Physics 1408-002 Principles of Physics Sung-Won Lee [email protected] Lecture 10 – Chapter 6 – February 12, 2008 Announcement I Lecture note is on the web Handout (6 slides/page) 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, 2/18 Announcement II SI session by Reginald Tuvilla No SI session on Thursday Next one: Monday 4:30 - 6:00pm - Holden SI sessions will be at the following times and location. Chapter 6 Gravitation & Newton’s Synthesis 1.! Newton’s Law of Universal Gravitation 2.! Gravity Near the Earth’s Surface; Geophysical Applications 3.! Satellites and “Weightlessness” 4.! Kepler’s Laws and Newton’s Synthesis 5.! Types of Forces in Nature 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.! The force is inversely proportional to the distance between the objects. 2.! The force is directly proportional to the product of the masses of the two objects. 1 21 on 2 2 on 12m mF F Gr= =[Universal gravitational constant] Moon’s Acceleration •! Newton looked at proportionality of accelerations between the Moon and objects on the Earth i.e. F ! acceleration ! (1/distance)2 Centripetal Acceleration •! The Moon experiences a centripetal acceleration as it orbits the Earth aM=v2rM= rM!2=4"2rMT2= 2.72 # 10$3m / s2Universal gravitation predicts aM = g(RE/rM)2 = g/3600 = 2.7e-3 m/s2 We know that rM = 60 RESurface Gravity •! Near the Earth’s surface, the distance to the center of the earth is roughly constant for heights h which is small compared to the radius of the earth:!RE m M!h Fg= GMEmRE2= m GMERE2!"#$%& = g! Experimentally, this is just as observed: |Fg| = mg = ma!!! a = g!g = GMERE2= 9.81 m / s2•! If an object is some distance h above the Earth’s surface, RE becomes RE + h Variation of g with Height The acceleration due to gravity varies over the Earth’s surface due to altitude, local geology, and the shape of the Earth 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=3/2g, 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 What is the force of gravity, FG, acting on a 2000-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)? The mass of the Earth is mE = 5.98 x 1024 kg. F ! acceleration ! (1/distance)2 Find the net force on the Moon (mM = 7.35 x 1022 kg) due to the gravitational attraction of both the Earth (mE = 5.98 x 1024 kg) and the Sun (mS =1.99 x 1030 kg) Estimate the effective value of g on the top of Mt. Everest, 8850 m (29,035 ft) above sea level. That is, what is the acceleration due to gravity of objects allowed to fall freely at this altitude?A Little History The Pre-History of Gravitation The study of the structure of the universe is called cosmology. The ancient Greeks developed a cosmological model (see the picture) that placed the earth at the center of the universe. The ancients observed that the stars were “fixed”, while the planets moved against the background of fixed stars. They were very interested in the stars because the movements of the stars were correlated with the seasons, growing cycles, etc. earth was at the center of a nested set of transparent spheres, with the fixed stars on the outer sphere and"the planets!Claudius Ptolemy (85-165) 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. Nicolaus Copernicus (1473-1543) From 1570 to 1600, !Danish astronomer Tycho Brahe compiled a set of extremely accurate astronomical observations.! Ptolemy’s cosmology became “the Standard Model” of the universe for about 1,400 years.!Aristotle (384 BC - 322 BC) Johannes Kepler (1571-1630) Kepler inherited Tycho’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 -1642) Newton hypothesized that the force of gravity acting on the planets is inversely proportional to their distances from the Sun.!Isaac Newton (1642 - 1727) 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. 6-5 Kepler’s Laws and Newton's Synthesis Kepler’s laws describe planetary motion. 1.! The orbit of each planet is an ellipse, with the Sun at one focus. 6-5 Kepler’s Laws and Newton's Synthesis 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, F1 and F2) remains constant. That is, the sum of the distances, F1P + F2P, is the same for all points on the curve.6-5 Kepler’s Laws and Newton's Synthesis 2. An imaginary line drawn from each planet to the Sun sweeps out equal areas in equal times. The two shaded regions have equal areas. The planet moves from point 1 to point 2 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. 6-5 Kepler’s Laws and Newton's Synthesis 3. The square of a planet’s orbital period is proportional to the cube of its mean distance from the Sun. Notes About Ellipses - Math •! F1 and F2 are each a focus of the ellipse –! They are located a distance c from the center •! The longest distance through the center is the major axis •! a is the semi-major axis •! The shortest distance through the center is the minor axis –! b is the semi-minor axis •! The eccentricity of the ellipse is defined as e = c /a –! For a circle, e = 0 –! The range of values of the eccentricity for ellipses is 0 < e < 1 Kepler’s First Law •! A