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Early Observations of Planetary Motion Just looking at the sky, one would assume that the earth is the center of the universe, and that everything we see is in orbit around us. Early models of the solar system were geocentric, but some of the planets were observed to exhibit retrograde motion: Kepler’s Laws: Early Observations of Planetary Motion The geocentric model was made more elaborate, with epicycles upon epicycles, but ultimately was unable to explain retrograde motion. Kepler’s Laws and Newton’s Synthesis Planets move in planar elliptical paths with the Sun at one focus of the ellipse. During equal time intervals the radius vector from the Sun to a planet sweeps out equal areas. If T is the time that it takes for a planet to make one full revolution around the Sun, and if S is half the major axis of the ellipse (S reduces to the radius of the planet’s orbit if that orbit is circular), then: where C is a constant whose value T2 ΣFn = G but S3 =C ∴G m1mSun v12 m = 1 r1 r12 2πr1 v1 = T1 mSun 4π 2 r1 = 2 r12 T1 ⇒ T12 4π 2 = r13 GmSun ⇒ C= 4π 2 GmSun is the same for all planets. 1 The asteroid belt between Mars and Jupiter consists of many fragments, once hypothesized to have been a planet. a) If the center of mass of the asteroid belt is about 3.0 times farther from the Sun than the Earth is, how long would have it taken for the hypothetical planet to orbit the Sun? b) Can we use this data to deduce the mass of that planet? Perturbations Kepler’s Laws are not exactly correct Perturbations in orbits were observed Newton expected this A body’s motion is influenced by the gravitational attraction of ALL other bodies in the universe. ⎛ m ⎞ m m F Earth = GmEarth ⎜⎜ 2 Sun rˆSun − Earth + 2 Mars rˆMars − Earth + 2 Venus rˆVenus − Earth + ... ⎟⎟ r r r Mars − Earth Venus − Earth ⎝ Sun − Earth ⎠ Newton’s Idea Force at a distance People were familiar with “contact” forces. This was new (and NOT universally accepted initially) Implications Causality Deterministic Universe Gravitational force per unit mass at point. “Caused” by a mass Taken from Electricity and Magnetism Gravity Electroweak Formally, electromagnetic and weak nuclear Strong nuclear Lead to “Modern Physics” Predominate view for about 200 years Force at a distance and causality caused problems Concept of a gravitational field Forces in Nature Relativity Quantum Mechanics Non-deterministic Causality is difficult New Ideas GUT – Grand Unified Theory TOE – Theory of Everything Sting Theory Holographic theory MORE SLIDES TO FOLLOW!!! 2 ConcepTest 12.1a Earth and Moon Which is stronger, Earth’s pull on the Moon, or the Moon’s pull on Earth? ConcepTest 12.2 Fly Me Away 1) the Earth pulls harder on the Moon You weigh yourself on a scale inside 2) the Moon pulls harder on the Earth an airplane that is flying with constant 3) they pull on each other equally speed at an altitude of 20,000 feet. 4) there is no force between the Earth and the Moon How does your measured weight in the 5) it depends upon where the Moon is in its orbit at that time measured on the surface of the Earth? airplane compare with your weight as 1) greater than 2) less than 3) same At a high altitude, you are farther away from the By Newton’s 3rd Law, the forces are center of Earth. Therefore, the gravitational force in equal and opposite. the airplane will be less than the force that you would experience on the surface of the Earth. ConcepTest 12.3 Two Satellites Two satellites A and B of the same mass are going around Earth in concentric orbits. The distance of satellite B from Earth’s center is twice that of satellite A. What is the ratio of the centripetal force acting on B compared to that acting on A? ConcepTest 12.4 Averting Disaster 1) It’s in Earth’s gravitational field 1) 1/8 2) 1/4 The Moon does not 3) 1/2 crash into Earth 4) it’s the same because: 5) 2 2) The net force on it is zero 3) It is beyond the main pull of Earth’s gravity 4) It’s being pulled by the Sun as well as by Earth 5) none of the above Using the Law of Gravitation: Mm F =G 2 R The Moon does not crash into Earth because of its high Note the 1/r2 factor speed. If it stopped moving, it would, of course, fall directly into Earth. With its high speed, the Moon would fly off into space if it weren’t for gravity we find that the ratio is 1/4. providing the centripetal force. Follow-up: What happens to a satellite orbiting Earth as it slows? Page 1 1 ConcepTest 12.5 In the Space Shuttle ConcepTest 12.6 Guess my Weight 1) They are so far from Earth that Earth’s gravity doesn’t act any more. If you weigh yourself at the equator Astronauts in the 2) Gravity’s force pulling them inward is cancelled by the centripetal force pushing them outward. space shuttle 3) While gravity is trying to pull them inward, they float because: are trying to continue on a straight-line path. of Earth, would you get a bigger, smaller or similar value than if you 1) bigger value 2) smaller value 3) same value weigh yourself at one of the poles? 4) Their weight is reduced in space so the force of gravity is much weaker. Astronauts in the space shuttle float because The weight that a scale reads is the normal force exerted by the floor (or the scale). At the equator, you are in circular motion, so they are in “free fall” around Earth, just like a satellite or the Moon. Again, it is gravity that there must be a net inward force toward Earth’s center. This means that the normal force must be slightly less than mg. So the scale would register something less than your actual weight. provides the centripetal force that keeps them in circular motion. Follow-up: How weak is the value of g at an altitude of 300 km? ConcepTest 12.7 Force Vectors A planet of mass m is a distance d from Earth. Another planet of mass 2m is a distance 2d from Earth. Which force vector best represents the direction of the total gravitation force on Earth? 2d 5 4 d 1 The force of gravity on the Earth due to m is greater than the force due to 2m, which means that the force component pointing down in the figure is greater than the component pointing to the right. 2m 2 3 m F2m = GME(2m) / (2d)2 = 1/2 GMm / d2 Fm = GME m / d2 = GMm / d2 Page 2 2