Physics 111. Ch 12: Gravity. Newton s Universal Gravity. R - hat. the equation. = Gm 1 m 2. F g 2 1. ˆr 2 1. Gravity G =

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1 ics Announcements day, embe 9, 004 Ch 1: Gavity Univesal Law Potential Enegy Keple s Laws Ch 15: Fluids density hydostatic equilibium Pascal s Pinciple This week s lab will be anothe physics wokshop - on fluids this time. No quiz this week. Gavity We ve mentioned last week that any two bodies that have gavitational mass exet an attactive foce on one anothe. We extend ou set of foces to include Newton s Univesal Law of Gavity. We also look at how this new consevative foce esults in a new Gavitational Potential enegy. Finally, we look biefly at Keple s Laws. So fa, we ve only looked at the fom this foce takes nea the suface of the Eath, namely Fg = W = m g Newton s Univesal Gavity What s this ˆ? But this fom has been deived fom a moe geneal fom known as Newton s Univesal Law of Gavity: F g 1 = Gm 1 m ˆ 1 whee m 1 and m ae the masses of the two objects, is the distance between them, and is a unit vecto pointing fom m to m 1. ˆ 1 the equation Univesal Gavitational Constant 11 Nm G = kg Well, we ve seen unit vectos fo the diections in a Catesian coodinate system. Remembe ˆx ŷ ẑ The unit vecto in the x-diection The unit vecto in the y-diection The unit vecto in the z-diection R - hat 1

2 What s this ˆ? x z ˆ y It s the unit vecto that points adially out fom the oigin of the coodinate system to the point of inteest in space. The quantity (the adial distance fom the oigin to the point of inteest) is one of thee coodinates in the spheical coodinate system. F Fg1 g 1 m 1 m Notice, these two foces ae equal in magnitude and opposite in diection. In fact, they ARE a 3d Law Pai! We also note that these foces act at a distance: that is, the two objects have no diect physical contact with one anothe. Woksheet #1 Such action-at-a-distance foces ae temed field foces. This means that we can define a new physical quantity known in this case as the gavitational field. G = Gm ˆ Evey mass has an associated gavitational field aound it. CQ1 univesal gavity Gavitational Fields How do we econcile these two foms? G = Gm If we put a mass m 1 in the field G ceated by the pesence of mass m, it feels a foce given by ˆ F g 1 = m 1G F g 1 = m 1G Tue univesally! G = Gm ˆ W = m g Tue only fo expeiments nea the suface of planet Eath. Let s see what value G has nea the suface of Eath. Fields & foces econciliation

3 G = Gm ˆ Let s see what value G has nea the suface of Eath. Woksheet # m will be the mass of the Eath and will be the adius of the Eath. Since ou expeiments ae at the suface of the Eath, the distance fom ou objects to the cente of the Eath applies. m = kg = m G = ( Nm )( kg) kg ( m) = 9.83 m s CQ obital speed We ve seen that nea the Eath s suface, the function fo gavitational potential enegy takes the fom U g = m g y Gm m U = 1 g But we noted that this fom is coect only fo poblems that take place at o nea the Eath s suface. Fo poblems in oute-space we need to use a moe geneal fom of this function... Gavitational Pot l Enegy This fom is deived using calculus and the elationship between foce and potential enegy (also fom calculus) with the assumption that U = 0 when = Fomula fo gavitational pot l enegy If we have a system that involves seveal masses, we can compute the total potential enegy of the system as the sum of the potential enegies between each pai of masses in the system. So, fo a 3-mass system... m 1 U tot = U 1 + U 13 + U 3 = G m m 1 + m m m m m body system 3 m3 U tot = U 1 + U 13 + U 3 = G m m 1 + m m m m Notice that this esult is simply the sum of the enegy changes that esult when each mass is bought fom infinity to its final location. Mass 1 is fee. m m 13 m3 Binging up mass in the pesence of mass 1 esults in the 1st tem. con t 3 3

4 U tot = U 1 + U 13 + U 3 = G m m 1 + m m m m How much enegy is equied to move a 1000-kg mass fom the Eath s suface to a distance that is twice the Eath s adius away fom the cente of the Eath? Use fo the Eath a mass of 6 X kg and a adius of 6400 km. The last two tems esult fom binging up mass 3 in the pesence of masses and 1. Mass 1 is fee. m m 13 m3 Binging up mass in the pesence of mass 1 esults in the 1st tem. Woksheet #3 The planets in ou sola system move aound the Sun in oughly cicula obits. Given that the Eath (m = 6 X kg) obits the Sun (m = X kg) in oughly a cicula obit ( = 1.5 X m) once pe yea, calculated the mean obital speed of the Eath. Woksheet #4 What foce is esponsible? This means that some foce must be acting on the planets causing a centipetal acceleation. The planets do not move aound the Sun in pefectly cicula obits. The fist peson to figue out the coect shape of the obits was Johannes Keple. foci The sum of the distance fom any point on the ellipse to each of the two foci is constant Majo axis Keple s Fist Law says that the planets move aound the Sun in elliptical obits, with the Sun at one focus. cente 4

5 Keple s Second Law says a line dawn connecting the Sun to a planet will sweep out an equal aea in the ellipse in equal time intevals. The law tells us how fast a planet moves at vaious points in its obit: close to the Sun the planet will have a geate speed than fa fom the Sun. Makes sense since gavity goes as 1/distance A B If it takes one month fo the planet to go fom A to B... C It will also take one month to go fom C to D, if the aeas of the blue & ed tiangles ae the same. D Skip deivation Keple s Thid Law says the squae of the peiod of the obit of a planet is popotional to the cube of the length of the semi-majo axis of the obit. Let s demonstate this law fo the case of a cicula obit. Simply use Newton s nd Law whee the Radial Foce is the Gavitational Foce. Fo a cicula obit, the semi-majo axis is simply the adius of the cicle (the diamete being the majo axis). F G = F m pl. = m v pl pl m pl. = m v pl pl m pl. = m v pl pl The planet will complete one obit in one peiod. The cicumfeence of the obit is the distance the planet will tavel in one peiod. So... v pl = C T = π T Now plug this value of velocity into the above equation... m pl. = m (π / T ) pl = 4π T T = 4π 3 5

6 T = K We can simplify this expession by calculating the constants... 3 Sun whee... K Sun = 4π = s m 3 T = K a 3 a = semi-majo axis Keple s Laws apply to the obits of planets about the Sun moons about a planet satellites about a planet comets about a sta.you name it! If it s in obit, consult Keple! Natually, the constant K depends upon the body being obited! The semi-majo axis of the obit of Pluto is about 4 times as geat as that of the obit of Satun. If Satun obits the Sun in about 30 yeas, how long does it take Pluto to obit the Sun once? 1) 30 yeas ) 10 yeas 3) 165 yeas 4) 5 yeas 5) 40 yeas Woksheet #5 Fluid Mechanics We ve spent a lot of time looking at systems of solid objects. But matte also comes in liquid and gaseous states. We can descibe the motions of such substances using the extension of Newtonian mechanics known as fluid dynamics. Let s stat by chaacteizing a solid mass of unifom composition. What quantities can we diectly measue? M V 0 = L 3 8M V = (L)3 V = 8L 3 = 8V 0 If ou block was made of coppe, and we doubled its size, what would happen to its mass? Mass & volume L L Remembe ou scale aguments (way back in the fist week of the couse)!!! scaling 6

7 m In fact, if we made a plot of the mass of ou coppe block vesus its volume, we d find slope = ρ V This line has a slope that chaacteizes the type of mateial fom which the block is made. We define the slope of this line as the density (ρ) of the mateial. How would the slope of the line fo a block of styofoam compae to that fo a block of lead? ρ M V The unit of mass, the gam, was chosen to be the mass of 1 cm 3 of liquid wate. So wate has a density of 1 g/cm 3 o 1 kg/l o 1000 kg/m 3. The tem specific gavity efes to the atio of the density of a given substance to that of wate. Objects with a specific gavity less than 1 will float in wate; those geate than 1 will sink. ρ M V [ρ] = kg m 3 [ρ] = [ M ] [V ] Note: density is often witten in gams pe cubic centimete o g/cc. Thee is a facto of 1000 diffeence between the two sets of units. units When a foce is applied ove an aea, we say that the object feels pessue. P F A [P] = [F] [ A] = N m = Pa pascal Usually, we talk about pessue of a fluid o a gas (like the atmosphee). pessue It s all aound you! It s all aound you! Ou geen fish is completely submeged. The wate exets a pessue on the fish. Fom which diection does the fish feel the pessue? Think about you own expeience walking aound outside (when thee s NOT a had wind blowing). You don t notice any diffeence in pessue ove the suface of you body. In fact, the pessue is exeted in the diection nomal to the body of the fish all ove the fish. What would happen if thee was a pessue diffeence acoss you hand? > Whee is pessue? Pessue gadient 7

8 It s all aound you! You hand would feel a net foce acting to the RIGHT in this case. Theefoe, you hand would stat to acceleate to the ight, o you would have to exet a foce though you am to counteact this foce known as a pessue gadient foce. F net 1PP> So, now let s look at how pessue changes with altitude. We know it s a lot hade to beathe at the top of a mountain than at sea level--thee ae fewe oxygen molecules and fewe molecules in geneal up thee. Let s look at the foce balance fo a little laye of atmosphee as we head up the mountain. At the Eath s suface, we sit at the bottom of an entie column of molecules in the atmosphee. These molecules exet a pessue on us at the suface of about kpa. We define this pessue to be 1 atmosphee (atm). As we go up though the atmosphee, what happens to the pessue we feel? WHY? The pessue of all the molecules above ou laye. W A = suface aea The pessue fom all the molecules below ou laye. The gavitational foce acting on the laye itself. < Foce on laye of atmosphee The pessue on the top suface of aea A esults in a foce downwad of F 1 = A. F = F 1 + W If ou system is in equilibium, the net foce must be 0. So... A = A + mg A = suface aea A = suface aea W The pessue on the bottom suface of aea A esults in a foce upwad of F = A. Δh W But what is the mass of ou little laye of atmosphee? m = ρv = ρ( AΔh) ρ is the density of ai. 8

9 If ou system is in equilibium, the net foce must be 0. So... A = A + mg A = A + ρ( AΔh)g = + ρg(δh) ΔP = ρ g( Δh) NOTE: ou deivation hee assumes a unifom density of molecules at a given laye in the atmosphee. In the eal atmosphee, density Deceases with altitude. Nevetheless, ou pessue and foce balance diagam applies so long as ou laye is sufficiently thin so that within it, the density is appoximately constant. A scuba dive exploes a eef 10 m below the suface. The density of wate is 1 g/cc. What is the extenal wate pessue on the dive? If the atmosphee is in equilibium (which would imply a unifom tempeatue and no winds blowing), the pessue at a given height above the suface would be the same aound the Eath. Woksheet #6 The same aguments can be made fo pessue unde wate. All othe things being equal, the pessue at a given depth below the suface is the same. In solving the last poblem, we applied a pinciple that we haven t even defined yet, but that pobably made good sense to us. We said that the suface pessue at the bottom of the atmosphee equaled the pessue in the suface laye of wate. If this ween t tue, the wate would fly out of the oceans o sink apidly towad the ocean floo! In fact, Pascal s Pinciple guaantees this will be tue. It states: The pessue applied to an enclosed liquid is tansmitted undiminished to evey point in the fluid and to the walls of the containe. Which means, that the pessue below the suface of the wate is equal to the suface pessue + the pessue due to the column of wate above a given level. 9