Lecture 22: Chapter 11 Gravity Lecture 2 2 Potential Energy r Gravitational potential energy Escape velocity of a point ass for ass distributions Discrete Rod Spherical shell Sphere Gravitational potential energy of a syste of particles Black holes 1 M 1 Work done to bring ass fro initial to final position. Zero point is arbitrary. Choose zero at infinity. 2 Gravitational potential energy Apollo 14, at lift-off Total energy E tot = K+U = ½ v 2 G (M )/r 3 4 1
Binding Energy The absolute value of the potential energy can be thought of as the binding energy At infinite separation, binding energy U=0, thus unbound. If an external agent applies a force larger than the binding energy, the excess energy will be in the for of kinetic energy of the particles when they are at infinite separation Escaping Gravity E>0: object is not bound E<0: object is bound to gravity. E=0: kinetic energy just enough to escape gravity (K=U) 5 6 Escaping Gravity Kinetic energy of the object ust be greater than its gravitational potential energy This defines the iniu velocity to escape KE+PE = constant Consider case when speed is just sufficient to escape to infinity with vanishing final velocity At infinity, KE+PE=0, therefore, on Earth, Quiz You are on the oon and you know how to calculate the escape velocity: You find that it is 2.37k/s A projectile fro the oon surface will escape even if it is shot horizontally, not vertically with a speed of at least 2.37k/s A) Correct B) Not correct 4/6/11 Phys 201, Spring 2011 7 4/7/11 Phys 201, Spring 2011 8 2
Gravity near Earth s surface... Gravity... Near the Earth s surface: R 12 = R E = 6371 k Won t change uch if we stay near the Earth's surface. since R E >> h, R E + h ~ R E. Near the Earth s surface... So F g = g = a =g a = g All objects accelerate with acceleration g, regardless of their ass! h R E M Choosing U(R E ) = 0, then U(h) = g h, Or: the equivalence principle: = g = i for h << R E 4/7/11 Phys 201, Spring 2011 9 10 Variation of g with Height Question This is twice the Earth radius: R E = 6000k We know F should drop with r 2 Indeed, g has dropped to 9.81/4 /s 2 E 11 X 4/7/11 Phys 201, Spring 2011 12 3
Gravitational force: it is a function of space-tie (r, t). Definition of the gravitational field that will act on any asspoint: Must be a function of space-tie (r, t) concept of field. If the field is caused by a ass distribution we need to su over all asspoints as the source. 13 The gravitational field vectors point in the direction of the acceleration for a particle would experience if placed in that field The agnitude is that of the freefall acceleration at that location The gravitational field describes the effect that any source object M has on the epty space around itself in ters of the force that would be present if a second object were soewhere in that space Gravitational field independent of, only on M! 14 Two source ass points M, fieldpoint in plane of syetry Magnitude of field due to each ass: Field due to rod of length L on a point along its axis. Field by one ass eleent d: Need to add x and y coponent of g 1 and g 2 X-coponent: Integrate over all ass eleents d: Y-coponent is zero for syetry reasons 15 16 4
Field due to spherical syetric ass distribution, a shell of ass M and radius R: Field of a spherical shell Field due to hoogeneous assive sphere Field inside the sphere (p.382) Geoetry: spherical shell is 0 anywhere inside (see p.384) 17 18 Systes with Three or More Particles The total gravitational potential energy of the syste is the su over all pairs of particles: siple scalar su Gravitational potential energy obeys the superposition principle Each pair of particles contributes a ter of U ij The absolute value of U total represents the work needed to separate the particles by an infinite distance Potential energy of a syste of asses What is the total potential energy of this ass syste? L L L 20 19 5
Four identical asses, each of ass M, are placed at the corners of a square of side L. The total potential energy of the asses is equal to xgm 2 /L, where x equals 21 A black hole is the reains of a star that has collapsed under its own gravitational force The escape speed for a black hole is very large due to the concentration of a large ass into a sphere of very sall radius If the escape speed exceeds the speed of light, radiation cannot escape and it appears black The critical radius at which the escape speed equals c is called the Schwarzschild radius, R S The iaginary surface of a sphere with this radius is called the event horizon This is the liit of how close you can approach the black hole and still escape Black Holes 22 Black Holes at Centers of Galaxies There is evidence that superassive black holes exist at the centers of galaxies (M=100illion solar asses) Theory predicts jets of aterials should be evident along the rotational axis of the black hole 4/6/11 An Hubble Space Telescope iage of the galaxy M87. The jet of aterial in the right frae is thought to be evidence of a superassive black hole at the galaxy s center. Phys 201, Spring 2011 23 6