. x.si ( 5 s [ ] CHAPER OSCILLAIONS x ax (.( ( 5 6. s s ( ( ( xax. 5.7 s s. x.si [] x. cos s Whe, x a x.5. s 5s.6 s x. x( x cos + si a f ( ( [ ] x.5cos +.59si. ( ( cos α β cosαcos β + siαsi β x Acos φ Acosφcos + Asiφsi x Α cos+βsi, Α Acosφ, Β Asiφ.5 x + kx x + kx k x x x x ( ( k x x x x ka x + kx xx xx A x + x + k x x xx x x xx A x
.6 l s.5s g 9.8 6.7 For sprigs ie i parallel: F x k x k x k + k x s ( ( ( k + k For sprigs ie i series: he upwar force is keq x. herefore, he owwar force o sprig k is keq x. he upwar force o he sprig k is kx where x is he isplacee of P, he poi a which he sprigs are ie. Sice he sprig is i equilibriu, kx k x. k eq Meawhile, he upwar force a P is kx. k x x. he owwar force a P is ( herefore, kx k ( xx A eq kx x k + k kx k x k k+ k k eq kk ( k+ k.8 For he syse ( M +, kx ( M + X he posiio a acceleraio of are he sae as for ( M k x x M + k k x Acos + δ cos M + M + he oal force o, F x gfr + :
Fr g+ k x k g+ cos k M + M + M + For he block o jus begi o leave he boo of he box a he op of he verical oscillaios, F r a x : k g M + g( M + k γ.9 x e Acos( φ x γ γ e A si ( φ γe Acos( φ x axia a si( φ + γ cos( φ γ a ( φ hus he coiio of relaive axiu occurs every ie ha icreases by : i+ i + γ For he i h axiu: x e i Acos( φ i i γ γ i + cos( i+ i+ φ xi x e A e xi x i+ γ γ e e. (a (b (c c γ s γ 6 s r 7 s F 8 Aax. C 6. ( r k 5s γ 7 r s γr γr r φ γ 7 a φ. γ
. (a 7 x + βx + β x 7 γ β a β γ β r r β (b A F ax γ A 5β γ β 5 5 β. (a So, e γ γ l f l ( γ ( + γ γ l f f + f + f.6hz (b ( γ r γ l fr f f 99.Hz fr γ. Sice he apliue iiishes by e γ ( e e e γ γ Now ( γ So ( γ + + i each coplee perio,
+ For large, + 8. (a r ( (b (c.77 γ ( γ Q.866 γ γ ( γ aφ φ a 6..66 D + F F A(.77 D ( ( ( (.5 A( for ( ax ( Aaxγ ( + γ γ A A, ( + γ ( + γ γ ± γ ± γ 5
.6 (b (c γ Q γ γ LC, γ R R L LC L L Q R RC L L C R Q γ R R i.7 Fex F si I Fe a x( is he iagiary par of he soluio o: i x + cx + kx F e ( i.e. x( I i φ Ae Asi ( φ where, as erive i he ex, F A ( k + c a γ aφ.8 Usig he hi, Fex Re( Fe β, where β α + i, a x( is he real par of he soluio o: x + cx + kx F e β. i Assuig a soluio of he for: x Ae β φ ( F c k x xe i φ β + β + A F α i α c α + ic + k ( cos isi A φ + φ F ( α cα + k cosφ A ( F α + c s A iφ 6
φ a ( c α ( Usig si φ + cos φ, a x F A α cα +k ( α cα + k + ( cα A F { ( α cα k + + ( cα } α ( Ae cos( φ +he rasie er..9 (a (b (c l A g 8 for A l,. g l g.8 l l Usig gives g, approxiaely 8% oo sall. g λ A B a λ 6 B A A 9 for A B,. A i. f ( c e, ±, ±,...,,,, f ( c cos + c isi i a c ( e f,, ±, ±,... i cos c f f ± ±... ( ( ( si ( he firs er o c is he sae for a ; he seco er chages sig for vs.. he sae hols rue for he rigooeric ers i f. herefore, whe ( ers ha cacel i he suaios are iscare: 7
f ( c + f ( cos( cos + f ( si ( si, ±, ±,..., a c f ( Now, ue o he equaliy of ers i ± : f ( c + f ( cos( cos + f ( si ( si,,,,... Equaios.9.9 a.9. follow irecly. i. f ( c e, c ( i f e i f e so c (, a, ±, ±, i i ( e e + i i e e i i + i i e e + i i i For eve, e e a he er i brackes is zero. i i For o, e e c i, ±, ±,... i f ( e, ±, ±,... i i i ( e e,,,5,... i si (,,,5,... f ( si si si 5 5 + + + 8
. I seay sae, x( Ae ( φ i F A ( + γ F Now F,,,5,... a Q 9 so γ γ, F A 9 ( 9 + F A F A 9 ( 9 9 + F A 7 F A5 5 ( 9 5 + ( 5 F A5 i.e., A : A : A 5 : 9.6 :.. (a x+ x y x hus y x x y y y x ivie hese wo equaios: x x y (b Solvig yy xx + Le C A y x + A A a Iegraig a ellipse y x + C 9
. he equaio of oio is F( x x x x. For sipliciy, le. he ` (a (b (c x xx. his is equivale o he wo firs orer equaios x y a y xx he equilibriu pois are efie by x x x x + x ( ( hus, he pois are: (-,, (, a (+,. We ca ell wheher or o he pois represe sable or usable pois of equilibriu by exaiig he phase space plos i he eighborhoo of he equilibriu pois. We ll o his i par (c. y y x x he eergy ca be fou by iegraig or x x y ( yy x x x+ C or y x x + C y x x I oher wors E + V + C. he oal eergy C is cosa. he phase space rajecories are give by soluios o he above equaio x y ± x + C. he upper righ quara of he rajecories is show i he figure below. he rajecories are syerically ispose abou he x a y axes. hey for close pahs for eergies C< abou he wo pois (-, a (+,. hus, hese are pois of sable equilibriu for sall excursios away fro hese pois. he rajecory passes hru he poi (, for C a is a sale poi. rajecories ever pass hru he poi (, for posiive eergies C>. hus, (, is a poi of usable equilibriu..5.5.5
.5 θ + siθ Iegraig: θ θ θ cosθ cosθ θ θ θ ( cosθ cosθ θ or θ cos ( θ cosθ ie for peulu o swig fro θ o θ θ is Now subsiue θ si siφ θ si so φ a θ θ θ θ a use he ieiy cosθ si θ θ si si φ θ a afer soe algebra or θ θ θ si si si (a φ θ where α si α si φ αsi φ + αsi φ + α si φ + 8 (b ( (c φ + α φ + α φ + si si 8 θ θ θ θ si 8 α +... θ α 9 α 6 + + + θ + + 6 ------------------------------------------------------------------------------------------------------