767:
3655:
780:
3948:
3538:
shows the period of oscillation is independent of the amplitude, though in practice the amplitude should be small. The above equation is also valid in the case when an additional constant force is being applied on the mass, i.e. the additional constant force cannot change the period of oscillation.
3249:
3943:
A Scotch yoke mechanism can be used to convert between rotational motion and linear reciprocating motion. The linear motion can take various forms depending on the shape of the slot, but the basic yoke with a constant rotation speed produces a linear motion that is simple harmonic in form.
1966:
33:
2748:
3057:
3381:
3654:
2565:
1837:
1248:
1470:
2242:
1358:
1079:
and starts going back to the equilibrium position. When the mass moves closer to the equilibrium position, the restoring force decreases. At the equilibrium position, the net restoring force vanishes. However, at
4106:
1091:
because of the acceleration that the restoring force has imparted. Therefore, the mass continues past the equilibrium position, compressing the spring. A net restoring force then slows it down until its
4167:
2923:
2034:
121:
145:
2162:
2105:
2637:
2997:
2617:
1832:
3749:
3640:
3536:
3448:
1024:
2372:
3853:
2858:
1758:
3929:, satisfying the definition of simple harmonic motion (that net force is directly proportional to the displacement from the mean position and is directed towards the mean position).
3258:
1515:
958:, consisting of a weight attached to one end of a spring, is shown. The other end of the spring is connected to a rigid support such as a wall. If the system is left at rest at the
3921:
2475:
2818:
2310:
1159:
1704:
916:
on the object at the end of the pendulum must be proportional to the displacement (and even so, it is only a good approximation when the angle of the swing is small; see
1637:
3642:
where l is the distance from rotation to center of mass of object undergoing SHM and g being gravitational acceleration. This is analogous to the mass-spring system.
1286:
1785:
1664:
1569:
1542:
1595:
3791:
3771:
3708:
811:
3244:{\displaystyle K(t)={\tfrac {1}{2}}mv^{2}(t)={\tfrac {1}{2}}m\omega ^{2}A^{2}\sin ^{2}(\omega t-\varphi )={\tfrac {1}{2}}kA^{2}\sin ^{2}(\omega t-\varphi ),}
400:
1373:
3773:, therefore a pendulum of the same length on the Moon would swing more slowly due to the Moon's lower gravitational field strength. Because the value of
2168:
519:
3793:
varies slightly over the surface of the earth, the time period will vary slightly from place to place and will also vary with height above sea level.
492:
1961:{\displaystyle x(t)=x_{0}\cos \left({\sqrt {\frac {k}{m}}}t\right)+{\frac {v_{0}}{\sqrt {\frac {k}{m}}}}\sin \left({\sqrt {\frac {k}{m}}}t\right).}
4040:
2875:
1973:
1071:
When the system is displaced from its equilibrium position, a restoring force that obeys Hooke's law tends to restore the system to equilibrium.
3753:
This shows that the period of oscillation is independent of the amplitude and mass of the pendulum but not of the acceleration due to gravity,
766:
4341:
3458:
4111:
1118:
plot are not co-linear, the phase space motion becomes elliptical. The area enclosed depends on the amplitude and the maximum momentum.
474:
4322:
4303:
4284:
2958:
804:
3713:
3684:, the motion of a simple pendulum is approximated by simple harmonic motion. The period of a mass attached to a pendulum of length
3604:
3500:
3390:
990:
3808:
2823:
927:
Simple harmonic motion provides a basis for the characterization of more complicated periodic motion through the techniques of
140:
77:
1131:
135:
3977:
395:
2423:), and the origin is set to be the equilibrium position. Each of these constants carries a physical meaning of the motion:
2111:
951:
on the line and whose magnitude is proportional to the displacement from the fixed point is called simple harmonic motion.
3885:
1033:
854:
2772:
2042:
390:
4386:
797:
784:
545:
468:
857:
to the distance of the object from an equilibrium position and acts towards the equilibrium position. It results in an
233:
464:
265:
2573:
1790:
4263:
684:
573:
499:
359:
292:
4372:
Geogebra applet for spring-mass, with 3 attached PDFs on SHM, driven/damped oscillators, spring-mass with friction
4396:
3967:
3681:
2316:
948:
917:
438:
1709:
1075:
Once the mass is displaced from its equilibrium position, it experiences a net restoring force. As a result, it
3972:
3548:
714:
558:
3992:
3987:
3552:
1475:
704:
664:
428:
3462:
4007:
1267:
959:
709:
2743:{\displaystyle a(t)={\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-A\omega ^{2}\cos(\omega t-\varphi ).}
1363:
1263:
1061:
719:
694:
380:
198:
2253:
3797:
3568:
1130:, for one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear
971:
893:
739:
734:
699:
607:
603:
595:
585:
375:
368:
124:
4002:
3962:
3470:
2395:
are two constants determined by the initial conditions (specifically, the initial position at time
1139:
1127:
955:
921:
514:
455:
433:
178:
173:
168:
68:
4391:
3598:
3005:(the period and frequency are independent of the amplitude and the initial phase of the motion).
1669:
881:
644:
385:
260:
228:
188:
3376:{\displaystyle U(t)={\tfrac {1}{2}}kx^{2}(t)={\tfrac {1}{2}}kA^{2}\cos ^{2}(\omega t-\varphi ).}
1600:
966:
acting on the mass. However, if the mass is displaced from the equilibrium position, the spring
908:
frequency. Other phenomena can be modeled by simple harmonic motion, including the motion of a
32:
4337:
4318:
4299:
4280:
3862:
3384:
2444:
1108:
843:
654:
611:
568:
563:
504:
280:
270:
163:
53:
45:
4036:
The choice of using a cosine in this equation is a convention. Other valid formulations are:
2560:{\displaystyle v(t)={\frac {\mathrm {d} x}{\mathrm {d} t}}=-A\omega \sin(\omega t-\varphi ),}
4017:
3252:
928:
749:
729:
674:
669:
615:
590:
445:
303:
248:
223:
41:
1763:
1642:
1547:
1520:
1151:
889:
850:
744:
689:
639:
634:
553:
17:
3597:
The motion of a body in which it moves to and from about a definite point is also called
1574:
3776:
3756:
3693:
3577:-plane, then its motion along each coordinate is simple harmonic motion with amplitude
3040:
1277:
1243:{\displaystyle F_{\mathrm {net} }=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-kx,}
1143:
1047:
975:
897:
771:
679:
580:
297:
243:
4357:
3947:
4380:
3997:
2454:
1465:{\displaystyle x(t)=c_{1}\cos \left(\omega t\right)+c_{2}\sin \left(\omega t\right),}
1051:
1037:
659:
486:
3457:
4361:
4012:
3796:
This approximation is accurate only for small angles because of the expression for
3664:
3494:
2469:
2237:{\displaystyle \sin \varphi ={\frac {c_{2}}{A}},\;\cos \varphi ={\frac {c_{1}}{A}}}
1076:
944:
724:
649:
338:
218:
3002:
1096:
reaches zero, whereupon it is accelerated back to the equilibrium position again.
3982:
3938:
1353:{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-{\frac {k}{m}}x,}
1115:
1103:
loss, the mass continues to oscillate. Thus simple harmonic motion is a type of
870:
858:
37:
56:
axes have been reversed from the standard convention to align the two diagrams)
901:
509:
183:
4366:
2430:
1706:; taking the derivative of that equation and evaluating at zero we get that
1367:
1104:
913:
905:
825:
531:
3660:
2465:
2461:
1093:
1088:
967:
940:
909:
866:
862:
450:
333:
308:
49:
2769:
is under SHM its acceleration is directly proportional to displacement.
3923:
which makes angular acceleration directly proportional and opposite to
1787:
is the initial speed of the particle divided by the angular frequency,
829:
423:
276:
193:
4371:
1100:
874:
846:
482:
328:
238:
3671:
3601:
or vibratory motion. The time period is able to be calculated by
3456:
963:
884:
for a variety of motions, but is typified by the oscillation of a
318:
313:
255:
31:
1255:
1147:
1107:
motion. If energy is lost in the system, then the mass exhibits
885:
323:
286:
3001:
These equations demonstrate that the simple harmonic motion is
3547:
Simple harmonic motion can be considered the one-dimensional
3383:
In the absence of friction and other energy loss, the total
4101:{\displaystyle x(t)=A\sin \left(\omega t+\varphi '\right),}
2918:{\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}},}
3805:
being proportional to the sine of the displacement angle:
2029:{\displaystyle x(t)=A\cos \left(\omega t-\varphi \right),}
1032:
is the restoring elastic force exerted by the spring (in
116:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
4162:{\displaystyle \tan \varphi '={\frac {c_{1}}{c_{2}}},}
3413:
3315:
3278:
3183:
3114:
3077:
2433:(maximum displacement from the equilibrium position),
1478:
4114:
4043:
3888:
3811:
3779:
3759:
3716:
3696:
3607:
3503:
3393:
3261:
3060:
2961:
2878:
2826:
2775:
2640:
2576:
2478:
2319:
2256:
2171:
2157:{\displaystyle \tan \varphi ={\frac {c_{2}}{c_{1}}},}
2114:
2045:
1976:
1840:
1793:
1766:
1712:
1672:
1645:
1603:
1577:
1550:
1523:
1376:
1289:
1162:
993:
80:
27:
To-and-fro periodic motion in science and engineering
3469:
The following physical systems are some examples of
920:). Simple harmonic motion can also be used to model
36:
Simple harmonic motion shown both in real space and
4207:The maximum displacement (that is, the amplitude),
2100:{\displaystyle A={\sqrt {{c_{1}}^{2}+{c_{2}}^{2}}}}
4161:
4100:
3915:
3847:
3785:
3765:
3743:
3702:
3634:
3530:
3442:
3375:
3243:
2991:
2917:
2852:
2812:
2742:
2611:
2559:
2366:
2304:
2236:
2156:
2099:
2028:
1960:
1826:
1779:
1752:
1698:
1658:
1631:
1589:
1563:
1536:
1509:
1464:
1352:
1242:
1018:
115:
865:which continues indefinitely (if uninhibited by
4313:Thornton, Stephen T.; Marion, Jerry B. (2003).
1970:This equation can also be written in the form:
1067:For any simple mechanical harmonic oscillator:
912:, although for it to be an accurate model, the
4275:Fowles, Grant R.; Cassiday, George L. (2005).
2992:{\displaystyle T=2\pi {\sqrt {\frac {m}{k}}}.}
2612:{\displaystyle {\omega }{\sqrt {A^{2}-x^{2}}}}
1827:{\displaystyle c_{2}={\frac {v_{0}}{\omega }}}
838:
3744:{\displaystyle T=2\pi {\sqrt {\frac {l}{g}}}}
3635:{\displaystyle T=2\pi {\sqrt {\frac {l}{g}}}}
3531:{\displaystyle T=2\pi {\sqrt {\frac {m}{k}}}}
805:
8:
4336:(9th ed.). Hoboken, New Jersey: Wiley.
3443:{\displaystyle E=K+U={\tfrac {1}{2}}kA^{2}.}
1019:{\displaystyle \mathbf {F} =-k\mathbf {x} ,}
4315:Classical Dynamics of Particles and Systems
2367:{\displaystyle \varphi =\arg(c_{1}+c_{2}i)}
3848:{\displaystyle -mgl\sin \theta =I\alpha ,}
3653:
2853:{\displaystyle \omega ^{2}={\frac {k}{m}}}
2204:
1753:{\displaystyle {\dot {x}}(0)=\omega c_{2}}
1138:coefficients, can be obtained by means of
812:
798:
59:
4367:Java simulation of spring-mass oscillator
4148:
4138:
4132:
4113:
4042:
3887:
3810:
3778:
3758:
3729:
3715:
3695:
3620:
3606:
3516:
3502:
3497:. The equation for describing the period
3431:
3412:
3392:
3343:
3333:
3314:
3296:
3277:
3260:
3211:
3201:
3182:
3152:
3142:
3132:
3113:
3095:
3076:
3059:
2974:
2960:
2900:
2885:
2877:
2840:
2831:
2825:
2798:
2774:
2707:
2685:
2676:
2665:
2660:
2656:
2639:
2601:
2588:
2582:
2577:
2575:
2507:
2497:
2494:
2477:
2352:
2339:
2318:
2294:
2285:
2272:
2263:
2255:
2223:
2217:
2190:
2184:
2170:
2143:
2133:
2127:
2113:
2089:
2082:
2077:
2067:
2060:
2055:
2052:
2044:
1975:
1935:
1906:
1900:
1877:
1860:
1839:
1813:
1807:
1798:
1792:
1771:
1765:
1744:
1714:
1713:
1711:
1690:
1677:
1671:
1666:is the initial position of the particle,
1650:
1644:
1623:
1602:
1576:
1555:
1549:
1528:
1522:
1497:
1492:
1487:
1485:
1477:
1431:
1396:
1375:
1334:
1319:
1310:
1299:
1294:
1290:
1288:
1216:
1207:
1196:
1191:
1187:
1168:
1167:
1161:
1008:
994:
992:
97:
91:
82:
81:
79:
3946:
3555:. If an object moves with angular speed
3487:attached to a spring of spring constant
4255:
4029:
1510:{\textstyle \omega ={\sqrt {{k}/{m}}}.}
131:
67:
3649:
880:Simple harmonic motion can serve as a
4279:(7th ed.). Thomson Brooks/Cole.
3882:and therefore the expression becomes
943:moving along a straight line with an
494:Newton's law of universal gravitation
7:
3916:{\displaystyle -mgl\theta =I\alpha }
2472:as a function of time can be found:
1366:above produces a solution that is a
981:Mathematically, the restoring force
947:whose direction is always towards a
849:an object experiences by means of a
4264:"Simple Harmonic Motion – Concepts"
3493:exhibits simple harmonic motion in
2813:{\displaystyle a(x)=-\omega ^{2}x.}
1064:from the equilibrium position (m).
475:Mechanics of planar particle motion
83:
2677:
2661:
2508:
2498:
1597:on the equation above we see that
1311:
1295:
1208:
1192:
1175:
1172:
1169:
904:in time and demonstrates a single
25:
3465:undergoes simple harmonic motion.
2305:{\displaystyle A=|c_{1}+c_{2}i|,}
892:when it is subject to the linear
3690:with gravitational acceleration
2411:, while the initial velocity is
1009:
995:
779:
778:
765:
98:
4053:
4047:
3367:
3352:
3308:
3302:
3271:
3265:
3235:
3220:
3176:
3161:
3107:
3101:
3070:
3064:
2785:
2779:
2734:
2719:
2650:
2644:
2551:
2536:
2488:
2482:
2361:
2332:
2295:
2264:
1986:
1980:
1850:
1844:
1731:
1725:
1613:
1607:
1386:
1380:
1132:ordinary differential equation
962:position then there is no net
1:
4317:(5th ed.). Brooks Cole.
1571:can be easily found: setting
1517:The meaning of the constants
1099:As long as the system has no
401:Koopman–von Neumann mechanics
4298:. University Science Books.
3667:in 60 s, a frequency of 0.41
853:whose magnitude is directly
469:Non-inertial reference frame
1699:{\displaystyle c_{1}=x_{0}}
1114:Note if the real space and
396:Appell's equation of motion
266:Inertial frame of reference
4413:
3936:
3561:around a circle of radius
3471:simple harmonic oscillator
1632:{\displaystyle x(0)=c_{1}}
956:simple harmonic oscillator
18:Simple harmonic oscillator
3978:Rayleigh–Lorentz pendulum
3968:Small-angle approximation
3682:small-angle approximation
3652:
3646:Mass of a simple pendulum
2763:By definition, if a mass
1280:for a mass on a spring).
1258:of the oscillating body,
918:small-angle approximation
896:restoring force given by
4294:Taylor, John R. (2005).
3973:Lorentz oscillator model
2460:Using the techniques of
1270:(or mean) position, and
559:Rotating reference frame
391:Hamilton–Jacobi equation
3993:Complex harmonic motion
3988:Uniform circular motion
3553:uniform circular motion
3543:Uniform circular motion
861:that is described by a
842:) is a special type of
836:(sometimes abbreviated
500:Newton's laws of motion
360:Newton's laws of motion
4358:Simple Harmonic Motion
4332:Walker, Jearl (2011).
4163:
4102:
4008:Pendulum (mathematics)
3952:
3917:
3849:
3787:
3767:
3745:
3704:
3636:
3583:and angular frequency
3532:
3466:
3444:
3377:
3245:
3048:of the system at time
2993:
2919:
2854:
2814:
2753:Maximum acceleration:
2744:
2631:(at equilibrium point)
2613:
2561:
2368:
2306:
2238:
2158:
2101:
2030:
1962:
1828:
1781:
1754:
1700:
1660:
1633:
1591:
1565:
1538:
1511:
1466:
1354:
1244:
1020:
834:simple harmonic motion
527:Simple harmonic motion
440:Euler's laws of motion
234:D'Alembert's principle
117:
57:
4334:Principles of Physics
4164:
4103:
3951:Scotch yoke animation
3950:
3918:
3850:
3788:
3768:
3746:
3705:
3637:
3533:
3460:
3445:
3387:has a constant value
3378:
3246:
2994:
2920:
2855:
2815:
2745:
2614:
2562:
2369:
2307:
2239:
2159:
2102:
2031:
1963:
1834:. Thus we can write:
1829:
1782:
1780:{\displaystyle c_{2}}
1755:
1701:
1661:
1659:{\displaystyle c_{1}}
1634:
1592:
1566:
1564:{\displaystyle c_{2}}
1539:
1537:{\displaystyle c_{1}}
1512:
1467:
1364:differential equation
1355:
1245:
1021:
381:Hamiltonian mechanics
199:Statistical mechanics
118:
35:
4277:Analytical Mechanics
4112:
4041:
3886:
3809:
3798:angular acceleration
3777:
3757:
3714:
3694:
3605:
3501:
3391:
3259:
3058:
2959:
2955:is the time period,
2876:
2824:
2773:
2638:
2574:
2476:
2317:
2254:
2169:
2112:
2043:
1974:
1838:
1791:
1764:
1710:
1670:
1643:
1601:
1575:
1548:
1521:
1476:
1374:
1287:
1160:
991:
604:Angular acceleration
596:Rotational frequency
376:Lagrangian mechanics
369:Analytical mechanics
125:Second law of motion
78:
4387:Classical mechanics
4296:Classical Mechanics
4003:Harmonic oscillator
3963:Newtonian mechanics
3663:making 25 complete
2759:(at extreme points)
1590:{\displaystyle t=0}
1368:sinusoidal function
1276:is a constant (the
1128:Newtonian mechanics
922:molecular vibration
456:Harmonic oscillator
434:Equations of motion
69:Classical mechanics
63:Part of a series on
4159:
4098:
3953:
3913:
3845:
3783:
3763:
3741:
3700:
3632:
3599:oscillatory motion
3593:Oscillatory motion
3528:
3467:
3463:spring–mass system
3440:
3422:
3373:
3324:
3287:
3241:
3192:
3123:
3086:
2989:
2915:
2850:
2810:
2740:
2609:
2557:
2364:
2302:
2234:
2154:
2097:
2026:
1958:
1824:
1777:
1750:
1696:
1656:
1629:
1587:
1561:
1534:
1507:
1462:
1350:
1240:
1109:damped oscillation
1016:
954:In the diagram, a
882:mathematical model
772:Physics portal
386:Routhian mechanics
261:Frame of reference
113:
58:
4343:978-0-470-56158-4
4154:
3863:moment of inertia
3786:{\displaystyle g}
3766:{\displaystyle g}
3739:
3738:
3703:{\displaystyle g}
3678:
3677:
3630:
3629:
3526:
3525:
3421:
3385:mechanical energy
3323:
3286:
3191:
3122:
3085:
2984:
2983:
2910:
2909:
2898:
2848:
2692:
2607:
2516:
2445:angular frequency
2377:In the solution,
2247:or equivalently
2232:
2199:
2149:
2095:
1945:
1944:
1922:
1921:
1920:
1887:
1886:
1822:
1722:
1502:
1342:
1326:
1223:
974:force that obeys
822:
821:
569:Centrifugal force
564:Centripetal force
520:Euler's equations
505:Relative velocity
281:Moment of inertia
111:
85:
16:(Redirected from
4404:
4397:Motion (physics)
4347:
4328:
4309:
4290:
4268:
4267:
4260:
4240:
4228:, and thus when
4227:
4215:
4205:
4197:
4191:
4189:
4188:
4185:
4182:
4168:
4166:
4165:
4160:
4155:
4153:
4152:
4143:
4142:
4133:
4128:
4107:
4105:
4104:
4099:
4094:
4090:
4089:
4034:
4018:String vibration
3928:
3922:
3920:
3919:
3914:
3881:
3870:
3860:
3854:
3852:
3851:
3846:
3804:
3792:
3790:
3789:
3784:
3772:
3770:
3769:
3764:
3750:
3748:
3747:
3742:
3740:
3731:
3730:
3709:
3707:
3706:
3701:
3689:
3670:
3657:
3650:
3641:
3639:
3638:
3633:
3631:
3622:
3621:
3588:
3582:
3576:
3567:centered at the
3566:
3560:
3537:
3535:
3534:
3529:
3527:
3518:
3517:
3492:
3486:
3477:Mass on a spring
3449:
3447:
3446:
3441:
3436:
3435:
3423:
3414:
3382:
3380:
3379:
3374:
3348:
3347:
3338:
3337:
3325:
3316:
3301:
3300:
3288:
3279:
3253:potential energy
3250:
3248:
3247:
3242:
3216:
3215:
3206:
3205:
3193:
3184:
3157:
3156:
3147:
3146:
3137:
3136:
3124:
3115:
3100:
3099:
3087:
3078:
3053:
3047:
3038:
3036:
3034:
3033:
3030:
3027:
3018:
2998:
2996:
2995:
2990:
2985:
2976:
2975:
2954:
2948:
2947:
2945:
2944:
2939:
2936:
2924:
2922:
2921:
2916:
2911:
2902:
2901:
2899:
2897:
2886:
2871:
2859:
2857:
2856:
2851:
2849:
2841:
2836:
2835:
2819:
2817:
2816:
2811:
2803:
2802:
2768:
2758:
2749:
2747:
2746:
2741:
2712:
2711:
2693:
2691:
2690:
2689:
2680:
2674:
2670:
2669:
2664:
2657:
2630:
2618:
2616:
2615:
2610:
2608:
2606:
2605:
2593:
2592:
2583:
2581:
2566:
2564:
2563:
2558:
2517:
2515:
2511:
2505:
2501:
2495:
2452:
2442:
2428:
2422:
2410:
2401:
2394:
2385:
2373:
2371:
2370:
2365:
2357:
2356:
2344:
2343:
2311:
2309:
2308:
2303:
2298:
2290:
2289:
2277:
2276:
2267:
2243:
2241:
2240:
2235:
2233:
2228:
2227:
2218:
2200:
2195:
2194:
2185:
2163:
2161:
2160:
2155:
2150:
2148:
2147:
2138:
2137:
2128:
2106:
2104:
2103:
2098:
2096:
2094:
2093:
2088:
2087:
2086:
2072:
2071:
2066:
2065:
2064:
2053:
2035:
2033:
2032:
2027:
2022:
2018:
1967:
1965:
1964:
1959:
1954:
1950:
1946:
1937:
1936:
1923:
1913:
1912:
1911:
1910:
1901:
1896:
1892:
1888:
1879:
1878:
1865:
1864:
1833:
1831:
1830:
1825:
1823:
1818:
1817:
1808:
1803:
1802:
1786:
1784:
1783:
1778:
1776:
1775:
1759:
1757:
1756:
1751:
1749:
1748:
1724:
1723:
1715:
1705:
1703:
1702:
1697:
1695:
1694:
1682:
1681:
1665:
1663:
1662:
1657:
1655:
1654:
1638:
1636:
1635:
1630:
1628:
1627:
1596:
1594:
1593:
1588:
1570:
1568:
1567:
1562:
1560:
1559:
1543:
1541:
1540:
1535:
1533:
1532:
1516:
1514:
1513:
1508:
1503:
1501:
1496:
1491:
1486:
1471:
1469:
1468:
1463:
1458:
1454:
1436:
1435:
1423:
1419:
1401:
1400:
1359:
1357:
1356:
1351:
1343:
1335:
1327:
1325:
1324:
1323:
1314:
1308:
1304:
1303:
1298:
1291:
1275:
1261:
1253:
1249:
1247:
1246:
1241:
1224:
1222:
1221:
1220:
1211:
1205:
1201:
1200:
1195:
1188:
1180:
1179:
1178:
1140:Newton's 2nd law
1086:
1059:
1045:
1031:
1025:
1023:
1022:
1017:
1012:
998:
986:
939:The motion of a
929:Fourier analysis
900:. The motion is
840:
814:
807:
800:
787:
782:
781:
774:
770:
769:
675:Johann Bernoulli
670:Daniel Bernoulli
591:Tangential speed
495:
471:
446:Fictitious force
441:
293:Mechanical power
283:
224:Angular momentum
122:
120:
119:
114:
112:
110:
102:
101:
92:
87:
86:
60:
21:
4412:
4411:
4407:
4406:
4405:
4403:
4402:
4401:
4377:
4376:
4354:
4344:
4331:
4325:
4312:
4306:
4293:
4287:
4274:
4271:
4262:
4261:
4257:
4253:
4247:
4245:
4242:
4235:
4229:
4217:
4214:
4208:
4202:
4199:
4186:
4183:
4180:
4179:
4177:
4171:
4144:
4134:
4121:
4110:
4109:
4082:
4072:
4068:
4039:
4038:
4031:
4027:
4022:
3958:
3941:
3935:
3924:
3884:
3883:
3872:
3866:
3856:
3807:
3806:
3800:
3775:
3774:
3755:
3754:
3712:
3711:
3692:
3691:
3685:
3674:
3668:
3648:
3603:
3602:
3595:
3584:
3578:
3572:
3562:
3556:
3545:
3499:
3498:
3488:
3482:
3479:
3455:
3427:
3389:
3388:
3339:
3329:
3292:
3257:
3256:
3207:
3197:
3148:
3138:
3128:
3091:
3056:
3055:
3049:
3043:
3031:
3028:
3025:
3024:
3022:
3020:
3014:
3011:
2957:
2956:
2950:
2940:
2937:
2934:
2933:
2931:
2926:
2890:
2874:
2873:
2863:
2827:
2822:
2821:
2794:
2771:
2770:
2764:
2754:
2703:
2681:
2675:
2659:
2658:
2636:
2635:
2622:
2621:Maximum speed:
2597:
2584:
2572:
2571:
2506:
2496:
2474:
2473:
2453:is the initial
2448:
2434:
2424:
2418:
2412:
2409:
2403:
2396:
2393:
2387:
2384:
2378:
2348:
2335:
2315:
2314:
2281:
2268:
2252:
2251:
2219:
2186:
2167:
2166:
2139:
2129:
2110:
2109:
2078:
2076:
2056:
2054:
2041:
2040:
2005:
2001:
1972:
1971:
1934:
1930:
1902:
1876:
1872:
1856:
1836:
1835:
1809:
1794:
1789:
1788:
1767:
1762:
1761:
1740:
1708:
1707:
1686:
1673:
1668:
1667:
1646:
1641:
1640:
1619:
1599:
1598:
1573:
1572:
1551:
1546:
1545:
1524:
1519:
1518:
1474:
1473:
1447:
1443:
1427:
1412:
1408:
1392:
1372:
1371:
1315:
1309:
1293:
1292:
1285:
1284:
1278:spring constant
1271:
1259:
1251:
1212:
1206:
1190:
1189:
1163:
1158:
1157:
1124:
1087:, the mass has
1081:
1055:
1048:spring constant
1041:
1027:
989:
988:
982:
937:
910:simple pendulum
851:restoring force
818:
777:
764:
763:
756:
755:
754:
629:
621:
620:
600:
554:Circular motion
548:
538:
537:
536:
493:
463:
460:
439:
418:
410:
409:
406:
405:
363:
353:
345:
344:
343:
302:
298:Mechanical work
291:
275:
213:
205:
204:
203:
158:
150:
127:
103:
93:
76:
75:
28:
23:
22:
15:
12:
11:
5:
4410:
4408:
4400:
4399:
4394:
4389:
4379:
4378:
4375:
4374:
4369:
4364:
4353:
4352:External links
4350:
4349:
4348:
4342:
4329:
4323:
4310:
4304:
4291:
4285:
4270:
4269:
4254:
4252:
4249:
4244:
4243:
4233:
4216:, occurs when
4212:
4206:
4200:
4158:
4151:
4147:
4141:
4137:
4131:
4127:
4124:
4120:
4117:
4097:
4093:
4088:
4085:
4081:
4078:
4075:
4071:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4046:
4035:
4028:
4026:
4023:
4021:
4020:
4015:
4010:
4005:
4000:
3995:
3990:
3985:
3980:
3975:
3970:
3965:
3959:
3957:
3954:
3937:Main article:
3934:
3931:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3820:
3817:
3814:
3782:
3762:
3737:
3734:
3728:
3725:
3722:
3719:
3699:
3676:
3675:
3658:
3647:
3644:
3628:
3625:
3619:
3616:
3613:
3610:
3594:
3591:
3544:
3541:
3524:
3521:
3515:
3512:
3509:
3506:
3478:
3475:
3454:
3451:
3439:
3434:
3430:
3426:
3420:
3417:
3411:
3408:
3405:
3402:
3399:
3396:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3346:
3342:
3336:
3332:
3328:
3322:
3319:
3313:
3310:
3307:
3304:
3299:
3295:
3291:
3285:
3282:
3276:
3273:
3270:
3267:
3264:
3240:
3237:
3234:
3231:
3228:
3225:
3222:
3219:
3214:
3210:
3204:
3200:
3196:
3190:
3187:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3155:
3151:
3145:
3141:
3135:
3131:
3127:
3121:
3118:
3112:
3109:
3106:
3103:
3098:
3094:
3090:
3084:
3081:
3075:
3072:
3069:
3066:
3063:
3041:kinetic energy
3010:
3007:
2988:
2982:
2979:
2973:
2970:
2967:
2964:
2914:
2908:
2905:
2896:
2893:
2889:
2884:
2881:
2847:
2844:
2839:
2834:
2830:
2809:
2806:
2801:
2797:
2793:
2790:
2787:
2784:
2781:
2778:
2761:
2760:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2710:
2706:
2702:
2699:
2696:
2688:
2684:
2679:
2673:
2668:
2663:
2655:
2652:
2649:
2646:
2643:
2633:
2632:
2619:
2604:
2600:
2596:
2591:
2587:
2580:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2514:
2510:
2504:
2500:
2493:
2490:
2487:
2484:
2481:
2416:
2407:
2391:
2382:
2375:
2374:
2363:
2360:
2355:
2351:
2347:
2342:
2338:
2334:
2331:
2328:
2325:
2322:
2312:
2301:
2297:
2293:
2288:
2284:
2280:
2275:
2271:
2266:
2262:
2259:
2245:
2244:
2231:
2226:
2222:
2216:
2213:
2210:
2207:
2203:
2198:
2193:
2189:
2183:
2180:
2177:
2174:
2164:
2153:
2146:
2142:
2136:
2132:
2126:
2123:
2120:
2117:
2107:
2092:
2085:
2081:
2075:
2070:
2063:
2059:
2051:
2048:
2025:
2021:
2017:
2014:
2011:
2008:
2004:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1957:
1953:
1949:
1943:
1940:
1933:
1929:
1926:
1919:
1916:
1909:
1905:
1899:
1895:
1891:
1885:
1882:
1875:
1871:
1868:
1863:
1859:
1855:
1852:
1849:
1846:
1843:
1821:
1816:
1812:
1806:
1801:
1797:
1774:
1770:
1747:
1743:
1739:
1736:
1733:
1730:
1727:
1721:
1718:
1693:
1689:
1685:
1680:
1676:
1653:
1649:
1626:
1622:
1618:
1615:
1612:
1609:
1606:
1586:
1583:
1580:
1558:
1554:
1531:
1527:
1506:
1500:
1495:
1490:
1484:
1481:
1461:
1457:
1453:
1450:
1446:
1442:
1439:
1434:
1430:
1426:
1422:
1418:
1415:
1411:
1407:
1404:
1399:
1395:
1391:
1388:
1385:
1382:
1379:
1349:
1346:
1341:
1338:
1333:
1330:
1322:
1318:
1313:
1307:
1302:
1297:
1239:
1236:
1233:
1230:
1227:
1219:
1215:
1210:
1204:
1199:
1194:
1186:
1183:
1177:
1174:
1171:
1166:
1123:
1120:
1073:
1072:
1015:
1011:
1007:
1004:
1001:
997:
936:
933:
820:
819:
817:
816:
809:
802:
794:
791:
790:
789:
788:
775:
758:
757:
753:
752:
747:
742:
737:
732:
727:
722:
717:
712:
707:
702:
697:
692:
687:
682:
677:
672:
667:
662:
657:
652:
647:
642:
637:
631:
630:
627:
626:
623:
622:
619:
618:
599:
598:
593:
588:
583:
581:Coriolis force
578:
577:
576:
566:
561:
556:
550:
549:
544:
543:
540:
539:
535:
534:
529:
524:
523:
522:
517:
507:
502:
497:
490:
479:
478:
477:
472:
459:
458:
453:
448:
443:
436:
431:
426:
420:
419:
416:
415:
412:
411:
408:
407:
404:
403:
398:
393:
388:
383:
378:
372:
366:
364:
357:
354:
351:
350:
347:
346:
342:
341:
336:
331:
326:
321:
316:
311:
306:
300:
295:
289:
284:
273:
268:
263:
258:
253:
252:
251:
246:
236:
231:
226:
221:
215:
214:
211:
210:
207:
206:
202:
201:
196:
191:
186:
181:
176:
171:
166:
160:
159:
156:
155:
152:
151:
149:
148:
143:
138:
132:
129:
128:
123:
109:
106:
100:
96:
90:
72:
71:
65:
64:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4409:
4398:
4395:
4393:
4390:
4388:
4385:
4384:
4382:
4373:
4370:
4368:
4365:
4363:
4359:
4356:
4355:
4351:
4345:
4339:
4335:
4330:
4326:
4324:0-534-40896-6
4320:
4316:
4311:
4307:
4305:1-891389-22-X
4301:
4297:
4292:
4288:
4286:0-534-49492-7
4282:
4278:
4273:
4272:
4265:
4259:
4256:
4250:
4248:
4239:
4232:
4225:
4221:
4211:
4204:
4201:
4195:
4175:
4169:
4156:
4149:
4145:
4139:
4135:
4129:
4125:
4122:
4118:
4115:
4095:
4091:
4086:
4083:
4079:
4076:
4073:
4069:
4065:
4062:
4059:
4056:
4050:
4044:
4033:
4030:
4024:
4019:
4016:
4014:
4011:
4009:
4006:
4004:
4001:
3999:
3998:Damping ratio
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3979:
3976:
3974:
3971:
3969:
3966:
3964:
3961:
3960:
3955:
3949:
3945:
3940:
3932:
3930:
3927:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3889:
3880:
3876:
3869:
3864:
3859:
3842:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3803:
3799:
3794:
3780:
3760:
3751:
3735:
3732:
3726:
3723:
3720:
3717:
3697:
3688:
3683:
3673:
3666:
3662:
3656:
3651:
3645:
3643:
3626:
3623:
3617:
3614:
3611:
3608:
3600:
3592:
3590:
3587:
3581:
3575:
3570:
3565:
3559:
3554:
3550:
3542:
3540:
3522:
3519:
3513:
3510:
3507:
3504:
3496:
3491:
3485:
3476:
3474:
3472:
3464:
3459:
3452:
3450:
3437:
3432:
3428:
3424:
3418:
3415:
3409:
3406:
3403:
3400:
3397:
3394:
3386:
3370:
3364:
3361:
3358:
3355:
3349:
3344:
3340:
3334:
3330:
3326:
3320:
3317:
3311:
3305:
3297:
3293:
3289:
3283:
3280:
3274:
3268:
3262:
3254:
3238:
3232:
3229:
3226:
3223:
3217:
3212:
3208:
3202:
3198:
3194:
3188:
3185:
3179:
3173:
3170:
3167:
3164:
3158:
3153:
3149:
3143:
3139:
3133:
3129:
3125:
3119:
3116:
3110:
3104:
3096:
3092:
3088:
3082:
3079:
3073:
3067:
3061:
3052:
3046:
3042:
3037:
3017:
3013:Substituting
3008:
3006:
3004:
2999:
2986:
2980:
2977:
2971:
2968:
2965:
2962:
2953:
2943:
2929:
2912:
2906:
2903:
2894:
2891:
2887:
2882:
2879:
2870:
2866:
2860:
2845:
2842:
2837:
2832:
2828:
2807:
2804:
2799:
2795:
2791:
2788:
2782:
2776:
2767:
2757:
2752:
2751:
2750:
2737:
2731:
2728:
2725:
2722:
2716:
2713:
2708:
2704:
2700:
2697:
2694:
2686:
2682:
2671:
2666:
2653:
2647:
2641:
2629:
2625:
2620:
2602:
2598:
2594:
2589:
2585:
2578:
2569:
2568:
2567:
2554:
2548:
2545:
2542:
2539:
2533:
2530:
2527:
2524:
2521:
2518:
2512:
2502:
2491:
2485:
2479:
2471:
2467:
2463:
2458:
2456:
2451:
2446:
2441:
2437:
2432:
2427:
2421:
2415:
2406:
2399:
2390:
2381:
2358:
2353:
2349:
2345:
2340:
2336:
2329:
2326:
2323:
2320:
2313:
2299:
2291:
2286:
2282:
2278:
2273:
2269:
2260:
2257:
2250:
2249:
2248:
2229:
2224:
2220:
2214:
2211:
2208:
2205:
2201:
2196:
2191:
2187:
2181:
2178:
2175:
2172:
2165:
2151:
2144:
2140:
2134:
2130:
2124:
2121:
2118:
2115:
2108:
2090:
2083:
2079:
2073:
2068:
2061:
2057:
2049:
2046:
2039:
2038:
2037:
2023:
2019:
2015:
2012:
2009:
2006:
2002:
1998:
1995:
1992:
1989:
1983:
1977:
1968:
1955:
1951:
1947:
1941:
1938:
1931:
1927:
1924:
1917:
1914:
1907:
1903:
1897:
1893:
1889:
1883:
1880:
1873:
1869:
1866:
1861:
1857:
1853:
1847:
1841:
1819:
1814:
1810:
1804:
1799:
1795:
1772:
1768:
1745:
1741:
1737:
1734:
1728:
1719:
1716:
1691:
1687:
1683:
1678:
1674:
1651:
1647:
1624:
1620:
1616:
1610:
1604:
1584:
1581:
1578:
1556:
1552:
1529:
1525:
1504:
1498:
1493:
1488:
1482:
1479:
1459:
1455:
1451:
1448:
1444:
1440:
1437:
1432:
1428:
1424:
1420:
1416:
1413:
1409:
1405:
1402:
1397:
1393:
1389:
1383:
1377:
1369:
1365:
1360:
1347:
1344:
1339:
1336:
1331:
1328:
1320:
1316:
1305:
1300:
1281:
1279:
1274:
1269:
1265:
1257:
1256:inertial mass
1237:
1234:
1231:
1228:
1225:
1217:
1213:
1202:
1197:
1184:
1181:
1164:
1155:
1153:
1149:
1145:
1141:
1137:
1133:
1129:
1121:
1119:
1117:
1112:
1110:
1106:
1102:
1097:
1095:
1090:
1084:
1078:
1070:
1069:
1068:
1065:
1063:
1058:
1053:
1049:
1044:
1039:
1035:
1030:
1013:
1005:
1002:
999:
985:
979:
977:
973:
969:
965:
961:
957:
952:
950:
946:
942:
934:
932:
930:
925:
923:
919:
915:
911:
907:
903:
899:
895:
891:
887:
883:
878:
876:
872:
869:or any other
868:
864:
860:
856:
852:
848:
845:
841:
835:
831:
827:
815:
810:
808:
803:
801:
796:
795:
793:
792:
786:
776:
773:
768:
762:
761:
760:
759:
751:
748:
746:
743:
741:
738:
736:
733:
731:
728:
726:
723:
721:
718:
716:
713:
711:
708:
706:
703:
701:
698:
696:
693:
691:
688:
686:
683:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
632:
625:
624:
617:
613:
609:
605:
602:
601:
597:
594:
592:
589:
587:
584:
582:
579:
575:
572:
571:
570:
567:
565:
562:
560:
557:
555:
552:
551:
547:
542:
541:
533:
530:
528:
525:
521:
518:
516:
513:
512:
511:
508:
506:
503:
501:
498:
496:
491:
488:
484:
481:
480:
476:
473:
470:
466:
462:
461:
457:
454:
452:
449:
447:
444:
442:
437:
435:
432:
430:
427:
425:
422:
421:
414:
413:
402:
399:
397:
394:
392:
389:
387:
384:
382:
379:
377:
374:
373:
371:
370:
365:
362:
361:
356:
355:
349:
348:
340:
337:
335:
332:
330:
327:
325:
322:
320:
317:
315:
312:
310:
307:
305:
301:
299:
296:
294:
290:
288:
285:
282:
278:
274:
272:
269:
267:
264:
262:
259:
257:
254:
250:
247:
245:
242:
241:
240:
237:
235:
232:
230:
227:
225:
222:
220:
217:
216:
209:
208:
200:
197:
195:
192:
190:
187:
185:
182:
180:
177:
175:
172:
170:
167:
165:
162:
161:
154:
153:
147:
144:
142:
139:
137:
134:
133:
130:
126:
107:
104:
94:
88:
74:
73:
70:
66:
62:
61:
55:
51:
47:
43:
39:
34:
30:
19:
4362:HyperPhysics
4333:
4314:
4295:
4276:
4258:
4246:
4237:
4230:
4223:
4219:
4209:
4203:
4193:
4173:
4037:
4032:
4013:Circle group
3942:
3925:
3878:
3874:
3867:
3857:
3801:
3795:
3752:
3710:is given by
3686:
3679:
3665:oscillations
3596:
3585:
3579:
3573:
3563:
3557:
3546:
3495:closed space
3489:
3483:
3480:
3468:
3461:An undamped
3050:
3044:
3021:
3015:
3012:
3000:
2951:
2941:
2927:
2868:
2864:
2861:
2765:
2762:
2755:
2634:
2627:
2623:
2470:acceleration
2459:
2449:
2439:
2435:
2425:
2419:
2413:
2404:
2397:
2388:
2379:
2376:
2246:
1969:
1362:Solving the
1361:
1282:
1272:
1264:displacement
1156:
1135:
1125:
1113:
1098:
1082:
1074:
1066:
1062:displacement
1056:
1042:
1028:
987:is given by
983:
980:
970:a restoring
953:
945:acceleration
938:
935:Introduction
926:
879:
855:proportional
837:
833:
823:
614: /
610: /
608:displacement
606: /
526:
467: /
429:Displacement
367:
358:
352:Formulations
339:Virtual work
279: /
219:Acceleration
212:Fundamentals
48:. (Here the
29:
3983:Isochronous
3939:Scotch yoke
3933:Scotch yoke
3003:isochronous
2925:and, since
1283:Therefore,
1268:equilibrium
1144:Hooke's law
1116:phase space
1077:accelerates
976:Hooke's law
960:equilibrium
949:fixed point
898:Hooke's law
871:dissipation
859:oscillation
750:von Neumann
417:Core topics
38:phase space
4381:Categories
4251:References
3871:is small,
3549:projection
1760:, so that
1639:, so that
902:sinusoidal
685:d'Alembert
665:Maupertuis
628:Scientists
510:Rigid body
184:Kinematics
4392:Pendulums
4123:φ
4119:
4084:φ
4074:ω
4066:
3911:α
3902:θ
3890:−
3840:α
3831:θ
3828:
3813:−
3727:π
3618:π
3514:π
3365:φ
3362:−
3356:ω
3350:
3233:φ
3230:−
3224:ω
3218:
3174:φ
3171:−
3165:ω
3159:
3130:ω
2972:π
2895:π
2829:ω
2796:ω
2792:−
2732:φ
2729:−
2723:ω
2717:
2705:ω
2698:−
2595:−
2579:ω
2549:φ
2546:−
2540:ω
2534:
2528:ω
2522:−
2431:amplitude
2330:
2321:φ
2212:φ
2209:
2179:φ
2176:
2122:φ
2119:
2016:φ
2013:−
2007:ω
1999:
1928:
1870:
1820:ω
1738:ω
1720:˙
1480:ω
1449:ω
1441:
1414:ω
1406:
1332:−
1266:from the
1229:−
1054:·m), and
1003:−
914:net force
826:mechanics
730:Liouville
612:frequency
532:Vibration
249:potential
174:Continuum
169:Celestial
146:Textbooks
4126:′
4087:′
3956:See also
3661:pendulum
3453:Examples
3251:and the
2570:Speed:
2466:velocity
2462:calculus
1136:constant
1122:Dynamics
1105:periodic
1094:velocity
1089:momentum
941:particle
906:resonant
867:friction
863:sinusoid
844:periodic
785:Category
710:Hamilton
695:Lagrange
690:Clairaut
655:Horrocks
616:velocity
586:Pendulum
574:reactive
546:Rotation
515:dynamics
465:Inertial
451:Friction
334:Velocity
309:Momentum
189:Kinetics
179:Dynamics
157:Branches
141:Timeline
54:position
50:velocity
46:periodic
4190:
4178:
3865:. When
3861:is the
3680:In the
3571:of the
3481:A mass
3035:
3023:
2946:
2932:
2443:is the
2429:is the
1262:is its
1254:is the
1060:is the
1046:is the
1036:units:
972:elastic
894:elastic
830:physics
745:Koopman
705:Poisson
700:Laplace
645:Huygens
640:Galileo
485: (
424:Damping
277:Inertia
271:Impulse
244:kinetic
194:Statics
164:Applied
136:History
4340:
4321:
4302:
4283:
4176:= sin(
4170:since
4108:where
3855:where
3569:origin
3039:, the
3009:Energy
2949:where
2862:Since
2820:where
2464:, the
2447:, and
2036:where
1472:where
1250:where
1152:spring
1146:for a
1101:energy
1026:where
968:exerts
890:spring
875:energy
847:motion
783:
735:Appell
720:Cauchy
715:Jacobi
660:Halley
650:Newton
635:Kepler
487:linear
483:Motion
329:Torque
304:Moment
239:Energy
229:Couple
40:. The
4360:from
4226:) = 1
4025:Notes
3672:Hertz
3019:with
2455:phase
1150:on a
1134:with
964:force
888:on a
740:Gibbs
725:Routh
680:Euler
319:Speed
314:Space
256:Force
42:orbit
4338:ISBN
4319:ISBN
4300:ISBN
4281:ISBN
4218:cos(
4172:cos
3873:sin
3054:is
2468:and
2386:and
1544:and
1148:mass
1142:and
886:mass
828:and
324:Time
287:Mass
52:and
4234:max
4213:max
4116:tan
4063:sin
3825:sin
3551:of
3341:cos
3255:is
3209:sin
3150:sin
2867:= 2
2714:cos
2531:sin
2438:= 2
2402:is
2400:= 0
2327:arg
2206:cos
2173:sin
2116:tan
1996:cos
1925:sin
1867:cos
1438:sin
1403:cos
1126:In
1085:= 0
1040:),
877:).
873:of
839:SHM
824:In
44:is
4383::
4236:=
4222:±
4220:ωt
4192:−
3877:≈
3659:A
3589:.
3574:xy
3473:.
2930:=
2872:,
2869:πf
2756:Aω
2628:ωA
2626:=
2457:.
2440:πf
1370::
1154:.
1111:.
1034:SI
978:.
931:.
924:.
832:,
4346:.
4327:.
4308:.
4289:.
4266:.
4241:.
4238:A
4231:x
4224:φ
4210:x
4198:.
4196:)
4194:θ
4187:2
4184:/
4181:π
4174:θ
4157:,
4150:2
4146:c
4140:1
4136:c
4130:=
4096:,
4092:)
4080:+
4077:t
4070:(
4060:A
4057:=
4054:)
4051:t
4048:(
4045:x
3926:θ
3908:I
3905:=
3899:l
3896:g
3893:m
3879:θ
3875:θ
3868:θ
3858:I
3843:,
3837:I
3834:=
3822:l
3819:g
3816:m
3802:α
3781:g
3761:g
3736:g
3733:l
3724:2
3721:=
3718:T
3698:g
3687:l
3669:6
3627:g
3624:l
3615:2
3612:=
3609:T
3586:ω
3580:r
3564:r
3558:ω
3523:k
3520:m
3511:2
3508:=
3505:T
3490:k
3484:m
3438:.
3433:2
3429:A
3425:k
3419:2
3416:1
3410:=
3407:U
3404:+
3401:K
3398:=
3395:E
3371:.
3368:)
3359:t
3353:(
3345:2
3335:2
3331:A
3327:k
3321:2
3318:1
3312:=
3309:)
3306:t
3303:(
3298:2
3294:x
3290:k
3284:2
3281:1
3275:=
3272:)
3269:t
3266:(
3263:U
3239:,
3236:)
3227:t
3221:(
3213:2
3203:2
3199:A
3195:k
3189:2
3186:1
3180:=
3177:)
3168:t
3162:(
3154:2
3144:2
3140:A
3134:2
3126:m
3120:2
3117:1
3111:=
3108:)
3105:t
3102:(
3097:2
3093:v
3089:m
3083:2
3080:1
3074:=
3071:)
3068:t
3065:(
3062:K
3051:t
3045:K
3032:m
3029:/
3026:k
3016:ω
2987:.
2981:k
2978:m
2969:2
2966:=
2963:T
2952:T
2942:f
2938:/
2935:1
2928:T
2913:,
2907:m
2904:k
2892:2
2888:1
2883:=
2880:f
2865:ω
2846:m
2843:k
2838:=
2833:2
2808:.
2805:x
2800:2
2789:=
2786:)
2783:x
2780:(
2777:a
2766:m
2738:.
2735:)
2726:t
2720:(
2709:2
2701:A
2695:=
2687:2
2683:t
2678:d
2672:x
2667:2
2662:d
2654:=
2651:)
2648:t
2645:(
2642:a
2624:v
2603:2
2599:x
2590:2
2586:A
2555:,
2552:)
2543:t
2537:(
2525:A
2519:=
2513:t
2509:d
2503:x
2499:d
2492:=
2489:)
2486:t
2483:(
2480:v
2450:φ
2436:ω
2426:A
2420:ω
2417:2
2414:c
2408:1
2405:c
2398:t
2392:2
2389:c
2383:1
2380:c
2362:)
2359:i
2354:2
2350:c
2346:+
2341:1
2337:c
2333:(
2324:=
2300:,
2296:|
2292:i
2287:2
2283:c
2279:+
2274:1
2270:c
2265:|
2261:=
2258:A
2230:A
2225:1
2221:c
2215:=
2202:,
2197:A
2192:2
2188:c
2182:=
2152:,
2145:1
2141:c
2135:2
2131:c
2125:=
2091:2
2084:2
2080:c
2074:+
2069:2
2062:1
2058:c
2050:=
2047:A
2024:,
2020:)
2010:t
2003:(
1993:A
1990:=
1987:)
1984:t
1981:(
1978:x
1956:.
1952:)
1948:t
1942:m
1939:k
1932:(
1918:m
1915:k
1908:0
1904:v
1898:+
1894:)
1890:t
1884:m
1881:k
1874:(
1862:0
1858:x
1854:=
1851:)
1848:t
1845:(
1842:x
1815:0
1811:v
1805:=
1800:2
1796:c
1773:2
1769:c
1746:2
1742:c
1735:=
1732:)
1729:0
1726:(
1717:x
1692:0
1688:x
1684:=
1679:1
1675:c
1652:1
1648:c
1625:1
1621:c
1617:=
1614:)
1611:0
1608:(
1605:x
1585:0
1582:=
1579:t
1557:2
1553:c
1530:1
1526:c
1505:.
1499:m
1494:/
1489:k
1483:=
1460:,
1456:)
1452:t
1445:(
1433:2
1429:c
1425:+
1421:)
1417:t
1410:(
1398:1
1394:c
1390:=
1387:)
1384:t
1381:(
1378:x
1348:,
1345:x
1340:m
1337:k
1329:=
1321:2
1317:t
1312:d
1306:x
1301:2
1296:d
1273:k
1260:x
1252:m
1238:,
1235:x
1232:k
1226:=
1218:2
1214:t
1209:d
1203:x
1198:2
1193:d
1185:m
1182:=
1176:t
1173:e
1170:n
1165:F
1083:x
1057:x
1052:N
1050:(
1043:k
1038:N
1029:F
1014:,
1010:x
1006:k
1000:=
996:F
984:F
813:e
806:t
799:v
489:)
108:t
105:d
99:p
95:d
89:=
84:F
20:)
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