Homework Set 3 Physics 319 Classical Mechanics
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1 Homewok Set 3 Phsics 319 lassical Mechanics Poblem 5.13 a) To fin the equilibium position (whee thee is no foce) set the eivative of the potential to zeo U 1 R U0 R U 0 at R R b) If R is much smalle than R 1/ 1 is, the secon oe epansion of neee to obtain the fist significant tem in the potential. R 1 U U0 R 1 / R Poblem 5.17 a) Suppose U0U0 U01 R R R U 0 U0 R U U k R mr 0 0 cos t A t cos t A t an / p/ q, whee p an q ae the lowest integes that specif the ational numbe atio. Because q p, one efines the peio of the common oscillation fequenc / q / p. Now afte the peio T pq p / q /, the motion epeats because t T A cos t T A cos t p A cos t t cos cos cos t T A t T A t q A t t b) One wa to chaacteize an iational numbe is as a numbe whose ecimal epansion neve epeats. Suppose one appoimates the fequenc atio fist b its 100 ecimal epansion, then its 00 ecimal epansion, an so foth. B pat a) the epetition peio 100 of the 100 ecimal epansion is q / 10 /, the epetition peio of the igit ecimal epansion is 10 /, an so foth. If the epansion of the fequenc
2 atio neve epeats, an the epetition peio gets longe the close one gets to the actual iational value, at the actual iational value the patten neve epeats. Poblem 5.3 E m k t t m k m b b equation 5.4. B the wok-eneg theoem, the ate that wok is issipate b the amping foce is W F b Poblem 5.4 g sec L If the eponential amping time is 8 hs = 8800 sec, the Q-value is 1 Q Poblem 5.51 Fo the focing function to be eal, the f n ae all eal. Now f t f n t f e f e g t n0 n0 n0 Fo an iniviual n, the solution to the iven oscillato poblem z z f n n n int int 0 zn e e t t is, b supeposing the solutions fom the iniviual tems on the RHS fn int fn zn t e e n in n in cos Re in t int n n Re n Re 0 0 fn int Re e 0 n in int If nt Re fne / 0 n in, summing ove all the ns iels the equie esult. int
3 Poblem 6.1 The stationa conition is, fom the Eule-Lagange equation L L L 1 0 B evaluating the pope eivatives, the equation fo the stationa solution is / whee the integation constant in is chosen to be fo futue convenience. The integal is the stana fom sinh / D whee D is the secon integation constant. 1 Poblem 6.16 The fomula fo the istance between two points on a sphee, Poblem 6.1, is D R 1sin 1 (This comes fom the Pthagoean Theoem applie to a small isplacement on the sphee. Then D R Rsin when. This comment is not pat of the solution.) D is stationa L L L 0 sin 1 sin is constant as a function of. Using the suggeste tick, if the fist point is aligne with the z- ais. An path passing though the pole necessail has 0 neab the pole an as sin 0 at the pole, 0. Theefoe, fo the geoesic, const as a function of. The geoesic is a longitue line at the pole, i.e., it is a geat cicle of the sphee. It s not too ba to just integate the equation fo
4 1 sin sin / sin sin 4 4 csc 4 1/ 1/ sin sin 1/ 1 cot 1 cot sin 1/ 1 1/ 1 1/ 0 1/ cos sin sin cos sin cos sin / 1 1/ These cooinates ae on the plane passing though the cente of the sphee. Poblem 6.18 The istance function in pola cooinates is 1 / / because 1 D D o 1 D. It tuns out using epession 1 is the eas wa, an epession is a bit moe involve. Using the Eule-Lagange equation on epession 1 iels / 0 1 / 4 1 1/ 1/ The integal is a stana one that ma be solve b the substitution ˆ ˆ / cos, sin / cos ˆ sin ˆ ˆ cos ˆ cos ˆ ˆ ˆ 1/ cos cos ˆ ˆ 1 0 cos cos 0 This epession is the pola equation fo a line: is ientifie as the istance of closest appoach of the line to the oigin an 0 is the angle the line makes with the -ais, positive being angle in counteclockwise oientation. Using epession fo the istance, the Eule-Lagange equation is 1/
5 , 1, 1 / / / / Because (this tick iscusse in Poblem 6.0) / L / / integating in iels L L L /, / / / / 4 / / / / / / 1 / / 4 1 cos 0 1/ Poblem 6.6 The integal to etemize is u S f u, u, u, u, uu As is the book s agument, assume a small eviation fom the stationa solution u u Fo the solution to be stationa u u u u u u
6 u u u S f u u, u u, u u, u u, uu 0 u S f u u, u u, u u, u u, uu 0 u u f f u u u 0 u f f u u u 0 u f f u uu 0 f f 0 u u u Because these final two integals must vanish fo all vaiations Lagange equations follow Poblem 6.7 The istance function to etemize is f f 0 u f f 0 u / / / D u u z u u u an u Appling the Eule-Lagange equations fo the thee cooinates in tun / u 0 u / u / u z / u / u u u u z u / / / z / u u u u z u / / / / u, / u, z / u is a constant vecto as a function of u. 0 0 u / u u whee / u / u 0 1 0, the Eule-
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