Physics Lecture 03: FRI 29 AUG

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1 Physics 23 Jonathan Dowling Isaac Newton ( ) Physics 23 Lecture 03: FRI 29 AUG CH3: Gravitation III Version: 8/28/4 Michael Faraday (79 867)

2 3.7: Planets and Satellites: Kepler s st Law. THE LAW OF ORBITS: All planets move in elliptical orbits, with the Sun at one focus. Laws were based on data fits! Tycho Brahe Johannes Kepler

3 3.7: Planets and Satellites: Kepler s 2 nd Law 2. THE LAW OF AREAS: A line that connects a planet to the Sun sweeps out equal areas in the plane of the planet s orbit in equal time intervals; that is, the rate da/dt at which it sweeps out area A is constant. Angular momentum, L: A t

3.7: Planets and Satellites: Kepler s 3 rd Law 3. THE LAW OF PERIODS: The square of the period of any planet is proportional to the cube of the semi-major axis of its orbit.

4 3.7: Planets and Satellites: Kepler s 3 rd Law 3. THE LAW OF PERIODS: The square of the period of any planet is proportional to the cube of the semi-major axis of its orbit. Consider a circular orbit with radius r (the radius of a circle is equivalent to the semimajor axis of an ellipse). Applying Newton s second law to the orbiting planet yields Using the relation of the angular velocity, ω, and the period, T, one gets: T = 2π ω

5 3.7: Planets and Satellites: Kepler s 3 rd Law ICPP: 2 (a) The larger the orbit the longer the period: SAT-2. T = 4π 2 r 3 GM r 3/2 (b) The smaller the orbit the greater the speed: SAT-. v = ωr = 2π T = GM r r r LEO = R Earth + a LEO = 0 7 m r GEO = R Earth + a GEO = m T LEO = 4π2 kg s m kg (0 7 m) 3 = 6307s T GEO = 4π2 kg s m kg ( m) 3 = s = 24hrs v LEO = m 3 kg s kg = 7.4km/s 0 7 m v GEO = m 3 kg s kg = 3km/s m

3.7: Newton Derived Kepler s Laws from Inverse Square Law! http://galileo.phys.

htm Kepler s Second Law First: Equal Areas Proportional to Equal Time!

6 3.7: Newton Derived Kepler s Laws from Inverse Square Law! Kepler s Second Law First: Equal Areas Proportional to Equal Time! Rate of sweeping out of area, da / dt = c Angular Momentum is proportional to the angular momentum L, and equal to L/2m = Constant = C. A t

7 3.7: Newton Derived Kepler s Laws from Inverse Square Law! Kepler s First Law: Ellipse with Sun at Focus This is equivalent to the standard (r, q ) equation of an ellipse of semi-major axis a and eccentricity e, with the origin the Sun at one focus. Note /L 2 is from inverse Square Law.

8 3.7: Newton Derived Kepler s Laws from Inverse Square Law! Kepler s 3 rd Law: For Ellipse T 2 a 3

9 Example, Halley s Comet ICPP: Estimate comet s speed at farthest distance? v = v = ωr ω = 2π / T v = 2πr T m 76y s/y 02 m 0 9 s,000m/s

10 3.8: Satellites: Orbits and Energy As a satellite orbits Earth in an elliptical path, the mechanical energy E of the satellite remains constant. Assume that the satellite s mass is so much smaller than Earth s mass. The potential energy of the system is given by For a satellite in a circular orbit, Thus, one gets: For an elliptical orbit (semimajor axis a),

11 ICPP E = GMm 2r de + r 2 dr r T = 4π 2 GM r 3 +r 3 dt +r 2 dr (a) path : As E decreases (de < 0); r decreases (dr < 0) (b) Less: As r decreases (dr < 0); T decreases (dt < 0)

NASA Gravity Recovery and Climate Experiment What Do the Two Satellites Measure? Changing g field! http://www.jpl.nasa.

Low g = GM r 2 + r 2 v = GM r +r 2 dg r 3 dr As g increases (dg > 0); r decreases (dr < 0). dv r 3 2 dr As r decreases (dr < 0); v increases (dv > 0).

12 NASA Gravity Recovery and Climate Experiment What Do the Two Satellites Measure? Changing g field! Earth is NOT a Uniform Sphere > Gravitational Field Changes in Orbit. Rocky Mtn. High ΔM mid-atl. Low g = GM r 2 + r 2 v = GM r +r 2 dg r 3 dr As g increases (dg > 0); r decreases (dr < 0). dv r 3 2 dr As r decreases (dr < 0); v increases (dv > 0). Changing field Δg give rise to changing velocity Δv. Changing Δv gives changing satellite-to-sattellite distance. Microwave link measures changing distance between satellites. Measuring Δg allows computation of ΔM Earth s Mass Distribution.

13 Example, Mechanical Energy of a Bowling Ball

14 3.9: Einstein and Gravitation: Curvature of Space

15 3.9: Einstein and Gravitation: Gravity Waves Two Orbiting Black Holes LIVINGSTON LASER INTERFEROMETER GRAVITATIONAL-WAVE OBSERVATORY Disturbances in the Gravitational Field Move Outward As Waves

HW0 DUE TONIGHT: :59PM FRI 29 AUG! WEB ASSIGN CLASS KEY FOR SECTION 2: lsu 88 373 Tutoring in Middleton & Nicholson (Starts NEXT Week): http://cas.lsu.edu/tutorial-centers Free online tutoring available NOW: http://cas.

16 HW0 DUE TONIGHT: :59PM FRI 29 AUG! WEB ASSIGN CLASS KEY FOR SECTION 2: lsu Tutoring in Middleton & Nicholson (Starts NEXT Week): Free online tutoring available NOW: Minority Student Tutoring via LAMP Program! Apply here: Student Athlete Tutoring: Private Tutors:

Chapter 13. Gravitation

Chapter 13. Gravitation Chapter 13 Gravitation 13.2 Newton s Law of Gravitation Here m 1 and m 2 are the masses of the particles, r is the distance between them, and G is the gravitational constant. G =6.67 x10 11 Nm 2 /kg 2