Transcript L23_gravity
Board Work 1. A satellite revolves around its planet in a perfectly circular orbit at a constant speed. a. At each of the four positions, draw a vector representing the net force on the satellite. Label all the force vectors F. b. At each position, draw a vector to represent the satellite’s velocity. Label each vector v. Board Work c. Are all F vectors the same magnitude? d. Are all v vectors the same magnitude? e. What is the angle between F and v at any position of the satellite? f. Is any component of F ever parallel to v? g. Is the KE of the satellite varying or constant? Gravity The laws of physics are universal § 12.1– 12.3 What’s the point? • Gravity is one of the fundamental forces. Historical Background • Earthly and celestial objects thought fundamentally different – Earthly objects seek a lowly position – Celestial objects move in perfect circles Newton’s Insight • The force pulling us to the ground is the same force curving the moon’s path: gravity! Poll Question Which is greater? A. B. C. D. The pull of gravity from the earth on the moon. The pull of gravity from the moon on the earth. Both forces are equally strong. Cannot tell without more information. Force of Gravity We have seen F = mg (g = gravitational field) This is true, but: • g varies with location • both in direction and magnitude Newton’s Law of Universal Gravitation Gravitational force between two particles: m1m2 F = –G r 2 r G: universal gravitational constant, 6.672 10–11 Nm2/kg2 m1, m2: masses of the particles r: distance between particles Properties of Gravity m1m2 F=G r2 • direction is toward the other object • magnitude decreases as (distance)2 increases • never becomes zero • infinite at zero separation (?!) Gravity between Real Objects .. The actual force acting on an object is the sum of all forces on all its particles from all the particles of the other object! A Lucky Break The gravitational field around a sphere is the same as would be around a point particle of the same mass at the center of the sphere. Poll Question The planet Saturn is about 100 times more massive than Earth, and about 10 times farther from the Sun. How does the gravitational attraction between Saturn and the Sun compare to that between Earth and the Sun? A. B. C. D. E. Saturn is attracted with 1/100 the force as Earth. Saturn is attracted with 1/10 the force as Earth. Their attractions to the Sun are about the same. Saturn is attracted with 10 times the force as Earth. Saturn is attracted with 100 times the force as Earth. Poll Question The planet Saturn is about 100 times more massive than Earth, and about 10 times farther from the Sun. How does the acceleration of Saturn toward the Sun compare to that of Earth toward the Sun? A. B. C. D. E. Saturn accelerates 1/100th as much as Earth. Saturn accelerates 1/10th as much as Earth. Their accelerations are about the same. Saturn accelerates 10 times as much as Earth. Saturn accelerates 100 times as much as Earth. Poll Question When two objects are separated by a distance D, the gravitational force between them is F. If they move closer, to 1/2 the distance (D/2), the force becomes F F D A. B. C. D. F/2. 2F. D/2 4F. Cannot tell without more information. Gravitational Field • Gives the force acting on an object of mass m GM F = m · field field = g = 2 r • M = mass of object creating the field • r = distance from the object • field is a vector! Gravity at Earth’s Surface • • • • G = 6.670 10–11 Nm2/kg2 Earth’s mass = 5.976 1024 kg = M Earth’s radius = 6378174 m = d so the field g is g = (6.670 10–11 Nm2/kg2 ) = 9.8 N/kg look familiar? (5.976 1024 kg ) (6378174 m )2 Group Work Find the tangential speed v of an object in circular orbit a distance r from an attractor of mass M. – Use the fact that the centripetal force is the force of gravity. Find the square of the orbital period T2 as well. – Use the fact that v = 2pr/T. Kepler’s Laws of planetary motion 1. Planets travel in elliptical paths with one focus at the Sun. 2. A planet’s path traces out equal areas in equal times. 3. The square of a planet’s orbital period is directly proportional to the cube of the semi-major axis of the orbit. Kepler’s Laws 2. Equal-area law 3. T2 a3 – For planets, y and AU are easy units Kepler’s laws • Work for other systems too (planets and moons, etc.) Example Problem Calculate the orbital distance of the Moon based on the observation that the Moon orbits the Earth once every 27.3 days. The mass of the Moon is 7.35 1022 kg and the mass of the Earth is 6.01024 kg. Double Systems • Both objects orbit system’s center of mass • Orbital radii < separation • Centripetal force = gravity