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Physics 231

Physics 231. Topic 9: Gravitation. Alex Brown October 24-29 2013. What’s up? (Friday Oct 24) 1 ) The correction exam will open at noon today and is due at 10 pm Tuesday Oct 28 th . The exam grades will be sent out after that on Wednesday Oct 29.

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Physics 231

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  1. Physics 231 Topic 9: Gravitation Alex Brown October 24-29 2013

  2. What’s up? (Friday Oct 24) 1) The correction exam will open at noon today and is due at 10 pm Tuesday Oct 28th. The exam grades will be sent out after that on Wednesday Oct 29. 2) Homework 07 is due Tuesday Nov 5th and covers Chapters 9 and 10. It is a little longer that usual so you may want to start early.

  3. Key Concepts: Gravitation Newton’s Law of Gravitation Gravitational Acceleration Planetary Motion Kepler’s Laws Gravitational Potential Energy Conservation of ME Artificial Satellites Covers chapter 9 in Rex & Wolfson

  4. The gravitational force Newton: G=6.673·10-11N m2/kg2 N m2/kg2 = m3/(kg s2) The gravitational force works between every two massive particles in the universe.

  5. Gravitation between two objects A B The gravitational force exerted by the spherical object A on B can be calculated as if all of A’s mass would is concentrated in its center and likewise for object B. Conditions: B must be outside of A A and B must be ‘homogeneous’

  6. Gravitation between two objects The force of the earth on the moon is equal and opposite to the force of the moon on the earth!

  7. Gravitation between two objects

  8. Clicker Quiz!Earth and Moon If the distance to the Moon were doubled, then the force of attraction between Earth and the Moon would be: a) one quarter b) one half c) the same d) two times e) four times

  9. Clicker Quiz!Earth and Moon If the distance to the Moon were doubled, then the force of attraction between Earth and the Moon would be: a) one quarter b) one half c) the same d) two times e) four times The gravitational force depends inversely on the distance squared. So if you increase the distance by a factor of 2, the force will decrease by a factor of 4.

  10. Gravitational acceleration at the surfaceof planet with mass M F=mg g = GM/R2

  11. Gravitational acceleration at the surface of the earth g=GMe/Re2 Me=5.97x1024 kg

  12. Gravitational potential energy So far, we used: PEgravity=mghOnly valid for h near earth’s surface. More general: PEgravity=-GMm/r Earlier we noted that we could define the zero of PEgravity anywhere we wanted. So the surface of the earth is as good as anywhere! R2=R1+h R1 < R2 Thus… PE1 < PE2 R1

  13. Gravitational potential energy So far, we used: PEgravity=mghOnly valid for h near earth’s surface. More general: PEgravity=-GMm/r Earlier we noted that we could define the zero of PEgravity anywhere we wanted. So the surface of the earth is as good as anywhere! R2=R+h R

  14. Gravitational potential energy PEgravity=mgh only valid for h near earth’s surface. More general: PEgravity=-GMm/r PE=0 at infinite distance from the center of the earth (r = ∞) Application: what should the minimum initial velocity of a rocket be if we want to make sure it will not fall back to earth? KEi+ PEi= ½mv2 - GMm/R KEf+ PEf= 0 v = (2GM/R) = (2gR) = 11.2 km/s

  15. Second cosmic speed Second cosmic speed: speed needed to break free from a planetof mass Mp and radius Rp (gp = GMp/Rp2) v2=(2GMp/Rp) = (2gpRp) For earth: g = 9.81 m/s2 R = 6.37x106m v2 = 11.2 km/s

  16. Orbital Velocities What does the word orbit mean? An orbit is the gravitationally curved path of an object around a point in space. To orbit the object, you need to satisfy the kinematic conditions of that type of orbit (more on this shortly…)

  17. launch speed 4 km/s 6 km/s 8 km/s

  18. First cosmic speed First cosmic speed: speed of a satellite of mass m on a low-lying circular around a planet with orbit of Mp and radius Rp(gp= GMp/Rp2) rsatellite≈ Rp F = mac mgp = mv2/Rp so v1=(gpRp) For earth:g = 9.81 m/s2 R = 6.37x106 m v1=7.91 km/s

  19. Period for orbits Consider an object in circular motion around a larger one

  20. Two common cases Planets and other objects orbiting the sun Moon and satellites orbiting the earth

  21. Synchronous orbit Synchronous orbit of a satellite: rotation period of satellite of mass m is the same as rotation period of the planet For earth: period T = 24 hours = 86 x 103 s r3 = T2/K = 75 x 1021 r= 42 x 106 m Re= 6.4 x 106 m (r/Re)= 6.6

  22. Total mechanical energy for Orbits Consider a planet in circular motion around the sun:

  23. launch speed = 10 km/s

  24. Kepler’s laws Johannes Kepler (1571-1630)

  25. Kepler’s First law p+q = constant An object A bound to another object B by a force that goes with 1/r2 moves in an elliptical orbit around B, with B being in one of the focus point of the ellipse.

  26. Kepler’s second law PEgravity=-GMEarthm/r A line drawn from the sun to the elliptical orbit of a planet sweeps out equal areas in equal time intervals. Area(D-C-SUN) = Area(B-A-SUN)

  27. Kepler’s second law PEgravity=-GMEarthm/r rmax speed and kinetic energy are smallest rmin rmax rmin speed and kinetic energy are largest

  28. Kepler’s third law PEgravity=-GMEarthm/r a same as for a circular orbit except r is replaced by the semi-major axis (red line) a = ½(rmin+rmax)

  29. An Example Two planets are orbiting a star. The orbit of A has a radius of 1x108km. The distance of closest approach of B to the star is 5x107 km and its maximum distance from the star is 1x109 km. If A has a rotational period of 1 year, what is the rotational period of B? B star A Need to use Kepler’s 3rd Law

  30. An Example B star A Rmin = distance of closest approach = 5x10710(perihelion) Rmax = maximum distance = 1x109 (aphelion) RA = 1x108km RB = ½(Rmin+Rmax) = ½(5x107 + 1x109) = 5.25x108 km R3/T2 = constant RA3/TA2= RB3/TB2 so TB2=(RB3/RA3)TA2 So TB=(5.253 x (1 yr)2) = 12 years

  31. Orbiting condition (angular momentum L) b a

  32. Quantities that are constant for a given orbit b a

  33. Quantities that depend on distance r b a r

  34. Clicker Quiz! Averting Disaster The Moon does not crash into Earth because: a) it’s in Earth’s gravitational field b) the net force on it is zero c) it is beyond the main pull of Earth’s gravity d) it’s being pulled by the Sun as well as by Earth e) its velocity is large enough to stay in orbit

  35. Clicker Quiz! Averting Disaster The Moon does not crash into Earth because: a) it’s in Earth’s gravitational field b) the net force on it is zero c) it is beyond the main pull of Earth’s gravity d) it’s being pulled by the Sun as well as by Earth e) its velocity is large enough to stay in orbit The Moon does not crash into Earth because of its high speed. If it stopped moving, it would fall directly into Earth. With its high speed, the Moon would fly off into space if it weren’t for gravity providing the centripetal force.

  36. Eccentricity: e

  37. Eccentricity: e circle when e = 0

  38. Kepler’s First law Eccentricity: e An object A bound to another object B by a force that goes with 1/r2 moves in an elliptical orbit around B, with B being in one of the focus point of the ellipse.

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