much gravitational force does the sun (150 million km away = 1 AU) exert on a 65 kg person? Msun = 2.00 x 1030 kg. Objectives • Be familiar with Kepler’s third law. • Understand how his law can be derived. • Perform calculations related to the law. A Brief History of Astromony • Ptolemy, Aristotle, and the Catholic Church: geocentric model • Aristarchus, Aryabhata, Copernicus: heliocentric model • Galileo: moons orbit Jupiter • Kepler develops 3 laws of orbital motion Kepler’s Third Law • Johannes Kepler (1619): r3/T2 = 1 for all planets in our solar system • r = # AU and T = # yrs Planet T (yrs) r (AU) T2 r3 Mercury 0.24 0.39 0.06 0.06 Venus 0.62 0.72 0.39 0.37 Earth 1.00 1.00 1.00 1.00 Mars 1.88 1.52 3.53 3.51 Jupiter 11.9 5.20 142 141 Saturn 29.5 9.54 870 868 Kepler’s 3rd Law Proof FG FC G mp M S r 2 m p 4 2 r T2 G MS r3 2 2 4 T 3 3 rA rB For any pair of satellites 2 2 TA TB orbiting the same star/planet. What is the orbital period of Jupiter if r = 5.2 AU? Objectives • Be able to explain why the same side of the moon always faces the earth. • Be able to explain how the force of gravity relates to ocean tides. • Understand the concept of a black hole. The Moon’s Orbit • center of mass ≠ center of gravity • as it orbits, the same side of moon must face the earth • rotational T = orbital T The Tides • FG ~ 1/r2, so FA > FB > FC, • tidal bulges form (not to scale!) • two high, two low tides daily (polar view) Tidal Forces • Fg of the sun is 180 X greater than the moon • but Fg from moon has 2X greater difference: SUN on EARTH Near side: 3.5456 x1022 N Far side: 3.5452 x1022 N Difference: 0.0004 x1022 N MOON on EARTH Near side: 0.0207 x1022 N Far side: 0.0198 x1022 N Difference: 0.0009 x1022 N Twice as much! Tides quarter moons new moon (most extreme) full moon Extreme Tides The tides are most extreme (higher and lower) at higher latitudes 15 m at Bay of Fundy, Nova Scotia