普通物理 General Physics 13 - Newtonian Gravitation 郭艷光Yen-Kuang Kuo 國立彰化師大物理系暨光電科技研究所 電子郵件: ykuo@cc.ncue.edu.tw 網頁: http://ykuo.ncue.edu.tw

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Outline 13-1 What Is Physics? 13-2 Newton’s Law of Gravitation 13-3 Gravitation and the Principle of Superposition 13-4 Gravitation Near Earth’s Surface 13-5 Gravitation Inside Earth 13-6 Gravitational Potential Energy 13-7 Planets and Satellites: Kepler’s Laws 13-8 Satellites: Orbits and Energy 13-9 Einstein and Gravitation 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 13-1 What Is Physics? When a star considerably larger than our Sun burns out, the gravitational force between all its particles can cause the star to collapse in on itself and thereby to form a black hole. The gravitational force at the surface of such a collapsed star is so strong that neither particles nor light can escape from the surface (thus the term “black hole”). 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-2 Newton’s Law of Gravitation Here, m1 and m2 are the masses of the two particles, r is their separation and G is the gravitational constant. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-2 Newton’s Law of Gravitation m1 and m2 are the masses of the two particles, r is their separation and G is the gravitational constant. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-2 Newton’s Law of Gravitation m1 m2 F12 F21 r m1 m2 r F1 Note: If the particle is inside the shell, the net force is zero. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-2 Newton’s Law of Gravitation Earth can be thought of as a nest of such shells, one within another and each shell attracting a particle outside Earth’s surface as if the mass of that shell were at the center of the shell. Thus, from the apple’s point of view, Earth does behave like a particle, one that is located at the center of Earth and has a mass equal to that of Earth. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-2 Newton’s Law of Gravitation Suppose that, Earth pulls down on an apple with a force of magnitude 0.80 N. The apple must then pull up on Earth with a force of magnitude 0.80 N, which we take to act at the center of Earth. Although the forces are matched in magnitude, they produce different accelerations when the apple is released. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-2 Newton’s Law of Gravitation The acceleration of the apple is about 9.8 m/s2, the familiar acceleration of a falling body near Earth’s surface. The acceleration of Earth, however, measured in a reference frame attached to the center of mass of the apple– Earth system, is only about 1 × 10-25 m/s2. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-3 Gravitation and the Principle of Superposition Given a group of particles, we find the net (or resultant) gravitational force on any one of them from the others by using the principle of superposition. This is a general principle that says a net effect is the sum of the individual effects. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-3 Gravitation and the Principle of Superposition For n interacting particles, we can write the principle of superposition for the gravitational forces on particle 1 as We can express this equation more compactly as a vector sum: 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-3 Gravitation and the Principle of Superposition The gravitational force on a particle from a real (extended) object becomes an integral and we have: In which the integral is taken over the entire extended object and we drop the subscript “net.” 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-3 Gravitation and the Principle of Superposition m1 dm r 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-1 An arrangement of three particles, particle 1 of mass m1=6.0 kg and particles 2 and 3 of mass m2 = m3 =4.0 kg, and distance a=2.0 cm. What is the net gravitational force on particle 1 due to the other particles? 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-1 Solutions: 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-1 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-2 Figure shows arrangement of five particles, with masses m1=8.0 kg, m2 = m3 = m4 = m5 = 2.0 kg, and with a = 2.0 cm and θ= 30°. What is the gravitational force 　　 on particle 1 due to the other particles? 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-2 Solutions: 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-2 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-4 Gravitation Near Earth’s Surface We measure g at various points on the surface of the earth, we find that its value is not constant. This is attributed to three reasons: 1. The earth’s mass isn’t uniformly distributed. 2. The earth is not a sphere. 3. The earth is rotating. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-4 Gravitation Near Earth’s Surface Earth’s mass is not uniformly distributed. The density (mass per unit volume) of Earth varies radially, and the density of the crust (outer section) varies from region to region over Earth’s surface. Thus, g varies from region to region over the surface. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-4 Gravitation Near Earth’s Surface Earth is not a sphere: Earth is approximately an ellipsoid, flattened at the poles and bulging at the equator. Its equatorial radius is greater than its polar radius by 21 km. Earth is rotating: The rotation axis runs through the north and south poles of Earth. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-4 Gravitation Near Earth’s Surface 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-4 Gravitation Near Earth’s Surface 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-3 (a) An astronaut whose height h is 1.70 m floats “feet down” in an orbiting space shuttle at distance r = m away from the center Earth. What is the distance between the gravitational acceleration at her feet and at her head? 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-3 (a) Solutions: 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-3 (b) If the astronaut is now “feet down” at the same orbital radius r = m about a black hole mass Mh = kg (10 times our Sun’s mass), what is the difference between the gravitational acceleration at her feet and at her head? 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-3 (b) The black hole has a mathematical surface (event horizon) of radius Rh= m . Nothing, not even light, can escape from that surface or any where inside it. Note the astronaut is well outside the surface (at r = 299 Rh). 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-3 (b) Solutions: 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-5 Gravitation Inside Earth If the particle is inside the shell, the net force is zero. If the particle is outside the shell, the force is given by: The net force on m is: Here, Mins us the mass of the part of the earth inside a sphere so radius r. F is linear with r. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-5 Gravitation Inside Earth If the particle is inside the shell, the net force is zero. If the particle is outside the shell, the force is given by: The net force on m is: 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-5 Gravitation Inside Earth Mins: mass of the part of the earth inside asphere so radius r. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-4 In Pole to Pole, an early science fiction story by George Griffith, three explorers attempt to travel by capsule through a naturally formed (and, of course, fictional) tunnel directly from the south pole to the north pole . According to the story, as the capsule approaches Earth’s center, the gravitational force on the explorers becomes alarmingly large and then, exactly at the center, it suddenly but only momentarily disappears. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-4 Then the capsule travels through the second half of the tunnel, to the north pole. Check Griffith’s description by finding the gravitational force on the capsule of mass m when it reaches a distance r from Earth’s center. Assume that the Earth is a sphere of uniform density (mass per unit volume). 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-5 Gravitation Inside Earth Solutions: 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-6 Gravitational Potential Energy The negative sign of U expresses the fact that the corresponding gravitational force is attractive. We take into account each pair once. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-6 Gravitational Potential Energy 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-6 Gravitational Potential Energy We move a baseball from point A to point G along a path consisting of three radial lengths and three circular arcs (centered on Earth). We are interested in the total work W done by Earth’s gravitational force on the ball as it moves from A to G. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-6 Gravitational Potential Energy The work done along each circular arc is zero, because the direction of is perpendicular to the arc at every point. Thus, W is the sum of only the works done by along the three radial lengths. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-6 Gravitational Potential Energy Path Independence: Since the work W done by a conservative force is independent of the actual path taken, the change U in gravitational potential energy is also independent of the path taken. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-6 Gravitational Potential Energy Potential Energy and Force Escape Speed Note: The escape speed does not depend on m. m B v = 0 v A 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-5 An asteroid, headed directly toward Earth, has a speed of 12 km/s relative to the planet when the asteroid is 10 Earth radii from Earth’s atmosphere on the asteroid, find the asteroid’s speed vf when it reaches Earth’s surface. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-5 Solutions: 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-7 Planets and Satellites: Kepler’s Laws Figure shows a planet of mass m moving in such an orbit around the Sun, whose mass is M. We assume that M >> m, so that the center of mass of the planet–Sun system is approximately at the center of the Sun. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-7 Planets and Satellites: Kepler’s Laws The orbit is described by giving its semimajor axis a and its eccentricity e, the latter defined so that ea is the distance from the center of the ellipse to either focus F or F’. An eccentricity of zero corresponds to a circle, in which the two foci merge to a single central point. The eccentricities of the planetary orbits are not large; so if the orbits are drawn to scale, they look circular. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-7 Planets and Satellites: Kepler’s Laws Kepler’s First Law: All planets move on elliptical orbits with the sun at one focus. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-7 Planets and Satellites: Kepler’s Laws Kepler’s Second Law : The line that connects a planet to the sun sweeps out equal areas ΔA in the plane of the orbit in equal time intervals. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-7 Planets and Satellites: Kepler’s Laws Kepler’s second law is equivalent to the law of conservation of angular momentum 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-7 Planets and Satellites: Kepler’s Laws Kepler’s Third law : The square of the period of any planet is proportional to the cube of the semi-major axis of its orbit. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-7 Planets and Satellites: Kepler’s Laws Note 1: The ratio does not depend in the mass m of the planet but only on the mass M of the central star. Note 2: For elliptical orbits, the ratio remains constant. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-6 (a) Comet Halley orbits the sun with a period of 76 years and, in 1986, had a distance of closest approach to the Sun, its perihelion distance Rp, of m. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-6 (a) Table 13-3 shows that this is between the orbits of Mercury and Venus. What is the comet’s farthest distance from the Sun, which is called it’s aphelion distance Ra? 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-6(a) Solutions: 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-6 (b) What is the eccentricity of the orbit of comet Halley? Solutions: 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-7 Let’s return to the story that opens this chapter. Figure shows the observed orbit of the star S2 as the star moves around a mysterious and unobserved object called Sagittarius A* (pronounced “A star”), which is at the center od the Milky Way garaxy. 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-7 S2 orbits Sagittarius A* with a period of T = 15.2 y and with a semimajor axis of a = 5.50 light-days ( = m ). What is the mass M of Sagittarius A*? What is the A*? 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-7 Planets and Satellites: Kepler’s Laws Solutions: 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-8 Satellites: Orbits and Energy 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-8 Satellites: Orbits and Energy 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-8 (a) A playful astronaut releases a bowling ball, of mass m=7.20 kg into circular orbit about Earth at an altitude h of 350 km. What is the mechanical energy E of the ball in its orbit? 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-8 (a) Solutions: 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-8 (b) What is the mechanical energy E0 of the ball on the launchpad at Cape Canaveral? From there to the orbit, what is the change ΔE in the ball’s mechanical energy? 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 Example 13-8 (b) Solutions: 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

13-9 Einstein and Gravitation Principle of Equivalence The fundamental postulate of this theory about gravitation (the gravitating of objects toward each other) is called the principle of equivalence. Curvature of Space 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授 End of chapter 13! 2018/11/7 普通物理講義-13 / 國立彰化師範大學物理系/ 郭艷光教授