From Essential University Physics 3 rd Edition by Richard Wolfson, Middlebury College 2016 by Pearson Education, Inc.

Transcription

1 PreClass Notes: Chapter 8 From Essential University Physics 3 rd Edition by Richard Wolfson, Middlebury College 2016 by Pearson Education, Inc. Narration and extra little notes by Jason Harlow, University of Toronto This video is meant for University of Toronto students taking PHY131. Toward a Law of Gravity Newton was not the first to discover gravity. Newton discovered that gravity is universal. Legend: Newton, sitting under an apple tree, realizes that the Earth s pull on an apple extends also to pull on the Moon Pearson Education, Inc. Slide 1-2 1

2 Outline 8.1,8.2 Newton s Law of Universal Gravitation 8.3 Orbital Motion 8.4 Gravitational Potential Energy 8.5 The Gravitational Field Image of the Moon from ] Newton s genius was to recognize that the motion of the apple and the motion of the Moon were the same, that both were falling toward Earth under the influence of the same force. R.Wolfson Toward a Law of Gravity In Aristotle s time, motion of planets and stars in the heavens was not expected to be governed by the same laws as objects on Earth. Newton recognized that a force directed toward the Sun must act on planets This is similar to force that Earth exerts on an apple that falls toward it. Newtonian synthesis: The same set of laws apply to both celestial and terrestrial objects Pearson Education, Inc. Slide 1-4 2

3 Universal Gravitation Law of universal gravitation: Everything pulls on everything else. Every body attracts every other body with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance separating them Pearson Education, Inc. Slide 1-5 Universal Gravitation Introduced by Isaac Newton, the Law of Universal Gravitation states that any two masses m 1 and m 2 attract with a force F that is proportional to the product of their distances and inversely proportional to the distance r between them. F Gm 1 m 2 r 2 Here G = N m 2 /kg 2 is the constant of universal gravitation. Newton invented calculus to show that this law applies to spherical masses using the centre-to-centre distance for r Pearson Education, Inc. Slide 1-6 3

4 Inverse Square Law 1 16 N 2012 Pearson Education, Inc. Slide 1-7 Got it? If the masses of two planets are each somehow doubled, the force of gravity between them A. doubles. B. quadruples. C. reduces by half. D. reduces by one-quarter. 4

5 Fast moving projectiles: Satellites! Sufficient tangential velocity is needed for orbit. With no air drag to reduce speed, a satellite goes around Earth indefinitely. Satellite motion is an example of a high-speed projectile. A satellite is simply a projectile that falls around Earth rather than into it. Images from and Orbits 5

6 Orbits The downward acceleration due to gravity is g 10 m/s 2. t = 1 second after a ball is thrown horizontally, it has fallen a distance No matter how fast the girl throws ball sideways, 1 second later it has fallen 5 m below the horizontal line Orbits Curvature of Earth Earth surface drops a vertical distance of 5 meters for every 8000 meters tangent to the surface 6

7 Orbits What speed will allow the ball to clear the gap? 8000 m per second: 8 km/s! Kepler s Laws of Planetary Motion 1 st Law: The path of each planet around the Sun is an ellipse with the Sun at one focus. 2 nd Law: The line from the Sun to any planet sweeps out equal areas of space in equal time intervals. 3 rd Law: The square of the orbital period of a planet is directly proportional to the cube of the average distance of the planet from the Sun (for all planets). 7

8 Orbits Ellipse specific curve, an oval path Example: A circle is a special case of an ellipse when its two foci coincide. Projectile Motion and Orbits The parabolic trajectories of projectiles near Earth s surface are actually sections of elliptical orbits that intersect Earth. The trajectories are parabolic only in the approximation that we can neglect Earth s curvature and the variation in gravity with distance from Earth s center. 8

9 Circular Orbits In a circular orbit, gravity provides the force of magnitude mv 2 /r needed to keep an object of mass m in its circular path about a much more massive object of mass M. Therefore, Orbital speed: Orbital period: Kepler s third law: For satellites in low-earth orbit, the period is about 90 minutes. Gravitational Potential Energy Because the gravitational force changes with distance, it s necessary to integrate to calculate potential energy changes over large distances. This integration gives 9

10 Gravitational Potential Energy It s convenient to take the zero of gravitational potential energy at infinity. Then the gravitational potential energy becomes Ur ( ) = GMm r This result holds regardless of whether the two points are on the same radial line. Gravitational Potential Energy 10

11 Energy and Orbits The total energy E = K + U determines the type of orbit an object follows: E < 0: The object is in a bound, elliptical orbit. Special cases include circular orbits and the straight-line paths of falling objects. E > 0: The orbit is unbound and hyperbolic. E = 0: The borderline case gives a parabolic orbit. Energy and Orbits 11

12 Got it? Suppose the paths in the figure are the paths of four projectiles. All four projectiles were launched from a common point at the top of the figure. Which projectile had the second-highest initial speed? A. The projectile with the closed path. B. The projectile with the hyperbolic path. C. The projectile with the parabolic path. D. The projectile with the elliptical path. Escape Speed An object with total energy E less than zero is in a bound orbit and can t escape from the gravitating center. With energy E greater than zero, the object is in an unbound orbit and can escape to infinitely far from the gravitating center. The minimum speed required to escape is given by Solving for v gives the escape speed: Escape speed from Earth s surface is about 11 km/s. 12

13 Energy in Circular Orbits In the special case of a circular orbit, kinetic energy and potential energy are precisely related: U 2K Thus in a circular orbit the total energy is E K U K 1 2 U GMm 2r This negative energy shows that the orbit is bound. The lower the orbit, the lower the total energy but the faster the orbital speed. This means an orbiting spacecraft needs to lose energy to gain speed. Got it? A moon is orbiting around Planet X. Which of the following statements is always true about its kinetic energy (K), and its gravitational potential energy (U)? A. K < 0 and U < 0 B. K < 0 and U > 0 C. K > 0 and U < 0 D. K > 0 and U > 0 E. K < 0 and U = 0 13

14 The Gravitational Field Fields are represented by field lines radiating into the object (Earth). The inward direction of arrows indicates that the force is always attractive to Earth. The crowding of arrows closer to Earth indicates that the magnitude of the force is larger closer to Earth. The Gravitational Field Inside a planet, it decreases to zero at the center because pull from the mass of Earth below you is partly balanced by what is above you. Outside a planet, it decreases to zero at infinity because you are farther away from planet. 14