The Gravitational Field Gravitational Potential Energyrfinn/phys130/P130-20-GFieldPE.pdf · The potential energy is negative because the force is attractive and we chose the potential

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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

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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

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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

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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

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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.