Lecture 22. PE = GMm r TE = GMm 2a. T 2 = 4π 2 GM. Main points of today s lecture: Gravitational potential energy: Total energy of orbit:

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Domenic Wright
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1 Lectue Main points of today s lectue: Gavitational potential enegy: Total enegy of obit: PE = GMm TE = GMm a Keple s laws and the elation between the obital peiod and obital adius. T = 4π GM a3

2 Midtem The aveage scoe was 3. The scoes wee highe than Midtem. Thus I kept the fomula fo the total midtem scoe given in the syllabus. The fomula is: MT = exam scoe + 0.3(coection pts exam scoe) Example: exam scoe = 6 pts, coection pts = 50 pts à MT = (50-6) = (4) à = = 33. = 66% instead of 5% Coections set closes on Tuesday at 0 pm

3 Satellite Obits View fom above Noth Pole, June Low obit satellites: Hubble, space sta@on, Iidium system, spy sats GPS, eseach sats Geosta@onay communica@on sats

4 Synchonous obit Synchonous obit of a satellite: ota@on peiod of satellite of mass m is the same as ota@on peiod of the planet Fo eath: peiod T = 4 hous = 86 x 0 3 s T = 4π GM eath 3 = 3 K Eath K(eath) = 4π GM eath = 9.9x0 4 s / m 3 3 = T /K = 75 x 0 m 3 =(75 x 0 m 3 ) /3 = 4 x 0 6 m R e = 6.4 x 0 6 m (/R e ) = 6.6, which is athe high

5 R B Conceptual question Two satellites A and B of the same mass ae going aound Eath in concentic cicula obits. The distance of satellite B fom Eath s cente is twice that of satellite A. What is the atio of the centipetal foce acting on B to that acting on A? a) /8 b) /4 c) / d) e) F c,b /F c,a? R A F = ma = GM m E c c F c,b F c,a = A B / GM E m s / / / B GM/ E m/ s A A = = = B = s B A = B A /4 i A = A A B A A = A B Hint: This is a atio poblem. 5

6 Example One satellite is in an obit about Jupite of adius and a peiod of 00 days. If a second satellite is placed in an obit with 4 times the adius (i.e. =4 ), what is the peiod fo the obit of the second satellite in days? a) 00 b) 400 c) 800 d) d T T? Hint: this can also be solved as a atio poblem T 4π 3 = GM Jupite T 4π 3 = GM Jupite 4π 3 GM T Jupite = T 4π 3 GM Jupite 3 3 = = = ( ) T = 8 T T = 8i T = 8 00 days = 800 days

7 Quiz A 00 kg man stands on a bathoom scale in an elevato. The scale measues weight not mass, but the eading is in mass assuming g=9.8 m/s. The cable suppoting the elevato beaks and both man and elevato begin falling feely down the shaft. What does the scale ead now? a) 00 kg b) 0 kg c) 00 kg d) none of the above Einstein s equivalence pinciple: If all objects in the local envionment ae in fee fall unde the influence of the same unifom gavitation attaction, it appeas to the obseve that thee is no gavitation foce acting at all. I.e. fo all pactical puposes, the objects ae weightless.

8 Conceptual quiz Suppose Eath had no atmosphee and a ball wee fied fom the top of Mt. Eveest in a diection tangent to the gound. If the initial speed wee high enough to cause the ball to tavel in a cicula tajectoy aound Eath, the ball s acceleation would a) be much less than g (because the ball doesn t fall to the gound). b) be appoximately g. Note: someone in a space ship in the same obit would c) depend on the ball s speed. feel weightless and many would call this a zeo g envionment. weightlessness is a sensation that occus when one feels no effects of gavity. It happens when g=0 o when someone is falling feely. Down, as in an elevato o sideways as in a satellite. 7

Gavitational Potential Enegy PE = mgδy is valid only nea the eath s suface Fo objects high above the eath s suface, an altenate expession is needed PE M Em = G Zeo potential enegy is defined to be

9 Gavitational Potential Enegy PE = mgδy is valid only nea the eath s suface Fo objects high above the eath s suface, an altenate expession is needed PE M Em = G Zeo potential enegy is defined to be the value infinitely fa fom the eath If the total mechanical enegy of an object exceeds zeo, the object can escape the gavitation field of the eath. The escape velocity is when the E mech =0. At the Eath's suface: GMEathm Emech = mv R Eath If E mech 0 and v v escape, the object can escape the Eath's gavitational field GM m GM 0= mv v = R R Eath Eath escape escape Eath Eath nd Cosmic speed 9

10 Escape Speed (Second cosmic speed) Second cosmic speed: speed needed to beak fee fom a planet of mass M p and adius R p (g p = GM p /R p ) Eh GM m E ( = 0) = mvii = 0 RE v II = GM E = GM E i = g E i = x0 6 m =.km / s Fo eath: g = 9.8 m/s = 6.37x0 6 m v =. km/s

11 Escape Speed The escape speed fo Eath is. km/s. Suppose we do not need to get all the way out of Eath s gavity, but just up to an altitude of h = 36,000 km above gound; what is the launch speed needed? Enegy consevation at the Eath s suface ( ) and at ( +h) : E(0) = mv 0 GM Em = E(h) = 0 GM Em + h v 0 = GM E + h = 0.3 km/s Compae to what is needed to each this altitude, it takes only a little highe velocity to escape to infinity! Fo escape: g = 9.8 m/s R = 6.37x0 6 m v =. km/s

Moe geneal: PE gavity =-GMm/ R =R+h Ealie we noted that we could define the zeo of

12 Gavitational potential enegy So fa, we used: PE gavity =mgh Only valid fo h nea eath s suface. Moe geneal: PE gavity =-GMm/ R =R+h Ealie we noted that we could define the zeo of PE gavity anywhee we wanted. So the suface of the eath is as good as anywhee! R MEARTH MEARTH U U = G m ( G m) R R MEARTH MEARTH = G m ( G m) ( R + h) R

13 Gavitational potential enegy Only valid fo y nea eath s suface: U gavity =mgy suface. U U = mgm EARTH ( + h ) R = mgm E + h EARTH + h h = mgm EARTH + h If R = and h<<, then +h, thus ( ) R ( E + h) ( ) m GM EARTH R E h R R =R +h U U m G M EARTH R h = mgh R

Centipetal foce povided by gavity Multiply both sides by /:

satellite is exactly -/ of its potential enegy PE = GMm K =

both spheical and elliptical obits Geneal fomula E = G Mm a

14 Centipetal foce povided by gavity Multiply both sides by /: Enegy of a Satellite We find that the kinetic enegy of the satellite is exactly -/ of its potential enegy PE = GMm K = PE Total mechanical enegy Moe geneal expession is valid fo both spheical and elliptical obits Geneal fomula E = G Mm a mv mv = G M Em = G M Em = PE E = KE + PE = PE + PE = PE = G Mm a

15 Compaison of elliptical obits Useful Equations: Total enegy TE = GMm TE = KE + PE a usually called E mech = KE + PE T = 4π PE = GMm GM angula momentum of mass about oigin L = mv sinθ a3

Chapter 13: Gravitation

Chapter 13: Gravitation 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