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Transcript
1 Physics: Mechanics Dr. Bill Pezzaglia Gravity Updated: 2012Jul10 2 Gravity A. The Law of Gravity B. Gravitational Field C. Tides A. Law of Gravity 1. Inverse Square Law 2. Newton’s 4th law 3. Acceleration of Gravity 3 1a. Inverse Square Law 4 1b. Inverse Square Law •Apparent Luminosity drops off inversely proportional to squared distance. •Sun at planet Saturn (10 further away than earth) would appear 1/100 as bright. •Sound behaves the same way •So do electric and magnetic forces 5 1c Gravity obeys inverse square law • Acceleration of gravity is inversely proportional to distance (from center of earth) • Example: At the surface of the earth (one earth radii distance) the acceleration of gravity is nearly g=10 m/s2 6 1 g 2 r • The moon is 60 further away • Acceleration of moon towards earth is hence 602 smaller (about a=0.003 m/s2). g g a 2 60 3600 2. Gravity: Newton’s 4th Law (a) The apple tree story "After dinner, the weather being warm, we went into the garden and drank tea, under the shade of some apple trees," wrote Stukeley, in the papers published in 1752 and previously available only to academics. "He told me, he was just in the same situation, as when formerly, the notion of gravitation came into his mind. It was occasion'd by the fall of an apple, as he sat in contemplative mood. Why should that apple always descend perpendicularly to the ground, thought he to himself." 7 (b) The Law of Gravitation • The mutual force between two bodies is proportional to their masses, and inversely proportional to square of distance. • Newton could not determine the Gravitation Constant “G” 8 (c) Cavendish Experiment: 1797 Over 100 years later Cavendish measures the constant: G=6.67×10-11 Nm2/kg2 Very Small! To have 1 N of force would need 1220 kg masses 1 cm apart! 9 3. The Acceleration of Gravity (a) Galileo’s Law of Falling Bodies • Combining Newton’s 2nd and 4th laws, we see that the mass of the test body cancels out! GmM ma F r2 GM ag 2 r • Hence we derive Galileo’s law that all test bodies fall at the same acceleration “g”, independent of their mass “m” 10 3b. Measure Mass of Earth • Hence if we measure “g”, and know the radius of the earth “r” (measured by ancient greeks), we can determine the mass of the earth! 2 GM g 2 r 9.8 6.4 10 m gr 24 M 6 10 kg 11 m 3 G 6.67 10 kg s 2 2 m s2 6 11 12 3c. Escape Speed • The “gravitational potential energy” is the amount of work we would have to do to lift a mass “m” from surface of earth to infinity. GmM U r • Equivalently, it’s the amount of Kinetic Energy an meteoroid would have if it fell to the earth. • Note mass “m” cancels out (all bodies fall at same rate!). • Hence, there is a minimum “escape speed” such that a body will not fall back to earth! [about 11 km/sec or 25,000 miles per hour] 1 2 GmM mv r 2 2GM v 2 gr r B. Gravity Field 1. Action at a Distance 2. Gravitational Field 3. Black Holes etc. 13 1. Action at a Distance “Action at a Distance” (no touching) • Huygens criticized: How can one believe that two distant masses attract one another when there is nothing between them? Nothing in Newton's theory explains how one mass can possible even know the other mass is there. • “actio in distans” (action at a distance), no mechanism proposed to transmit gravity • Newton himself writes: "...that one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity that, I believe no man, who has in philosophic matters a competent faculty of thinking, could ever fall into it." 14 2. The field concept •1821 Faraday proposes ideas of “Lines of Force” • Example: iron filings over a magnetic show field lines Michael Faraday 1791 - 1867 •Gravitational Analogy: –Earth’s mass “M” creates a gravity field “g” –Force of field on mass “m” is: F=mg –(i.e. “weight”) –This eliminates “action at a distance” 15 2b. Definition of Mass There are 3 ways to think about mass 1. Inertial Mass F=ma 2. Passive Gravitational Mass F=mg 3. Active Gravitational Mass GM g 2 r The “Weak Equivalence principle” says that inertial mass equals passive gravitational mass 16 3a. The 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. 17 3b. Bending of Starlight • Newton: Light is NOT affected by gravity • Einstein: Elevator example shows light must be affected by gravity. • Predicts starlight will be bent around sun! • 1919 Measured by Eddington! 18 3c. Black Hole 19 If the mass of a star is very big and its size shrinks very small the escape speed becomes bigger than the speed of light, and not even light can escape! Any mass is compressed into a size smaller than the “Schwarzschild Radius” Rs, it will become a black hole This can happen during a supernova explosion, or later by additional mass falling on a neutron star. Anything that comes closer than the Schwarzschild Radius, will fall in and never escape. 3d. Observing a Black Hole 20 If black how do we see them? Material shed from another star falls towards black hole. Not all the material falls into the hole. Some is ejected at very high energies out “jets” along the axis of the black hole. 3e. Radio Lobes from galaxy Centaurus A 21 Black holes at the center of galaxies have a mass of over a billion stars combined! C. Tidal Forces 1. History 2. Tidal Force 3. Cycle of Tides 22 1. Discovery of Tides Alexander the Great knew nothing about tides and his entire fleet was stranded on a sand bar in the Indian Ocean 23 2. Tidal Forces 24 This animation illustrates the origin of tidal forces. Imagine three identical billiard balls placed some distance from a planet and released. The closer a ball is to the planet, the more gravitational force the planet exerts on it. Thus, a short time after the balls are released, the yellow 1-ball has moved a short distance, the green 2-ball has moved a longer distance, and the red 3-ball has moved a still longer distance. From the perspective of the center ball (the 2-ball), a force seems to have pushed the 1-ball away from the planet, and a force seems to have pulled the 3-ball toward the planet. These forces are called tidal forces. 2b. Two Tides! 25 You get a high tide on BOTH sides of the earth! 2c Tides (continued) • The friction of tides are slowing the earth down (day is getting longer!), causes the moon to move further away by 1 cm a year (to conserve angular momentum) 26 2d. Tidal Strength Sun’s tides are only half as strong because its further away 27 3a Cycle of Tides: Daily • High tides twice a day, at (near) transit (upper culmination) and lower culmination of moon • (Lunar) Tides are 25/2= 12.5 hours apart • Tides can be early/late by half hour or more because of influence of sun pulling it off to side. 28 3b Tides from BOTH moon AND sun 29 3c Spring Tides At Full Moon the tidal forces add, and you get a really BIG “spring” or “king” tide. Since tides are created on both sides of earth, you also get a spring tide at New Moon when the sun and moon are on same side of earth 30 3d Neap Tides At First Quarter and Last Quarter Moon, the weaker tidal force of the sun partially cancels out the lunar tide, and you get a really small “NEAP” tide. 31 Notes • Hewitt does escape speed in chap 10. 32