Tidal forces. m r. m 1 m 2. x r 2. r 1

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Andrea Walsh
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1 Tidal foces Befoe we look at fee waves on the eath, let s fist exaine one class of otion that is diectly foced: astonoic tides. Hee we will biefly conside soe of the tidal geneating foces fo -body systes. 1 1 x Conside the two asses in the diaga above. Each of these two asses, denoted by 1 &, otate about one anothe and unde the action of gavity ae in a stable binay obit. The cente of otation is on a line between the two asses at a position denoted by x. The foce balance on these two point asses is given as: g 1v 1 1 v 1, g The fist is the gavitational foce of attaction (G is a univesal constant) and the second is the obital balance eflecting a balance between centifugal foce and gavity, which ust be in balance fo a stable obit. The total distance between the two asses having velocities v is denoted by. If the otation ate is given by ω, then the above can be ewitten as v 1 1 1ω 1, ω v g The second allows us to calculate the cente of ass, whee the cente of otation lies knowing the asses. o the case of the eath and oon, we have the following:

2 e e 5.983*10 1.8* *10 kg e With these we can calculate that the distance fo the cente of the eath to the cente of otation of the eath-oon binay syste is 4500 k, within the eath (since the eath s ean adius is 6371 k). Now we ae inteested in the details of the foces elative to the cente of ass of the eath due to anothe ass, be it the oon o the eath. So conside the pictue below: b R a The distance fo the cente of the eath (on the left) to the othe ass is (~60R). At the points a, b, we ae inteested in the gavitational foces on paticles of unit ass due to the othe ass (of ass ). We also note that the two ae in obital balance, so that the centifugal foce, c, fo evey paticle on the eath is pescibed by the obital balance given above fo point asses. Thus the foce balance pe unit ass fo paticles diectly unde the othe ass but on eithe side of the eath is as follows: a ( R ) b c a b ( + R ) c ( R ) c ( + R ) R 3 R 3

3 The foce balance between gavity and centifugal foces will be unbalanced except at the cente of ass of the eath and will lead to esulting foce that is to the ight at point a and to the left at point b. These foces elative to the cente of ass ae given fo these two points above: a,b c. These foces ae equal and opposite, ae popotional to the ass of the extenal body, and depend on the invese cube of the distance. The net effect on the suface of the eath is shown in the figue below. This is taken fo the book by Knauss and, to y disay, has the oon on the left, not the ight as plotted above. Although the sun has a ass that is lage than the oon by a facto of.5x10 7, because the distance between the eath/sun is about 400 ties that of the eath/oon, the effective tidal foces ae lage fo the oon than the sun by about a facto of. The tidal foces ae weak copaed to local gavity (1 pat in 9 illion) but the hoizontal coponents of the foce will dive pessue gadients that will foce fluid to ove lateally. The foce ibalance will cause, on a wate coveed globe, the suface to defo into an ellipsoidal shape which, unde a theoy of equilibiu tides, should cause a bulge on eithe side of the eath of about 55 c due to the luna focing. Recall that the eath otates abou t an axis which is 3.5 oon o sun

4 degees fo the plane of its otation about the sun. The oon otates aound the eath (in 7.3 days) in an obit that is tilted 5 degees fo the ecliptic plane: thus the declination of the oon can vay between 0 to 8 degees ove tie. If, fo exaple, the oon is diectly above the equato, then noth is up in the above figue and while the eath otates, thee will be two high equilibiu tides pe luna day with a peiod of 1.4 hous: a luna seidiunal tide. If, howeve, the oon is at its axiu declination and found above a latitude of 8 degees, thee will be only one high tide pe luna day (4.84 hs.) at the latitude of the sub-luna point: a luna diunal tide. The sola and luna tides can einfoce one anothe duing peiods of a new o full oon. This is when the sun and oon ae in line. In the figue below, I have plotted two sinusoidal fluctuations having aplitudes of 1 (0.5) ete and peiods of 1.4 (1) hs, epesenting the seidiunal tides of luna (sola) oigin. The patten has a axiu o sping tide when the two ae in phase o anti-phase and a weak o neap tide when they ae 90 o 70 degees out of phase. At a latitude of 45N/S, the speed at which the equilibiu tide popagates as the eath evolves is quite lage, ove 300 s -1. This is faste than the phase

5 speed of a suface gavity wave. iction will cause the eath to dag the tidal bulge in the diection of the otation causing a toque that acts to slow down the otation of the eath at a ate of ca..3 s/centuy. A futhe inteesting fact is that the solid eath has tides as well: equilibiu tidal displaceents of the solid eath can actually educe the fluid tides by about 30%. Since the eath is not wate coveed, the equilibiu tide is not obseved. Instead, the detail and shape of the continental boundaies play a ajo ole in deteining the actual tidal esponse to focing. So does the wavelike natue of the long suface waves. In any pats of the ocean, focing is within the fequency band whee feely popagating waves can exist and thus the esulting pattens if tidal flow ae elated to natual standing odes of long wavelength suface waves and can vay consideably with the fequency chaacteistics of the vaious tidal coponents. An exaple of a global tidal odel fo the M (luna seidiunal: S would be the coesponding sola seidiunal) tide will be shown in class. The Topex-oseidon altiete has been successfully used to ipove global odels of tides beyond pevious studies like the one shown. Iage eoved due to copyight concens.

6 We will now investigate soe aspects of fee waves: waves which natually occu on and below the ocean suface.

SPH4U Unit 6.3 Gravitational Potential Energy Page 1 of 9

SPH4U Unit 6.3 Gravitational Potential Energy Page 1 of 9 SPH4 nit 6.3 Gavitational Potential negy Page of Notes Physics ool box he gavitational potential enegy of a syste of two (spheical) asses is diectly popotional to the poduct of thei asses, and invesely