The Gravitational Field Gravitational Potential Energy
Transcript of The Gravitational Field Gravitational Potential Energy
1
The Gravitational Field
• Exists at every point in space• The gravitational force
experienced by a test particleplaced at that point divided bythe mass of the test particle• magnitude of the freefall
acceleration at that location
• Points in the direction of theacceleration a particle wouldexperience if placed in that field
Gravitational Potential Energy
• The gravitational force is conservative
• The gravitational force is a central force• It is directed along a radial line toward the center
• Its magnitude depends only on r
• A central force can be represented by
Grav. Potential Energy – Work
• A particle moves from A to Bwhile acted on by a centralforce F
• The path is broken into aseries of radial segmentsand arcs
• Because the work donealong the arcs is zero, thework done is independent ofthe path and depends onlyon rf and ri
Grav. Potential Energy – Work
• The work done by F along any radial segment is
• The work done by a force that is perpendicular to thedisplacement is 0
• The total work is
• Therefore, the work is independent of the path
2
Gravitational Potential Energy
• As a particle moves from A toB, its gravitational potentialenergy changes by
• This is the general form, weneed to look at gravitationalforce specifically
Grav. Potential Energy for the Earth
• Choose the zero for the gravitational potentialenergy where the force is zero• This means Ui = 0 where ri = ∞
• This is valid only for r ≥ RE and not valid for r < RE
• U is negative because of the choice of Ui
Grav. Potential Energy for the Earth
• Graph of the gravitationalpotential energy U versus rfor an object above theEarth’s surface
• The potential energy goes tozero as r approaches infinity
• What about inside Earth?• only mass interior is important
Gravitational Potential Energy• For any two particles, the gravitational potential energy
function becomes
• The potential energy is negative because the force isattractive and we chose the potential energy to be zero atinfinite separation
• An external agent must do positive work to increase theseparation between two objects
3
Binding Energy
• The absolute value of the potential energycan be thought of as the binding energy
• If an external agent applies a force largerthan the binding energy, the excess energywill be in the form of kinetic energy of theparticles when they are at infinite separation
Systems with Three or MoreParticles
• The total gravitational potential energy ofthe system is the sum over all pairs ofparticles
• Gravitational potential energy obeys thesuperposition principle
• Assuming three particles:
• The absolute value of Utotal represents thework needed to separate the particles byan infinite distance
Energy and Satellite Motion• Total energy E = K +U
• The absolute value of E is equal to the bindingenergy of the system• E < 0
• a bound system
• E = 0• barely unbound
• E > 0• unbound
Energy in a Circular Orbit
• An object of mass m ismoving in a circularorbit about M
• The gravitational forcesupplies a centripetalforce
4
Escape Speed from Earth
• Use energy considerations to find the minimumvalue of the initial speed needed to allow theobject to move infinitely far away from the Earth
• This minimum speed is called the escape speed
• Note:• vesc is independent of the mass of the object
• independent of the direction of the velocity
• ignores air resistance
Escape Speed, General
• The Earth’s result can beextended to any planet
• The table at right gives someescape speeds from variousobjects
Planetary Atmospheres
EscapeSpeed
SurfaceTemperature
Black Holes• A black hole is the remains of a star that has
collapsed under its own gravitational force• The escape speed is very large due to the
concentration of a large mass into a sphereof very small radius• If the escape speed exceeds the speed of
light, radiation cannot escape and it appearsblack
• Schwarzschild radius, RS
• The critical radius at which the escape speedequals c
• event horizon• The imaginary surface of a sphere with radius,
Rs
• the limit of how close you can approach theblack hole and still escape
5
Black Holes and Accretion Disks
• Although light from a black holecannot escape, light from eventstaking place near the black holeshould be visible
• If a binary star system has a blackhole and a normal star, the materialfrom the normal star can be pulledinto the black hole
• This material forms an accretiondisk around the black hole
• Friction among the particles in thedisk transforms mechanical energyinto internal energy
Black Holes and Accretion Disks• The orbital height of the material above the
event horizon decreases and the temperaturerises
• The high-temperature material emitsradiation, extending well into the x-ray region
• These x-rays are characteristics of blackholes
• Examples:• X-ray binaries
• MBH ~ a few times Msun
• death state of very massive star
Black Holes at Centers of Galaxies• There is evidence that supermassive
black holes exist at the centers ofgalaxies, probably all galaxies
• Theory predicts jets of materialsshould be evident along the rotationalaxis of the black hole
• Examples• Active Galactic Nuclei
• MBH ~ a few million times Msun
• found in centers of galaxies• originated from first stars, grew
through merging
• Quasars• MBH ~ a few billion times Msun
• also found in galactic centers• originated from first stars, more
merging?• outshine entire galaxy!
• An HST image of the galaxyM87. The jet of material in theright frame is thought to beevidence of a supermassiveblack hole at the galaxy’scenter.