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9 Pages

### Lecture22

Course: PHYS 2101, Fall 2007
School: LSU
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Word Count: 675

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22 Lecture Gravitation Phys 2101 Gabriela Gonzlez Kepler's laws: I Three laws, but all consequences of just one: Newton's law of gravitation! LAW OF ORBITS: All planets move in elliptical orbits, with the Sun at one focus. 2 x2 + 2 =1 2 a a = semimajor axis b = semiminor axis Sun's distance from center =ea e= eccentricity If e=0, orbit is circular. Animations by by Bill Drennon, Physics Teacher Central Valley...

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22 Lecture Gravitation Phys 2101 Gabriela Gonzlez Kepler's laws: I Three laws, but all consequences of just one: Newton's law of gravitation! LAW OF ORBITS: All planets move in elliptical orbits, with the Sun at one focus. 2 x2 + 2 =1 2 a a = semimajor axis b = semiminor axis Sun's distance from center =ea e= eccentricity If e=0, orbit is circular. Animations by by Bill Drennon, Physics Teacher Central Valley Christian High School Visalia, CA USA , http://home.cvc.org/science/kepler.htm 2 Solar system orbits: not circular, but close 1 AU = 149 598 000 000 meters 3 Kepler's laws: II LAW OF AREAS: A line that connects the planet to the Sun sweeps out equal areas in the plane of the planet's orbit in equal times. "Areal velocity" dA/dt is constant dA ( / 2) 1 2 = = dt 2 Angular momentum: L = = = = 2 dA = dt 2 Conservation of angular momentum! 4 Kepler's laws: III LAW OF PERIODS: The square of the period of any planet is proportional to the cube of the semimajor axis of the orbit. 4 2 3 T2 = 5 A very real example "In spring 2002 S2 was passing with the extraordinary velocity of more than 5000 km/s at a mere 17 light hours distance -- about three times the size of our solar system -- through the perinigricon, the point of closest approach to the black hole. By combining all measurements of the position of S2 made between spring 1992 and summer 2002, we have obtained enough data in order to determine a unique keplerian orbit for this star, presented in Figure 1." "It is highly elliptical (eccentricity 0.87), has a semimajor axis of 5.5 light days, a period of 15.2 years and an inclination of 46 degrees with respect to the plane of the sky. From Kepler's 3rd law we can determine the enclosed mass in a straightforward manner to be 3.71.5 million solar masses. Therefore at least 2.2 million solar masses have to be enclosed in a region with a radius of 17 light hours." http://www.mpe.mpg.de/www_ir/GC/gc.html Schdel, R. et al. A star in a 15.2-year orbit around supermassive the black hole at the centre of the Milky Way. Nature, 419, 694 - 696, (2002). 6 Orbits and Energy Energy conservation: kinetic + potential = constant! Newton's Gravitational law (circular orbits): U = - GMm/r F=ma GMm/r2=m(v2/r) Then, K = m v2 = GMm/2r = - U/2 E = K+U = - U/2+U= U/2 = - GMm/2r Elliptical orbits: E=- GMm/2a Total energy does not depend on eccentricity, only on semimajor axis! 7 Examples (a) What is the gravitational acceleration on Earth, on the equator? (Use g = ag- 2R = GM/R2 - 2R) (b) How fast would the Earth had to rotate to produce weightlessness at the Equator? (c)What is the escape speed in Earth? (Use Ve2 = 2GM/R) (d)What would be the escape velocity is the Earth had the same mass concentrated in only a radius of 5mm? (e) How far from the surface will a particle thrown at 112m/s go? E=K+U = (1/2) mv^2 -GMm/r = constant (f) With what speed will an object hit the Earth if it is dropped from the space shuttle orbiting at 400km above the Earth? (g) How long does the space shuttle take to orbit once around the Earth? (law of periods) (h) How high should a satellite be above the Earth to have a circular geosynchronous orbit? 8 Example Neutron stars are extr...

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LSU - PHYS - 2101
Lecture 23: FluidsPhys 2101 Gabriela GonzlezDensityDensity: mass per unit volume = m/V. If uniform, = m/V SI units: kg/m3QuickTimef and a TIFF (LZW) decompressor are needed to see this picture.2PressurePressure: force per unit area p = F
LSU - PHYS - 2101
Lecture 24: FluidsPhys 2101 Gabriela GonzlezDensityDensity: mass per unit volume = m/V. If uniform, = m/V SI units: kg/m32PressurePressure: force per unit area p = F/A SI units: N/m2 = Pa Non-SI units: 1 atm = 1.01x105 Pa = 760 torr =760 mm
LSU - PHYS - 2101
OscillationsPhys 2101 Gabriela GonzlezOscillationsOscillations = anything that &quot;swings&quot; back and forth. Harmonic oscillations: described with trigonometric functions. Three kinds of oscillations: free, damped and forced.2Examples3Simpl
LSU - PHYS - 2101
OscillationsPhys 2101 Gabriela Gonzlez2 Force: F = m a = - 2 x = - x m kThe force law for SHMDisplacement: x(t) = xm cos( t + ) Velocity: v(t) = dx(t)/dt = - xm sin( t + ) Acceleration: a(t) = dv(t)/dt = - 2 xm cos( t + ) = - 2 x(t)Simpl
LSU - PHYS - 2101
OscillationsPhys 2101 Gabriela Gonzlez2 Force: F = m a = - 2 x = - x m kThe force law for SHMDisplacement: x(t) = xm cos( t + ) Velocity: v(t) = dx(t)/dt = - xm sin( t + ) Acceleration: a(t) = dv(t)/dt = - 2 xm cos( t + ) = - 2 x(t)2Si
LSU - PHYS - 2101
Temperature and heatPhys 2101 Gabriela GonzlezTemperatureZeroth law of thermodynamics: If bodies A and B are each in thermal equilibrium with a third body T, then they are in thermal equilibrium with each other. Or: every body in thermal equilibr
LSU - PHYS - 2101
Ludwig Boltzmann 1866- 1906Lecture 34: ThermodynamicsPhys 2101 Gabriela GonzlezWork done by an ideal gaspV=nRTW= Constant temperature: keep temperature constant, change the volume. The pressure will change, following: p = nRT/V How muc
LSU - PHYS - 2101
Ludwig Boltzmann 1866- 1906Lecture 35: ThermodynamicsPhys 2101 Gabriela GonzlezIdeal gases so farpV = n R T Eint = Q W Eint= (3/2) n R T = n CV T CP = CV + R ; CV=(3/2)R Constant volume: W=0, Q = n CV T Eint= n CV TConstant pressure: W=
LSU - PHYS - 2101
Ludwig Boltzmann 1866- 1906Lecture 36: ThermodynamicsPhys 2101 Gabriela GonzlezIrreversible processesWe can easily tell the direction of the &quot;arrow of time&quot; in irreversible processes which, if spontaneous, only happen one way: spilling fluids
LSU - PHYS - 2101
Final Review 15 weeks in 50 minutes!Phys 2101 Gabriela GonzlezA course overview Ch 2,4 - Kinematics: motion in 1,2 and 3D: x(t), v(t), a(t) Ch 5,6 - Dynamics: F = ma Ch 7,8 - Energy: potential, kinetic, Work-Energy theorem Ch 9,10 - Particles
LSU - PHYS - 2101
Phys 2101 Review problemsIn the figure below, block 1 of mass m1 slides from rest along a frictionless ramp from height h = 3.00 m and then collides with stationary block 2, which has mass m2 = 2m1. After the collision, block 2 slides into a region
LSU - PHYS - 2101
Lecture 1: Introduction Motion along a straight linePhys 2101 Fall 2003 Gabriela GonzlezThe basics about the courseInstructor: Gabriela Gonzlez (www.phys.lsu.edu/faculty/gonzalez/) Office hours: MF, 10:30am - 12:00pm , 266A Nicholson Hall Course
LSU - PHYS - 2101
Lecture 2: Motion in one and two dimensionsPhys 2101 Fall 2003 Gabriela GonzlezMotion in one dimension Position is a function of time: x(t) Velocity is also a function of time, defined as:dx(t ) v(t ) = dt x(t ) = v(t )0 t Acceleration is
LSU - PHYS - 2101
Lecture 4: Friction, circular motionPhys 2101 Fall 2003 Gabriela GonzlezFriction forceFriction force: parallel to surface; opposes motion; proportional to normal force: F= s N F = k NmA=10kg=300What's should mB be for mA to move? What
LSU - PHYS - 2101
Lecture 6: Work, Energy and PowerPhys 2101 Fall 2003 Gabriela GonzlezWork-Energy TheoremThe change in kinetic energy of a particle is equal to the net work done on the particle by all external forces. A constant force moves a body with mass 3kg a
LSU - PHYS - 2101
Lecture 7: Potential EnergyPhys 2101 Fall 2003 Gabriela GonzlezPotential energyWhen a conservative force does work W on an object, it changes the amount of potential energy of that object byU = - WIf the force is constant, W= Fd and U=-Fd. If
LSU - PHYS - 2101
Lecture 12: Linear momentum and energyPhys 2101 Fall 2003 Gabriela GonzlezRocketsThe last stage of a rocket, which is traveling at a speed of 7600 m/s, consists of two parts that are clamped together: a rocket case with a mass of 290.0 kg and a p
LSU - PHYS - 2101
Lecture 18: RollingPhys 2101 Fall 2003 Gabriela GonzlezRolling = Translation + Rotation2Rotation around which axis?It depends on the reference frame!If we sit at the center of the wheelIf we sit on the road3Kinetic energy2 KE = 1 I
LSU - PHYS - 2101
Lecture 19: Exam 2 ReviewPhys 2101 Fall 2003 Gabriela GonzlezTopics Ch8: Conservation of energy: potential energy curves, work-energy theorem Ch9: Systems of particles :center of mass, Newton's laws, linear momentum, conservation of momentum;
LSU - PHYS - 2101
Lecture 20: Torque and Angular momentumPhys 2101 Fall 2003 Gabriela GonzlezTorqueOur definition for torque produced by a given force F about a point P as = F r sin allows us to add up torques only when the forces are all in the same plane. We no
LSU - PHYS - 2101
Phys 2101 Section 3 Fall 2003 Quiz 4 Oct 13, 2003Argentinian gauchos used &quot;boleadoras&quot; to hunt pampa animals, as shown in the figure above. The weapon consists of three balls connected to a common point by a sturdy string. The gaucho first held one
LSU - PHYS - 2101
Lecture 22: Equilibrium and ElasticityPhys 2101 Fall 2003 Gabriela GonzlezEquilibrium: exampleA uniform plank with a length L=6m and weight 500N, rests on the ground an against a frictionless roller at the top of a wall of height h=3m. The plank
LSU - PHYS - 2101
Lecture 23: GravitationPhys 2101 Fall 2003 Gabriela GonzlezThe Law of GravitationNewton's gravitation law (not F=ma!):m 1 m2 F =G 2 rG = 6.67 x 10 -11 N m2/kg22Superposition principleWe can add gravitational forces. We have to add them as
LSU - PHYS - 2101
Lecture 24: GravitationPhys 2101 Fall 2003 Gabriela GonzlezPotential EnergyPotential energy = -work done by gravitational forcesU = W = Z1 rF dx = Z1rGM m GM m dx = x2 rWe were using before Ug=mgh. Was that wrong?Ug = = = = GM
LSU - PHYS - 2101
Lecture 25: FluidsPhys 2101 Fall 2003 Gabriela GonzlezFluidsFluids are substances that can &quot;flow&quot;. Density= mass per unit volume: = m/V (kg/m3) Pressure= force per unit area: p = F/A (N/m2= Pa) Other units for pressure: 1 atm = 1.01 x 105 Pa (
LSU - PHYS - 2101
Lecture 27: OscillationsPhys 2101 Fall 2003 Gabriela GonzlezOscillationsOscillations = anything that &quot;swings&quot; back and forth. Harmonic oscillations: described with trigonometric functions. Three kinds of oscillations: free, damped and forced.
LSU - PHYS - 2101
Lecture 28: OscillationsPhys 2101 Fall 2003 Gabriela GonzlezSimple Harmonic Motion: EnergyIf F=-kx like for a spring, potential energy = -work done by the force is U= k x2. Total mechanical energy is conserved: E= U + K = k x2 + m v2 = k (xm c
LSU - PHYS - 2101
Lecture 29: WavesPhys 2101 Fall 2003 Gabriela GonzlezWavesParticles moving back and forth, forming a wave which is traveling and carries energy away from the source. Examples: Sound waves Seismic waves Water wavesSometimes, it is not particl
LSU - PHYS - 2101
Lecture 35: Temperature and HeatPhys 2101 Fall 2003 Gabriela GonzlezFirst law of ThermodynamicsThe work W done by the system during a transformation from an initial state to a final state depends on the path taken. The heat Q absorbed by the sys
LSU - PHYS - 2101
Lecture 36: The kinetic theory of gasesPhys 2101 Fall 2003 Gabriela Gonzlez Ludwig Boltzmann 1866- 1906Kinetic theory of gasesAt the microscopic level, temperature is a measure of the kinetic energy in the elemental units (atoms or molecules);
LSU - PHYS - 2101
Lecture 37: Kinetic Theory of gasesPhys 2101 Fall 2003 Gabriela GonzlezPressure, temperature and speedMolecules in a gas are colliding with each other, and with the walls. When colliding with walls, they transfer momentum, and the effect is measu
LSU - PHYS - 2101
Lecture 38: Kinetic Theory of gasesPhys 2101 Fall 2003 Gabriela GonzlezIdeal gases so farpV = n R T Eint = Q W Eint= (3/2) n R T = n CV T CP = CV + R Constant volume: W=0, Q = n CV T Eint= n CV TConstant pressure: W= p V = nRT Q = n CP T
LSU - PHYS - 2101
Lecture 39: Entropy: The Second law of ThermodynamicsPhys 2101 Fall 2003 Gabriela GonzlezIrreversible processesWe can easily tell the direction of the &quot;arrow of time&quot; in irreversible processes which, if spontaneous, only happen one way: spilling
LSU - PHYS - 2101
Lecture 40: Heat enginesPhys 2101 Fall 2003 Gabriela GonzlezIdeal heat enginesIdeal heat engines use a cycle of reversible thermodynamic processes. A heat engine transforms energy extracted as heat from thermal reservoirs, into mechanical work. C
LSU - PHYS - 2101
Lecture 41: Final Review 15 weeks in 50 minutes!Phys 2101 Fall 2003 Gabriela GonzlezA course overview Ch 2,4 - Kinematics: motion in 1,2 and 3D: x(t), v(t), a(t) Ch 5,6 - Dynamics: F = ma Ch 7,8 - Energy: potential, kinetic, Work-Energy theore
LSU - PHYS - 2101
SID 323 376 524 1293 1526 1796 1902 2383 2508 2726 2736 3704 3722 3974 4166 4288 4303 4640 5137 5255 6348 6503 6597 6733 6752 7349 7490 7569 7657 7729 7731 7779 7792 7850 7879 7947 7975 8312 8384 8712 8715 8879 8927 9325 9382 9395 9779 9971Exam1 63
LSU - PHYS - 2101
SID 323 376 524 1293 1526 1796 1902 2383 2508 2726 2736 3704 3722 3974 4166 4288 4303 4640 5137 5255 6348 6503 6597 6733 6752 7349 7490 7569 7657 7731 7779 7792 7850 7879 7947 7975 8312 8384 8712 8715 8879 8927 9325 9382 9395 9779P1 10 5 5 11 12 13
LSU - PHYS - 2101
SID 106 323 376 524 857 918 1293 1341 1526 1796 1902 2383 2485 2508 2543 2715 2726 2736 3023 3357 3704 3722 3748 3974 4166 4288 4303 4640 5054 5137 5255 6276 6348 6503 6597 6733 6752 6897 7160 7349 7490 7549 7569 7657 7729 7731 7779 7792 7850 7879 78
LSU - PHYS - 2101
SID 106 323 376 524 857 918 1293 1341 1526 1796 1902 2383 2485 2508 2543 2715 2726 2736 3023 3357 3704 3722 3748 3974 4166 4288 4303 4640 5054 5137 5255 6276 6348 6503 6597 6733 6752 6897 7160 7349 7490 7549 7569 7657 7729 7731 7779 7792 7850 7879 78
LSU - PHYS - 2101
Assignment Name: Due: Totals SID 106 323 376 524 857 918 1293 1341 1526 1796 1902 2383 2485 2508 2543 2715 2726 2736 3023 3357 3704 3722 3748 3974 4166 4288 4303 4640 5054 5137 5255 6276 6348 6503 6597 6733 6752 6897 7160 7349 7490 7549 7569 7657 772
LSU - PHYS - 2101
SID 106 323 376 524 857 918 1293 1341 1526 1796 1902 2383 2485 2508 2543 2715 2726 2736 3023 3357 3704 3722 3748 3974 4166 4288 4303 4640 5054 5137 5255 6276 6348 6503 6597 6733 6752 6897 7160 7349 7490 7549 7569 7657 7729 7731 7779 7792 7850 7879 78
LSU - PHYS - 2101
Lecture 2: Motion in 2-D (projectile motion) Newton's laws Newton'Phys 2101 Gabriela GonzlezNewton's lawsPhysics having predictive value! Newton's laws: two qualitative, one quantitative: 1. If no net force, no acceleration (either at rest, or mo
LSU - PHYS - 2101
Lecture 9: RotationPhys 2101 Gabriela GonzlezTranslation and Rotation21Angular displacement, velocity, accelerationPure rotation of a reference line in a solid body around an axis is described with an angle as a function of time. =s/r ccw:
LSU - PHYS - 2101
Lecture 11: Rolling, angular momentumPhys 2101 Gabriela GonzlezWork, energy, power. linear and circular motion2ExampleA meter stick is held vertically with one end on the floor and is then allowed to fall. Assuming that the end on the floor d
LSU - PHYS - 2101
Lecture 17: FluidsPhys 2101 Gabriela GonzlezExample How deep in the sea is the pressure twice as much the pressure at the surface? Would the answer be different in a mountain lake?Is Scuba diving safe? http:/www.mtsinai.org/pulmonary/books/scu
LSU - PHYS - 2101
Lecture 24: Waves on strings TemperaturePhys 2101 Gabriela GonzlezSuperposition of wavesIf we have two waves traveling in the same string, they will overlap and add up to a resultant wave. Overlapping waves do not in any way alter the travel of e
LSU - PHYS - 2101
Lecture 27: Thermodynamics, EntropyPhys 2101 Gabriela GonzlezIdeal gases so farFor any process (1, 2, 3 or 4): pV = n R T Eint = Q W Eint= (f/2) n R T = n CV T Also, CP = CV + R ; CV=(f/2)R Constant volume (4): W=0, Q = n CV T Eint= n C
Western Michigan - ECE - 3510
DISPLAY UNITPCU-6C Display ModulePCU-6C is a small, lightweight, driver display module designed to present the driver with the information he requires in a clearly legible and simple format via LED displays and LED lamps. Interface to a remote ECU
Western Michigan - ECE - 3510
LIU-4LVDT Interface UnitThe LIU-4 provides signal conditioning for four 5 wire LVDT positioning sensors. When the LVDT sensors are being used in latency critical control systems, the conditioned signals are passed to the host ECU as four 0 to 5V an
Western Michigan - ECE - 3510
SYSTEM MONITORVEHICLE TUNING AND CONFIGURATION TOOLSystem Monitor is a software package for configuring and tuning automotive control systems. It has built-in support for motor sport systems and can be tailored to support third party control units.
Western Michigan - ECE - 3510
ATLASADVANCED TELEMETRY LINKED ACQUISITION SYSTEMATLAS is a software package which is used to obtain, display and analyse data from motorsport and automotive control systems. Familiar controls and extensive use of the mouse, menus and accelerator k
UNC Marching Tar Heels / CheerleadersProposed Continental Tire Bowl ScheduleTuesday, December 28 (see appropriate schedule) FUCHS' GROUP 12:30p Luggage Drop Off (location TBA) (parking provided) 1:15p Report to Carmichael (uniform black and pul
MEMORANDUM TO: FROM: Carolina Law Students Louise W. Harris Assistant Dean for Alumni &amp; Special Programs September 8, 2008 2008 Law Alumni Weekend, October 1012, 2008DATE: RE:Carolina Law will host Law Alumni Weekend on October 1012, 2008. On be