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8. Gravity

8. Gravity. Kepler’s Laws. Copernicus to Newton. Heliocentric Model. Kepler’s 1 st Law. All planets move in elliptical orbits with the sun at one focus. Kepler’s 2 nd Law. A line joining any planet to the sun sweeps out equal areas in equal times. Kepler’s 3 rd Law.

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8. Gravity

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  1. 8. Gravity

  2. Kepler’s Laws

  3. Copernicus to Newton

  4. Heliocentric Model

  5. Kepler’s 1st Law All planets move in elliptical orbits with the sun at one focus

  6. Kepler’s 2nd Law A line joining any planet to the sun sweeps out equal areas in equal times

  7. Kepler’s 3rd Law semi-major axis The square of the period of a planet is proportional to the cube of the semi-major axis of its orbit

  8. Example – Kepler’s 3rd Law What is the orbital period of Jupiter? Mean distance between Jupiter and the sun is 5.2 AU Kepler’s 3rd Law Astronomical Unit (AU) = Mean Earth-Sun distance

  9. Example – Kepler’s 3rd Law Divide equation for Jupiter by the one for Earth

  10. Kepler’s 3rd Law

  11. Kepler’s 3rd Law

  12. Universal Gravitation

  13. Universal Gravitation (1) Newton reasoned that the motion of a projectile and that of an object orbiting the Earth, for example the Moon, are essentially the same. He suggested that a universal force, he called gravity, governed the motion of both.

  14. Universal Gravitation (2) Newton’s task was to guess the precise mathematical form of the law of universal gravitation. His main clue is that this law had to be consistent with the three laws of planetary motion discovered by Kepler.

  15. Universal Gravitation (3) Newton concluded that the force of gravity is proportional to the product of the masses of the two objects and the inverse square of their separation

  16. Universal Gravitation (4) The negative sign implies that the force is attractive

  17. Universal Gravitation (5) Consider motion in a circle where the centripetal force is provided by gravity where T is the period Since we obtain Kepler’s 3rd Law:

  18. M m r Acceleration of Gravity, g (1) According to F = ma, the gravitational force Fg of an object of mass M on one of mass m causes the acceleration

  19. M m r Acceleration of Gravity, g (2) This result shows, consistent with observation, that all objects a distance r from an object of mass M have the same acceleration due to gravity, regardless of mass, m

  20. ME m r Acceleration of Gravity, g (3) At the Earth’s surface, r = RE, the acceleration due to gravity is given by

  21. Orbital Motion

  22. Example – The Space Station The International Space Station is in a circular orbit of altitude 380 km. What are its orbital speed and period ? For a circular orbit = 7.7 km/s and = 5.5 x 103 s where G = 6.67 x 10-11 Nm2/kg2 M = 5.97 x 1024 kg r = 6.37 x 106 m + 380 x 103 m

  23. Aphelion and Perihelion The expression suggests that the orbital speed is lower when a planet is at aphelion than when it is at perihelion Earth here ~Jan. 4

  24. Closed and Open Orbits For elliptical orbits (of which a circular orbit is a special case) the motion repeats indefinitely. However, if the initial speed is large enough, the orbit becomes open, that is, it does not repeat. The form of the orbit depends on the total energy of the orbiting object.

  25. Geosynchronous Orbit What altitude is required for geosynchronous orbit? By definition, the period of a geosynchronous orbit is 24 h or 86,400 s. From we obtain = 42,200 km, that is, an altitude of 36,000 km (or 22,000 miles)

  26. Gravitational Energy

  27. A B Gravitational Energy (1) Recall that for a conservative force we can define the potential difference as the negative of the work done on the object in going from point A to point B. Note:

  28. A B Gravitational Energy (2) Therefore, the potential energy difference in moving an object from A to B is given by

  29. A B Gravitational Energy (3) If we move an object from radius r = r1 to r =r2 = infinity then the gravitational energy of the object increases by That is, the gravitational energy at infinity is greater than that at finite values of r.

  30. A B Gravitational Energy (4) So, if we choose the gravitational energy to be zero at infinity, then its value at any other radius r must be negative: However, sometimes, it is more convenient to choose the zero energy at the Earth’s surface.

  31. m ME RE Escape Speed (1) If we throw a ball upwards its speed decreases, while its gravitational energy increases. If the ball could reach infinity its gravitational energy would have increased by

  32. m ME RE Escape Speed (2) Suppose that the ball reaches infinity with zero kinetic energy. Conservation of energy implies that the ball’s initial kinetic energy has been transformed into gravitational energy. Hence, Note that this is just an application of the work kinetic energy theorem:

  33. m ME RE Escape Speed (3) This condition defines the escape velocity from an object of mass M and radius R. For the Earth, setting M = ME and R = RE, we get 11.2 km/s

  34. Escape Speed – Black Holes According to current physics, the maximum speed of an object is the speed of light, c. This suggests the possible existence of objects from which nothing can escape. These objects are called black holes. There is mounting evidence that such objects exist! Event horizon Schwarzschild radius Karl Schwarzschild 1873 - 1916

  35. Energy in Circular Orbits The kinetic energy of an object in a circular orbit is given by while the gravitational energy is Therefore, the total energy K + U is given by

  36. Energy in Circular Orbits We see that the total energy E = –GMm/2r is negative, that is, less than that of an object that can reach infinity with zero kinetic energy. Orbits with total energy < 0 are closed, or bound, while those with total energy > 0 are open, or unbound.

  37. The Gravitational Field What is gravity? One view is that gravity is a physical entity called a field that is created by an object. A field is something that has values at every point in space. In the case of the gravitational field, a vector is attached to every point in space. For a spherical (or point) mass M, the vectors are given by The gravitational force can then be written as

  38. Summary • Universal gravitation is an attractive force that acts between any two objects. The magnitude of the force is Fg = Gm1m2 / r2 where m1 and m2 are the masses and r their separation. G is the gravitational constant. • If we choose the gravitational energy to be zero at infinity then the gravitational energy can be written as U(r) = –GMm / r. • Closed orbits: E < 0; open orbits E > 0.

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