Announcement I Physics 1408-00 Lecture note is on the web Handout (6 slides/page) Principles of Physics Lecture 10 Chapter 6 February 1, 008 Sung-Won Lee Sungwon.Lee@ttu.edu Announcement II SI session by Reginald uvilla SI sessions will be at the following times and location. No SI session on hursday Next one: Monday 4:30-6:00pm - Holden http://highenergy.phys.ttu.edu/~slee/1408/ *** Class attendance is strongly encouraged and will be taken randomly. Also it will be used for extra credits. HW Assignment #4 is placed on MateringPHYSICS, and is due by 11:59pm on Wendseday, /18 Chapter 6 Gravitation & Newton s Synthesis 1.! Newton s Law of Universal Gravitation.! Gravity Near the Earth s Surface; Geophysical Applications 3.! Satellites and Weightlessness 4.! Kepler s Laws and Newton s Synthesis 5.! ypes of Forces in Nature Moon s Acceleration Newton s Law of Universal Gravitation Newton proposed that every object in the universe attracts every other object with a force that has the following properties: 1.! he force is inversely proportional to the distance between the objects..! he force is directly proportional to the product of the masses of the two objects. F1 on mm = F on 1 = G 1 r [Universal gravitational constant]! Newton looked at proportionality of accelerations between the Moon and objects on the Earth i.e. F! acceleration! (1/distance) Centripetal Acceleration! he Moon experiences a centripetal acceleration as it orbits the Earth am = v 4" rm = rm! = =.7 # 10 $3 m / s rm We know that rm = 60 RE Universal gravitation predicts am = g(re/rm) = g/3600 =.7e-3 m/s
Surface Gravity Variation of g with Height! Near the Earth s surface, the distance to the center of the earth is roughly constant for heights h which is small compared to the radius of the earth:! Fg = G! M $ MEm = m # G E & RE " RE % = g! Experimentally, this is just as observed: Fg = mg = ma! m!! a = g! g=g h ME = 9.81 m / s RE RE M!! If an object is some distance h above the Earth s surface, RE becomes RE + h 6-4 Satellites and Weightlessness (a)! An object in an elevator at rest exerts a force on a spring scale equal to its weight; F = w-mg = 0, w=mg (b)! In an elevator accelerating upward at! g, the object s apparent weight is 1! times larger than its true weight; F = w mg = ma, w = mg + ma, w=3/g, where a =! g (a)! In a freely falling (a=-g) elevator, the object experiences weightlessness : the scale reads zero; w = mg + ma = mg + m(-g) = 0 Find the net force on the Moon (mm = 7.35 x 10 kg) due to the gravitational attraction of both the Earth (me = 5.98 x 104 kg) and the Sun (ms =1.99 x 1030 kg) he acceleration due to gravity varies over the Earth s surface due to altitude, local geology, and the shape of the Earth What is the force of gravity, FG, acting on a 000-kg spacecraft when it orbits two Earth radii from the Earth s center (that is, a distance re = 6380 km above the Earth s surface)? he mass of the Earth is me = 5.98 x 104 kg. F! acceleration! (1/distance) Estimate the effective value of g on the top of Mt. Everest, 8850 m (9,035 ft) above sea level. hat is, what is the acceleration due to gravity of objects allowed to fall freely at this altitude?
he Pre-History of Gravitation A Little History he study of the structure of the universe is called cosmology. he ancient Greeks developed a cosmological model (see the picture) that placed the earth at the center of the universe. he ancients observed that the stars were fixed, while the planets moved against the background of fixed stars. hey were very interested in the stars because the movements of the stars were correlated with the seasons, growing cycles, etc. Aristotle (384 BC - 3 BC) Ptolemy s cosmology became the Standard Model of the universe for about 1,400 years.! Claudius Ptolemy (85-165) earth was at the center of a nested set of transparent spheres, with the fixed stars on the outer sphere and" the planets! Nicolaus Copernicus (1473-1543) argued that the Sun was the center of the universe, and that the Earth was" one of the planets revolved about it in circular orbits. From 1570 to 1600,! Danish astronomer ycho Brahe compiled a set of extremely accurate astronomical observations.! Johannes Kepler (1571-1630) Kepler inherited ycho s observations and tried to make sense! of them, using algebra and geometry. He deduced three laws of! planetary motion (we will see these later)! Galileo Galilei (1564-164) He discovered the telescope, used this to view the stars and planets. He discovered that the planet Venus has phases, like the Moon, that Saturn had rings, and that four tiny points of light can be seen around Jupiter. Newton hypothesized that the force of gravity acting on the planets is inversely proportional to their distances from the Sun.! Isaac Newton (164-177) 6-5 Kepler s Laws and Newton's Synthesis Kepler s laws describe planetary motion. 6-5 Kepler s Laws and Newton's Synthesis 1.! he orbit of each planet is an ellipse, with the Sun at one focus. An ellipse is a closed curve such that the sum of the distances from any point P on the curve to two fixed points (called the foci, F 1 and F ) remains constant. hat is, the sum of the distances, F 1 P + F P, is the same for all points on the curve.
6-5 Kepler s Laws and Newton's Synthesis. An imaginary line drawn from each planet to the Sun sweeps out equal areas in equal times. 6-5 Kepler s Laws and Newton's Synthesis 3. he square of a planet s orbital period is proportional to the cube of its mean distance from the Sun. he two shaded regions have equal areas. he planet moves from point 1 to point in the same time as it takes to move from point 3 to point 4. Planets move fastest in that part of their orbit where they are closest to the Sun. Notes About Ellipses - Math! F 1 and F are each a focus of the ellipse! hey are located a distance c from the center! he longest distance through the center is the major axis! a is the semi-major axis! he shortest distance through the center is the minor axis! b is the semi-minor axis! he eccentricity of the ellipse is defined as e = c /a! For a circle, e = 0! he range of values of the eccentricity for ellipses is 0 < e < 1 Kepler s First Law! A circular orbit is a special case of the general elliptical orbits! Direct result of the inverse square nature of the gravitational force! Elliptical (and circular) orbits are allowed for bound objects! A bound object repeatedly orbits the center! An unbound object would pass by and not return! hese objects could have paths that are parabolas (e = 1) and hyperbolas (e > 1)! Pluto has the highest eccentricity of any planet (a)! e Pluto = 0.5! Halley s comet has an orbit with high eccentricity (b)! e Halley s comet = 0.97 he satellite s angular momentum! L = m r v = m r v sin! 1 1 1 Kepler s nd Law Fig. shows a satellite moving in an elliptical orbit around a star or planet at one focus. he tangential component of v:! v = v sin! he satellite moves forward a small distance!s during!t. his motion defines the triangle of area!a. he triangle area:! " A = # base# altitude = # r # " s sin! = r vsin! " t he rate at which the area is swept out by! he satellite as it moves is! " A 1 L = r v! = = " t sin constant m Kepler s nd Law is a consequence of the conservation of angular momentum! Angle between r and v Area Law from Conservation of Angular Momentum Central force (along radius) implies angular momentum conserved.! L =! r!! p = m! r!! v = constant da = 1! r! d r! = 1! r! v! dt = L m dt " " A 1 L = r v! = = " t sin constant m da dt = L m = constant
Squaring both side and solving for gives! 4! = r GM 3 Kepler s 3 rd Law An important parameter of circular motion is the period. Recall that the period is the time to complete one full orbit: see the following relationship.! 1 circumference! r GM v = = = 1 period r m V R = mr! = G mm R he square of the period is proportional to the cube of the radius: Kepler s 3 rd Law!!! herefore, Kepler s 3 rd Law is a direct consequence of Newton s Law of Gravity.! In the log vs log r plot, the data for the planets of the solar system fall on a power-law straight line specified by:! log 10 = 1.500 log 10 r - 9.64!! = " R 3 = GM 4" he Solar System 4! = r GM 3 Extra-solar Planets Astronomers, using the most advanced telescopes, have recently seen evidence of planets orbiting nearby stars. hese are called extra-solar planets. Suppose a planet is observed to have a 100 day ( = 1.037x 10 8 s) period as it orbits a star at the same distance that Jupiter is from the Sun. What is the mass of the star in solar masses? 1 solar mass is defined to be the mass of the sun. 3 4! 3 4! r 31 = r M = =.59" 10 kg GM G 31.59! 10 kg M / M S = = 13 solar masses 30 1.99! 10 kg Example, Mass of the Sun! Using the distance between the Earth and the Sun, and the period of the Earth s orbit, Kepler s third Law can be used to find the mass of the Sun! Similarly, the mass of any object being orbited can be found if you know information about objects orbiting it Example 6-8: Where is Mars? Mars period (its year ) was first noted by Kepler to be about 687 days (Earth-days), which is (687 d/365 d) = 1.88 yr (Earth years). Determine the mean distance of Mars from the Sun using the Earth as a reference. Example 6-9: he Sun s mass determined. Determine the mass of the Sun given the Earth s distance from the Sun as r ES = 1.5 x 10 11 m.
6-6 Gravitational Field he gravitational field is the gravitational force per unit mass: he gravitational field due to a single mass M is given by: he Gravitational Field! he gravitational field vectors point in the direction of the acceleration a particle would experience if placed in that field! he magnitude is that of the free fall acceleration at that location 6-7 ypes of Forces in Nature Modern physics now recognizes four fundamental forces: 1.!Gravity.!Electromagnetism 3.!Weak nuclear force (responsible for some types of radioactive decay) 4.!Strong nuclear force (binds protons and neutrons together in the nucleus) 6-7 ypes of Forces in Nature So, what about friction, the normal force, tension, and so on? Except for gravity, the forces we experience every day are due to electromagnetic forces acting at the atomic level. 6-8 Principle of Equivalence; Curvature of Space; Inertial mass: the mass that appears in Newton s second law Gravitational mass: the mass that appears in the universal law of gravitation Principle of equivalence: inertial mass and gravitational mass are the same 6-8 Principle of Equivalence; Curvature of Space; (a) Light beam goes straight across an elevator that is not accelerating. (b) he light beam bends (exaggerated) in an elevator accelerating in an upward direction. herefore, light should be deflected by a massive object:
6-8 Principle of Equivalence; Curvature of Space; his bending has been measured during total solar eclipses: (a)!hree stars in the sky. (b)!if the light from one of these stars passes very near the Sun, whose gravity bends the light beam, the star will appear higher than it actually is. 6-8 Principle of Equivalence; Curvature of Space; One way to visualize the curvature of space (a two-dimensional analogy): If the gravitational field is strong enough, even light cannot escape, and we have a black hole.