v m m F G Chapte 13: Gavitation The foce that makes an apple fall is the same foce that holds moon in obit. Newton s law of gavitation: Evey paticle attacts any othe paticle with a gavitation foce given by: ( ˆ ) 1 G: gavitation constant, G 6.67 x 10-11 N. m /kg The minus sign means this foce is always attactive.
v m m F G ( ˆ ) 1 This foce depends on the masses *and* the distance squaed between them. Between the eath and a 60 kg peson standing on the eath s suface: F 588 N If the peson moved to twice the eath s adius, the foce will now be divided by (o 4). F R 588N/4 147N Gavitational foce between two 60 kg pesons standing 1 m apat: F.4x10-7 N
The pinciple of supeposition: net effect is the sum of the individual effects F 1, net F 1 + F 13 + F 14 + + F 1 n n i F 1i Sample 13-1: What is the net gavitational foce F 1 that act on paticle 1due to the othe two paticles? F + F1, net 1 F13
Shell Theoem Shell theoem: a unifom spheical shell of matte attacts a paticle that is outside the shell as if all the shell s mass wee concentated at its cente F is the same as
Gavitation nea Eath s Suface Assume the Eath is a unifom sphee of mass M, the gavitation foce on a paticle of mass m located outside eath a distance fom Eath s cente is F G g Mm since F m a a G g M a g vaies with attitude, Nea eath s suface: a g 9.801 m/s at an altitude 35700 km, a g 0.5 m/s
Gavitation inside eath Conside a unifom sphee of matte. We want to find the foce on m a adial distance a fom the cente Opposing foces cancel fo masses > a. Net foce on m comes fom the mass that is inside < a. R Minsidem F G a a π π whee ρ M θ θ inside d sin d dϕ 0 0 0 π Fo a unifom density, the integal educes to times the a 3 volume inside a. M inside ρ 4 3 a d ()(π) ρπa M total 3 3 0 R
Gavitational Potential Enegy Gavitational potential enegy of a system of two paticles M and m: U() W F()d F() d GMm d F()d cos180 GMm o + constant
U GMm U () + constant 0 Let U() 0 when then constant 0 M m so we have U() G fo any finite value of, U is negative
U Escape speed: the minimum initial speed v fo a pojectile (e.g. ocket) to keep moving upwad foeve, i.e., v > 0 0 Fom enegy consevation: K i + U i ½ mv + ( GMm/R ) K f + U f E tot > 0 This yields: v GM R Eath: M 5.98x10 4 kg, R 6.37x10 6 m, v 11. km/s
Planets and Satellites: Keple s laws The law of obits: All planets move in elliptical obits, with the Sun at one focus.
The law of aeas: A line that connects a planet to the Sun sweeps out equal aeas in the plane of the planet s obit in equal times; that is, the ate da/dt at which it sweeps out aea A is constant. v v v L p L p da dt 1 ()(mv ) dθ dt m 1 ω ω Angula momentum is conseved da dt L m
Keple s Laws The law of peiods: the squae of the peiod of any planet is popotional to the cube of the semimajo axis of its obit. Cicula obit e 0 T 4π GM 3
Keple s Laws The law of peiods: the squae of the peiod of any planet is popotional to the cube of the semimajo axis of its obit. Elliptical obit e > 0 T 4π GM a 3 a is the majo axis
A Quiz At which point is m moving the fastest? 1) 1 ) 3) 3 4) 4 5) always moves at the same speed 6) some othe point on the obit 1 3 4
Daily Quiz, Mach 18, 004 Reason: m sweeps equal aeas in equal times. Anothe way of looking at it: U() is most negative at 1, so K must be geatest thee to keep E constant. At which point is m moving the fastest? 1) 1 ) 3) 3 4) 4 5) always moves at the same speed 6) some othe point on the obit 1 3 4
Poblem 13-0 Two concentic sphees M 1 and M. Find F at adii a, b, and c. Mm F g G
Thee masses. Move B fom nea A to nea C. Find wok done by a) you, b) by gavity. Poblem 13-31
Thee masses. Move B fom nea A to nea C. Find wok done by a) you, b) by gavity. Poblem 13-31
Find distance between the foci of the Eath s obit. Poblem 13-44
Poblem 13-46 Find distance fo geosynchonous obit. http://science.nasa.gov/realtime/jtack/3d/jtack3d.html
Satellites and Obits Potential enegy Centipetal foce Mm U() G v Mm F c m G Kinetic enegy K 1 mv G Mm 1 U Total enegy E 1 Mm K + U U + U G elliptical a
Elliptical Obits Total enegy E G Mm a
A Quiz All thee obits intesect at P. Which path has the geate total enegy? 1) 1 ) 3) 3 4) all have the same total enegy
A Quiz Total Enegy E G Mm a a 1 < a 3 < a > E is least negative. All thee obits intesect at P. Which path has the geate total enegy? 1) 1 ) 3) 3 4) all have the same total enegy 3