756:
3644:
769:
3937:
22:
3527:
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.
3238:
3932:
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.
1955:
2737:
3046:
3370:
2554:
1826:
1237:
1459:
2231:
1347:
1068:
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
4095:
1080:
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
4156:
2912:
2023:
110:
2151:
2094:
2626:
2986:
2606:
1821:
3738:
3629:
3525:
3437:
1013:
3643:
3447:
2361:
3842:
2847:
1747:
3918:, 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).
3247:
1504:
947:, 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
3910:
2464:
2807:
2299:
134:
1148:
1693:
905:
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
1626:
3631:
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.
1275:
1774:
1653:
1558:
1531:
1584:
3780:
3760:
3697:
3233:{\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 ),}
1362:
3762:, 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
2157:
3782:
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.
1950:{\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).}
4029:
2864:
1962:
1060:
When the system is displaced from its equilibrium position, a restoring force that obeys Hooke's law tends to restore the system to equilibrium.
3742:
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,
800:
4100:
389:
1107:
plot are not co-linear, the phase space motion becomes elliptical. The area enclosed depends on the amplitude and the maximum momentum.
508:
2947:
3702:
3673:, the motion of a simple pendulum is approximated by simple harmonic motion. The period of a mass attached to a pendulum of length
3593:
3489:
481:
3379:
979:
3797:
2812:
916:
Simple harmonic motion provides a basis for the characterization of more complicated periodic motion through the techniques of
66:
4330:
2412:), and the origin is set to be the equilibrium position. Each of these constants carries a physical meaning of the motion:
2100:
940:
on the line and whose magnitude is proportional to the displacement from the fixed point is called simple harmonic motion.
3874:
2761:
2031:
463:
3961:
846:
to the distance of the object from an equilibrium position and acts towards the equilibrium position. It results in an
4311:
4292:
4273:
793:
2562:
1779:
4252:
4361:
Geogebra applet for spring-mass, with 3 attached PDFs on SHM, driven/damped oscillators, spring-mass with friction
755:
2305:
129:
1698:
1064:
Once the mass is displaced from its equilibrium position, it experiences a net restoring force. As a result, it
1120:
384:
124:
3966:
1022:
843:
1464:
4375:
786:
773:
534:
457:
379:
222:
2732:{\displaystyle a(t)={\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-A\omega ^{2}\cos(\omega t-\varphi ).}
453:
254:
232:
2242:
673:
562:
488:
348:
281:
4385:
3956:
3670:
1119:, for one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear
937:
906:
427:
3537:
2384:
are two constants determined by the initial conditions (specifically, the initial position at time
703:
547:
3981:
3976:
3541:
2994:(the period and frequency are independent of the amplitude and the initial phase of the motion).
1658:
1266:
653:
417:
3365:{\displaystyle U(t)={\tfrac {1}{2}}kx^{2}(t)={\tfrac {1}{2}}kA^{2}\cos ^{2}(\omega t-\varphi ).}
1589:
955:
acting on the mass. However, if the mass is displaced from the equilibrium position, the spring
693:
3996:
3451:
1256:
948:
897:
frequency. Other phenomena can be modeled by simple harmonic motion, including the motion of a
698:
4025:
The choice of using a cosine in this equation is a convention. Other valid formulations are:
2549:{\displaystyle v(t)={\frac {\mathrm {d} x}{\mathrm {d} t}}=-A\omega \sin(\omega t-\varphi ),}
1352:
1252:
1050:
708:
683:
369:
187:
3786:
3557:
1752:
1631:
1536:
1509:
960:
882:
728:
688:
596:
592:
584:
574:
364:
357:
113:
723:
8:
3991:
3951:
3586:
The motion of a body in which it moves to and from about a definite point is also called
3459:
1563:
1128:
1116:
944:
910:
503:
444:
422:
167:
162:
157:
57:
4380:
3765:
3745:
3682:
3587:
3566:-plane, then its motion along each coordinate is simple harmonic motion with amplitude
1232:{\displaystyle F_{\mathrm {net} }=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-kx,}
870:
633:
374:
249:
217:
177:
4346:
4326:
4307:
4288:
4269:
3851:
3373:
2433:
1454:{\displaystyle x(t)=c_{1}\cos \left(\omega t\right)+c_{2}\sin \left(\omega t\right),}
1097:
832:
643:
600:
557:
552:
493:
269:
259:
152:
42:
34:
4006:
3785:
This approximation is accurate only for small angles because of the expression for
3241:
2226:{\displaystyle \sin \varphi ={\frac {c_{2}}{A}},\;\cos \varphi ={\frac {c_{1}}{A}}}
917:
738:
718:
663:
658:
604:
579:
434:
292:
237:
212:
30:
2991:
1085:
reaches zero, whereupon it is accelerated back to the equilibrium position again.
1140:
878:
839:
733:
678:
628:
623:
542:
1342:{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-{\frac {k}{m}}x,}
1092:
loss, the mass continues to oscillate. Thus simple harmonic motion is a type of
3029:
1132:
1036:
964:
886:
760:
668:
569:
286:
45:
axes have been reversed from the standard convention to align the two diagrams)
21:
4369:
4355:
3986:
2443:
1040:
1026:
648:
475:
1695:; taking the derivative of that equation and evaluating at zero we get that
4350:
4001:
3653:
3483:
2458:
1065:
933:
713:
638:
327:
207:
3971:
3927:
1244:
1104:
859:
847:
26:
3936:
2758:
is under SHM its acceleration is directly proportional to displacement.
3912:
which makes angular acceleration directly proportional and opposite to
3446:
1776:
is the initial speed of the particle divided by the angular frequency,
890:
498:
172:
4360:
2419:
1356:
1093:
902:
894:
814:
520:
3649:
2454:
2450:
1082:
1077:
956:
929:
898:
855:
851:
439:
322:
297:
38:
3590:
or vibratory motion. The time period is able to be calculated by
873:
for a variety of motions, but is typified by the oscillation of a
818:
412:
265:
182:
1096:
motion. If energy is lost in the system, then the mass exhibits
2990:
These equations demonstrate that the simple harmonic motion is
1089:
863:
835:
471:
317:
227:
3660:
3536:
Simple harmonic motion can be considered the one-dimensional
952:
307:
302:
244:
3372:
In the absence of friction and other energy loss, the total
4090:{\displaystyle x(t)=A\sin \left(\omega t+\varphi '\right),}
2907:{\displaystyle f={\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}},}
1136:
874:
312:
275:
3794:
being proportional to the sine of the displacement angle:
2018:{\displaystyle x(t)=A\cos \left(\omega t-\varphi \right),}
1021:
is the restoring elastic force exerted by the spring (in
105:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
4151:{\displaystyle \tan \varphi '={\frac {c_{1}}{c_{2}}},}
3402:
3304:
3267:
3172:
3103:
3066:
2422:(maximum displacement from the equilibrium position),
1467:
4103:
4032:
3877:
3800:
3768:
3748:
3705:
3685:
3596:
3492:
3382:
3250:
3049:
2950:
2867:
2815:
2764:
2629:
2565:
2467:
2308:
2245:
2160:
2146:{\displaystyle \tan \varphi ={\frac {c_{2}}{c_{1}}},}
2103:
2034:
1965:
1829:
1782:
1755:
1701:
1661:
1634:
1592:
1566:
1539:
1512:
1365:
1278:
1151:
982:
69:
16:
To-and-fro periodic motion in science and engineering
3458:
The following physical systems are some examples of
909:). Simple harmonic motion can also be used to model
25:
Simple harmonic motion shown both in real space and
4196:The maximum displacement (that is, the amplitude),
2089:{\displaystyle A={\sqrt {{c_{1}}^{2}+{c_{2}}^{2}}}}
4150:
4089:
3904:
3836:
3774:
3754:
3732:
3691:
3623:
3519:
3431:
3364:
3232:
2980:
2906:
2841:
2801:
2731:
2600:
2548:
2355:
2293:
2225:
2145:
2088:
2017:
1949:
1815:
1768:
1741:
1687:
1647:
1620:
1578:
1552:
1525:
1498:
1453:
1341:
1231:
1007:
104:
4367:
854:which continues indefinitely (if uninhibited by
4302:Thornton, Stephen T.; Marion, Jerry B. (2003).
1959:This equation can also be written in the form:
1056:For any simple mechanical harmonic oscillator:
901:, although for it to be an accurate model, the
4301:
4264:Fowles, Grant R.; Cassiday, George L. (2005).
4263:
2981:{\displaystyle T=2\pi {\sqrt {\frac {m}{k}}}.}
2601:{\displaystyle {\omega }{\sqrt {A^{2}-x^{2}}}}
1816:{\displaystyle c_{2}={\frac {v_{0}}{\omega }}}
827:
3733:{\displaystyle T=2\pi {\sqrt {\frac {l}{g}}}}
3624:{\displaystyle T=2\pi {\sqrt {\frac {l}{g}}}}
3520:{\displaystyle T=2\pi {\sqrt {\frac {m}{k}}}}
794:
4325:(9th ed.). Hoboken, New Jersey: Wiley.
3634:
3432:{\displaystyle E=K+U={\tfrac {1}{2}}kA^{2}.}
1008:{\displaystyle \mathbf {F} =-k\mathbf {x} ,}
4304:Classical Dynamics of Particles and Systems
2356:{\displaystyle \varphi =\arg(c_{1}+c_{2}i)}
3837:{\displaystyle -mgl\sin \theta =I\alpha ,}
3642:
3531:
2842:{\displaystyle \omega ^{2}={\frac {k}{m}}}
2193:
1742:{\displaystyle {\dot {x}}(0)=\omega c_{2}}
1127:coefficients, can be obtained by means of
801:
787:
4356:Java simulation of spring-mass oscillator
3486:. The equation for describing the period
1655:is the initial position of the particle,
3935:
3544:. If an object moves with angular speed
3476:attached to a spring of spring constant
3445:
20:
1499:{\textstyle \omega ={\sqrt {{k}/{m}}}.}
4368:
4320:
4282:
869:Simple harmonic motion can serve as a
4268:(7th ed.). Thomson Brooks/Cole.
3871:and therefore the expression becomes
3581:
932:moving along a straight line with an
483:Newton's law of universal gravitation
3905:{\displaystyle -mgl\theta =I\alpha }
2461:as a function of time can be found:
1355:above produces a solution that is a
970:Mathematically, the restoring force
936:whose direction is always towards a
838:an object experiences by means of a
4253:"Simple Harmonic Motion – Concepts"
3482:exhibits simple harmonic motion in
3465:
2802:{\displaystyle a(x)=-\omega ^{2}x.}
1053:from the equilibrium position (m).
464:Mechanics of planar particle motion
72:
13:
2666:
2650:
2497:
2487:
1586:on the equation above we see that
1300:
1284:
1197:
1181:
1164:
1161:
1158:
893:in time and demonstrates a single
14:
4397:
4340:
3454:undergoes simple harmonic motion.
2294:{\displaystyle A=|c_{1}+c_{2}i|,}
881:when it is subject to the linear
3679:with gravitational acceleration
2400:, while the initial velocity is
998:
984:
768:
767:
754:
87:
923:
4245:
4042:
4036:
3921:
3356:
3341:
3297:
3291:
3260:
3254:
3224:
3209:
3165:
3150:
3096:
3090:
3059:
3053:
2774:
2768:
2723:
2708:
2639:
2633:
2540:
2525:
2477:
2471:
2350:
2321:
2284:
2253:
1975:
1969:
1839:
1833:
1720:
1714:
1602:
1596:
1375:
1369:
1121:ordinary differential equation
951:position then there is no net
1:
4306:(5th ed.). Brooks Cole.
4239:
1560:can be easily found: setting
1506:The meaning of the constants
1088:As long as the system has no
390:Koopman–von Neumann mechanics
4287:. University Science Books.
3656:in 60 s, a frequency of 0.41
842:whose magnitude is directly
458:Non-inertial reference frame
7:
3944:
3441:
1688:{\displaystyle c_{1}=x_{0}}
1110:
1103:Note if the real space and
385:Appell's equation of motion
255:Inertial frame of reference
10:
4402:
4190:
4019:
3925:
3550:around a circle of radius
3460:simple harmonic oscillator
1621:{\displaystyle x(0)=c_{1}}
945:simple harmonic oscillator
3967:Rayleigh–Lorentz pendulum
3957:Small-angle approximation
3671:small-angle approximation
3641:
3635:Mass of a simple pendulum
2997:
2752:By definition, if a mass
1269:for a mass on a spring).
1247:of the oscillating body,
907:small-angle approximation
885:restoring force given by
4283:Taylor, John R. (2005).
4013:
3962:Lorentz oscillator model
2449:Using the techniques of
1259:(or mean) position, and
548:Rotating reference frame
380:Hamilton–Jacobi equation
3982:Complex harmonic motion
3977:Uniform circular motion
3542:uniform circular motion
3532:Uniform circular motion
850:that is described by a
831:) is a special type of
825:(sometimes abbreviated
489:Newton's laws of motion
349:Newton's laws of motion
4347:Simple Harmonic Motion
4321:Walker, Jearl (2011).
4152:
4091:
3997:Pendulum (mathematics)
3941:
3906:
3838:
3776:
3756:
3734:
3693:
3625:
3572:and angular frequency
3521:
3455:
3433:
3366:
3234:
3037:of the system at time
2982:
2908:
2843:
2803:
2742:Maximum acceleration:
2733:
2620:(at equilibrium point)
2602:
2550:
2357:
2295:
2227:
2147:
2090:
2019:
1951:
1817:
1770:
1743:
1689:
1649:
1622:
1580:
1554:
1527:
1500:
1455:
1343:
1233:
1009:
823:simple harmonic motion
516:Simple harmonic motion
429:Euler's laws of motion
223:D'Alembert's principle
106:
46:
4323:Principles of Physics
4153:
4092:
3940:Scotch yoke animation
3939:
3907:
3839:
3777:
3757:
3735:
3694:
3626:
3522:
3449:
3434:
3376:has a constant value
3367:
3235:
2983:
2909:
2844:
2804:
2734:
2603:
2551:
2358:
2296:
2228:
2148:
2091:
2020:
1952:
1823:. Thus we can write:
1818:
1771:
1769:{\displaystyle c_{2}}
1744:
1690:
1650:
1648:{\displaystyle c_{1}}
1623:
1581:
1555:
1553:{\displaystyle c_{2}}
1528:
1526:{\displaystyle c_{1}}
1501:
1456:
1353:differential equation
1344:
1234:
1010:
370:Hamiltonian mechanics
188:Statistical mechanics
107:
24:
4266:Analytical Mechanics
4101:
4030:
3875:
3798:
3787:angular acceleration
3766:
3746:
3703:
3683:
3594:
3490:
3380:
3248:
3047:
2948:
2944:is the time period,
2865:
2813:
2762:
2627:
2563:
2465:
2306:
2243:
2158:
2101:
2032:
1963:
1827:
1780:
1753:
1699:
1659:
1632:
1590:
1564:
1537:
1510:
1465:
1363:
1276:
1149:
980:
593:Angular acceleration
585:Rotational frequency
365:Lagrangian mechanics
358:Analytical mechanics
114:Second law of motion
67:
4376:Classical mechanics
4285:Classical Mechanics
3992:Harmonic oscillator
3952:Newtonian mechanics
3652:making 25 complete
2748:(at extreme points)
1579:{\displaystyle t=0}
1357:sinusoidal function
1265:is a constant (the
1117:Newtonian mechanics
911:molecular vibration
445:Harmonic oscillator
423:Equations of motion
58:Classical mechanics
52:Part of a series on
4148:
4087:
3942:
3902:
3834:
3772:
3752:
3730:
3689:
3621:
3588:oscillatory motion
3582:Oscillatory motion
3517:
3456:
3452:spring–mass system
3429:
3411:
3362:
3313:
3276:
3230:
3181:
3112:
3075:
2978:
2904:
2839:
2799:
2729:
2598:
2546:
2353:
2291:
2223:
2143:
2086:
2015:
1947:
1813:
1766:
1739:
1685:
1645:
1618:
1576:
1550:
1523:
1496:
1451:
1339:
1229:
1098:damped oscillation
1005:
943:In the diagram, a
871:mathematical model
761:Physics portal
375:Routhian mechanics
250:Frame of reference
102:
47:
4332:978-0-470-56158-4
4143:
3852:moment of inertia
3775:{\displaystyle g}
3755:{\displaystyle g}
3728:
3727:
3692:{\displaystyle g}
3667:
3666:
3619:
3618:
3515:
3514:
3410:
3374:mechanical energy
3312:
3275:
3180:
3111:
3074:
2973:
2972:
2899:
2898:
2887:
2837:
2681:
2596:
2505:
2434:angular frequency
2366:In the solution,
2236:or equivalently
2221:
2188:
2138:
2084:
1934:
1933:
1911:
1910:
1909:
1876:
1875:
1811:
1711:
1491:
1331:
1315:
1212:
963:force that obeys
811:
810:
558:Centrifugal force
553:Centripetal force
509:Euler's equations
494:Relative velocity
270:Moment of inertia
100:
74:
4393:
4386:Motion (physics)
4336:
4317:
4298:
4279:
4257:
4256:
4249:
4229:
4217:, and thus when
4216:
4204:
4194:
4186:
4180:
4178:
4177:
4174:
4171:
4157:
4155:
4154:
4149:
4144:
4142:
4141:
4132:
4131:
4122:
4117:
4096:
4094:
4093:
4088:
4083:
4079:
4078:
4023:
4007:String vibration
3917:
3911:
3909:
3908:
3903:
3870:
3859:
3849:
3843:
3841:
3840:
3835:
3793:
3781:
3779:
3778:
3773:
3761:
3759:
3758:
3753:
3739:
3737:
3736:
3731:
3729:
3720:
3719:
3698:
3696:
3695:
3690:
3678:
3659:
3646:
3639:
3638:
3630:
3628:
3627:
3622:
3620:
3611:
3610:
3577:
3571:
3565:
3556:centered at the
3555:
3549:
3526:
3524:
3523:
3518:
3516:
3507:
3506:
3481:
3475:
3466:Mass on a spring
3438:
3436:
3435:
3430:
3425:
3424:
3412:
3403:
3371:
3369:
3368:
3363:
3337:
3336:
3327:
3326:
3314:
3305:
3290:
3289:
3277:
3268:
3242:potential energy
3239:
3237:
3236:
3231:
3205:
3204:
3195:
3194:
3182:
3173:
3146:
3145:
3136:
3135:
3126:
3125:
3113:
3104:
3089:
3088:
3076:
3067:
3042:
3036:
3027:
3025:
3023:
3022:
3019:
3016:
3007:
2987:
2985:
2984:
2979:
2974:
2965:
2964:
2943:
2937:
2936:
2934:
2933:
2928:
2925:
2913:
2911:
2910:
2905:
2900:
2891:
2890:
2888:
2886:
2875:
2860:
2848:
2846:
2845:
2840:
2838:
2830:
2825:
2824:
2808:
2806:
2805:
2800:
2792:
2791:
2757:
2747:
2738:
2736:
2735:
2730:
2701:
2700:
2682:
2680:
2679:
2678:
2669:
2663:
2659:
2658:
2653:
2646:
2619:
2607:
2605:
2604:
2599:
2597:
2595:
2594:
2582:
2581:
2572:
2570:
2555:
2553:
2552:
2547:
2506:
2504:
2500:
2494:
2490:
2484:
2441:
2431:
2417:
2411:
2399:
2390:
2383:
2374:
2362:
2360:
2359:
2354:
2346:
2345:
2333:
2332:
2300:
2298:
2297:
2292:
2287:
2279:
2278:
2266:
2265:
2256:
2232:
2230:
2229:
2224:
2222:
2217:
2216:
2207:
2189:
2184:
2183:
2174:
2152:
2150:
2149:
2144:
2139:
2137:
2136:
2127:
2126:
2117:
2095:
2093:
2092:
2087:
2085:
2083:
2082:
2077:
2076:
2075:
2061:
2060:
2055:
2054:
2053:
2042:
2024:
2022:
2021:
2016:
2011:
2007:
1956:
1954:
1953:
1948:
1943:
1939:
1935:
1926:
1925:
1912:
1902:
1901:
1900:
1899:
1890:
1885:
1881:
1877:
1868:
1867:
1854:
1853:
1822:
1820:
1819:
1814:
1812:
1807:
1806:
1797:
1792:
1791:
1775:
1773:
1772:
1767:
1765:
1764:
1748:
1746:
1745:
1740:
1738:
1737:
1713:
1712:
1704:
1694:
1692:
1691:
1686:
1684:
1683:
1671:
1670:
1654:
1652:
1651:
1646:
1644:
1643:
1627:
1625:
1624:
1619:
1617:
1616:
1585:
1583:
1582:
1577:
1559:
1557:
1556:
1551:
1549:
1548:
1532:
1530:
1529:
1524:
1522:
1521:
1505:
1503:
1502:
1497:
1492:
1490:
1485:
1480:
1475:
1460:
1458:
1457:
1452:
1447:
1443:
1425:
1424:
1412:
1408:
1390:
1389:
1348:
1346:
1345:
1340:
1332:
1324:
1316:
1314:
1313:
1312:
1303:
1297:
1293:
1292:
1287:
1280:
1264:
1250:
1242:
1238:
1236:
1235:
1230:
1213:
1211:
1210:
1209:
1200:
1194:
1190:
1189:
1184:
1177:
1169:
1168:
1167:
1129:Newton's 2nd law
1075:
1048:
1034:
1020:
1014:
1012:
1011:
1006:
1001:
987:
975:
928:The motion of a
918:Fourier analysis
889:. The motion is
829:
803:
796:
789:
776:
771:
770:
763:
759:
758:
664:Johann Bernoulli
659:Daniel Bernoulli
580:Tangential speed
484:
460:
435:Fictitious force
430:
282:Mechanical power
272:
213:Angular momentum
111:
109:
108:
103:
101:
99:
91:
90:
81:
76:
75:
49:
48:
4401:
4400:
4396:
4395:
4394:
4392:
4391:
4390:
4366:
4365:
4343:
4333:
4314:
4295:
4276:
4260:
4251:
4250:
4246:
4242:
4236:
4234:
4231:
4224:
4218:
4206:
4203:
4197:
4191:
4188:
4175:
4172:
4169:
4168:
4166:
4160:
4137:
4133:
4127:
4123:
4121:
4110:
4102:
4099:
4098:
4071:
4061:
4057:
4031:
4028:
4027:
4020:
4016:
4011:
3947:
3930:
3924:
3913:
3876:
3873:
3872:
3861:
3855:
3845:
3799:
3796:
3795:
3789:
3767:
3764:
3763:
3747:
3744:
3743:
3718:
3704:
3701:
3700:
3684:
3681:
3680:
3674:
3663:
3657:
3637:
3609:
3595:
3592:
3591:
3584:
3573:
3567:
3561:
3551:
3545:
3534:
3505:
3491:
3488:
3487:
3477:
3471:
3468:
3444:
3420:
3416:
3401:
3381:
3378:
3377:
3332:
3328:
3322:
3318:
3303:
3285:
3281:
3266:
3249:
3246:
3245:
3200:
3196:
3190:
3186:
3171:
3141:
3137:
3131:
3127:
3121:
3117:
3102:
3084:
3080:
3065:
3048:
3045:
3044:
3038:
3032:
3020:
3017:
3014:
3013:
3011:
3009:
3003:
3000:
2963:
2949:
2946:
2945:
2939:
2929:
2926:
2923:
2922:
2920:
2915:
2889:
2879:
2874:
2866:
2863:
2862:
2852:
2829:
2820:
2816:
2814:
2811:
2810:
2787:
2783:
2763:
2760:
2759:
2753:
2743:
2696:
2692:
2674:
2670:
2665:
2664:
2654:
2649:
2648:
2647:
2645:
2628:
2625:
2624:
2611:
2610:Maximum speed:
2590:
2586:
2577:
2573:
2571:
2566:
2564:
2561:
2560:
2496:
2495:
2486:
2485:
2483:
2466:
2463:
2462:
2442:is the initial
2437:
2423:
2413:
2407:
2401:
2398:
2392:
2385:
2382:
2376:
2373:
2367:
2341:
2337:
2328:
2324:
2307:
2304:
2303:
2283:
2274:
2270:
2261:
2257:
2252:
2244:
2241:
2240:
2212:
2208:
2206:
2179:
2175:
2173:
2159:
2156:
2155:
2132:
2128:
2122:
2118:
2116:
2102:
2099:
2098:
2078:
2071:
2067:
2066:
2065:
2056:
2049:
2045:
2044:
2043:
2041:
2033:
2030:
2029:
1994:
1990:
1964:
1961:
1960:
1924:
1923:
1919:
1895:
1891:
1889:
1866:
1865:
1861:
1849:
1845:
1828:
1825:
1824:
1802:
1798:
1796:
1787:
1783:
1781:
1778:
1777:
1760:
1756:
1754:
1751:
1750:
1733:
1729:
1703:
1702:
1700:
1697:
1696:
1679:
1675:
1666:
1662:
1660:
1657:
1656:
1639:
1635:
1633:
1630:
1629:
1612:
1608:
1591:
1588:
1587:
1565:
1562:
1561:
1544:
1540:
1538:
1535:
1534:
1517:
1513:
1511:
1508:
1507:
1486:
1481:
1476:
1474:
1466:
1463:
1462:
1436:
1432:
1420:
1416:
1401:
1397:
1385:
1381:
1364:
1361:
1360:
1323:
1308:
1304:
1299:
1298:
1288:
1283:
1282:
1281:
1279:
1277:
1274:
1273:
1267:spring constant
1260:
1248:
1240:
1205:
1201:
1196:
1195:
1185:
1180:
1179:
1178:
1176:
1157:
1156:
1152:
1150:
1147:
1146:
1113:
1076:, the mass has
1070:
1044:
1037:spring constant
1030:
1016:
997:
983:
981:
978:
977:
971:
926:
899:simple pendulum
840:restoring force
807:
766:
753:
752:
745:
744:
743:
618:
610:
609:
589:
543:Circular motion
537:
527:
526:
525:
482:
452:
449:
428:
407:
399:
398:
395:
394:
352:
342:
334:
333:
332:
291:
287:Mechanical work
280:
264:
202:
194:
193:
192:
147:
139:
116:
92:
86:
82:
80:
71:
70:
68:
65:
64:
17:
12:
11:
5:
4399:
4389:
4388:
4383:
4378:
4364:
4363:
4358:
4353:
4342:
4341:External links
4339:
4338:
4337:
4331:
4318:
4312:
4299:
4293:
4280:
4274:
4259:
4258:
4243:
4241:
4238:
4233:
4232:
4222:
4205:, occurs when
4201:
4195:
4189:
4147:
4140:
4136:
4130:
4126:
4120:
4116:
4113:
4109:
4106:
4086:
4082:
4077:
4074:
4070:
4067:
4064:
4060:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4024:
4017:
4015:
4012:
4010:
4009:
4004:
3999:
3994:
3989:
3984:
3979:
3974:
3969:
3964:
3959:
3954:
3948:
3946:
3943:
3926:Main article:
3923:
3920:
3901:
3898:
3895:
3892:
3889:
3886:
3883:
3880:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3771:
3751:
3726:
3723:
3717:
3714:
3711:
3708:
3688:
3665:
3664:
3647:
3636:
3633:
3617:
3614:
3608:
3605:
3602:
3599:
3583:
3580:
3533:
3530:
3513:
3510:
3504:
3501:
3498:
3495:
3467:
3464:
3443:
3440:
3428:
3423:
3419:
3415:
3409:
3406:
3400:
3397:
3394:
3391:
3388:
3385:
3361:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3335:
3331:
3325:
3321:
3317:
3311:
3308:
3302:
3299:
3296:
3293:
3288:
3284:
3280:
3274:
3271:
3265:
3262:
3259:
3256:
3253:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3203:
3199:
3193:
3189:
3185:
3179:
3176:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3144:
3140:
3134:
3130:
3124:
3120:
3116:
3110:
3107:
3101:
3098:
3095:
3092:
3087:
3083:
3079:
3073:
3070:
3064:
3061:
3058:
3055:
3052:
3030:kinetic energy
2999:
2996:
2977:
2971:
2968:
2962:
2959:
2956:
2953:
2903:
2897:
2894:
2885:
2882:
2878:
2873:
2870:
2836:
2833:
2828:
2823:
2819:
2798:
2795:
2790:
2786:
2782:
2779:
2776:
2773:
2770:
2767:
2750:
2749:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2699:
2695:
2691:
2688:
2685:
2677:
2673:
2668:
2662:
2657:
2652:
2644:
2641:
2638:
2635:
2632:
2622:
2621:
2608:
2593:
2589:
2585:
2580:
2576:
2569:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2503:
2499:
2493:
2489:
2482:
2479:
2476:
2473:
2470:
2405:
2396:
2380:
2371:
2364:
2363:
2352:
2349:
2344:
2340:
2336:
2331:
2327:
2323:
2320:
2317:
2314:
2311:
2301:
2290:
2286:
2282:
2277:
2273:
2269:
2264:
2260:
2255:
2251:
2248:
2234:
2233:
2220:
2215:
2211:
2205:
2202:
2199:
2196:
2192:
2187:
2182:
2178:
2172:
2169:
2166:
2163:
2153:
2142:
2135:
2131:
2125:
2121:
2115:
2112:
2109:
2106:
2096:
2081:
2074:
2070:
2064:
2059:
2052:
2048:
2040:
2037:
2014:
2010:
2006:
2003:
2000:
1997:
1993:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1946:
1942:
1938:
1932:
1929:
1922:
1918:
1915:
1908:
1905:
1898:
1894:
1888:
1884:
1880:
1874:
1871:
1864:
1860:
1857:
1852:
1848:
1844:
1841:
1838:
1835:
1832:
1810:
1805:
1801:
1795:
1790:
1786:
1763:
1759:
1736:
1732:
1728:
1725:
1722:
1719:
1716:
1710:
1707:
1682:
1678:
1674:
1669:
1665:
1642:
1638:
1615:
1611:
1607:
1604:
1601:
1598:
1595:
1575:
1572:
1569:
1547:
1543:
1520:
1516:
1495:
1489:
1484:
1479:
1473:
1470:
1450:
1446:
1442:
1439:
1435:
1431:
1428:
1423:
1419:
1415:
1411:
1407:
1404:
1400:
1396:
1393:
1388:
1384:
1380:
1377:
1374:
1371:
1368:
1338:
1335:
1330:
1327:
1322:
1319:
1311:
1307:
1302:
1296:
1291:
1286:
1228:
1225:
1222:
1219:
1216:
1208:
1204:
1199:
1193:
1188:
1183:
1175:
1172:
1166:
1163:
1160:
1155:
1112:
1109:
1062:
1061:
1004:
1000:
996:
993:
990:
986:
925:
922:
809:
808:
806:
805:
798:
791:
783:
780:
779:
778:
777:
764:
747:
746:
742:
741:
736:
731:
726:
721:
716:
711:
706:
701:
696:
691:
686:
681:
676:
671:
666:
661:
656:
651:
646:
641:
636:
631:
626:
620:
619:
616:
615:
612:
611:
608:
607:
588:
587:
582:
577:
572:
570:Coriolis force
567:
566:
565:
555:
550:
545:
539:
538:
533:
532:
529:
528:
524:
523:
518:
513:
512:
511:
506:
496:
491:
486:
479:
468:
467:
466:
461:
448:
447:
442:
437:
432:
425:
420:
415:
409:
408:
405:
404:
401:
400:
397:
396:
393:
392:
387:
382:
377:
372:
367:
361:
355:
353:
346:
343:
340:
339:
336:
335:
331:
330:
325:
320:
315:
310:
305:
300:
295:
289:
284:
278:
273:
262:
257:
252:
247:
242:
241:
240:
235:
225:
220:
215:
210:
204:
203:
200:
199:
196:
195:
191:
190:
185:
180:
175:
170:
165:
160:
155:
149:
148:
145:
144:
141:
140:
138:
137:
132:
127:
121:
118:
117:
112:
98:
95:
89:
85:
79:
61:
60:
54:
53:
15:
9:
6:
4:
3:
2:
4398:
4387:
4384:
4382:
4379:
4377:
4374:
4373:
4371:
4362:
4359:
4357:
4354:
4352:
4348:
4345:
4344:
4334:
4328:
4324:
4319:
4315:
4313:0-534-40896-6
4309:
4305:
4300:
4296:
4294:1-891389-22-X
4290:
4286:
4281:
4277:
4275:0-534-49492-7
4271:
4267:
4262:
4261:
4254:
4248:
4244:
4237:
4228:
4221:
4214:
4210:
4200:
4193:
4184:
4164:
4158:
4145:
4138:
4134:
4128:
4124:
4118:
4114:
4111:
4107:
4104:
4084:
4080:
4075:
4072:
4068:
4065:
4062:
4058:
4054:
4051:
4048:
4045:
4039:
4033:
4022:
4018:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3988:
3987:Damping ratio
3985:
3983:
3980:
3978:
3975:
3973:
3970:
3968:
3965:
3963:
3960:
3958:
3955:
3953:
3950:
3949:
3938:
3934:
3929:
3919:
3916:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3869:
3865:
3858:
3853:
3848:
3831:
3828:
3825:
3822:
3819:
3816:
3813:
3810:
3807:
3804:
3801:
3792:
3788:
3783:
3769:
3749:
3740:
3724:
3721:
3715:
3712:
3709:
3706:
3686:
3677:
3672:
3662:
3655:
3651:
3645:
3640:
3632:
3615:
3612:
3606:
3603:
3600:
3597:
3589:
3579:
3576:
3570:
3564:
3559:
3554:
3548:
3543:
3539:
3529:
3511:
3508:
3502:
3499:
3496:
3493:
3485:
3480:
3474:
3463:
3461:
3453:
3448:
3439:
3426:
3421:
3417:
3413:
3407:
3404:
3398:
3395:
3392:
3389:
3386:
3383:
3375:
3359:
3353:
3350:
3347:
3344:
3338:
3333:
3329:
3323:
3319:
3315:
3309:
3306:
3300:
3294:
3286:
3282:
3278:
3272:
3269:
3263:
3257:
3251:
3243:
3227:
3221:
3218:
3215:
3212:
3206:
3201:
3197:
3191:
3187:
3183:
3177:
3174:
3168:
3162:
3159:
3156:
3153:
3147:
3142:
3138:
3132:
3128:
3122:
3118:
3114:
3108:
3105:
3099:
3093:
3085:
3081:
3077:
3071:
3068:
3062:
3056:
3050:
3041:
3035:
3031:
3026:
3006:
3002:Substituting
2995:
2993:
2988:
2975:
2969:
2966:
2960:
2957:
2954:
2951:
2942:
2932:
2918:
2901:
2895:
2892:
2883:
2880:
2876:
2871:
2868:
2859:
2855:
2849:
2834:
2831:
2826:
2821:
2817:
2796:
2793:
2788:
2784:
2780:
2777:
2771:
2765:
2756:
2746:
2741:
2740:
2739:
2726:
2720:
2717:
2714:
2711:
2705:
2702:
2697:
2693:
2689:
2686:
2683:
2675:
2671:
2660:
2655:
2642:
2636:
2630:
2618:
2614:
2609:
2591:
2587:
2583:
2578:
2574:
2567:
2558:
2557:
2556:
2543:
2537:
2534:
2531:
2528:
2522:
2519:
2516:
2513:
2510:
2507:
2501:
2491:
2480:
2474:
2468:
2460:
2456:
2452:
2447:
2445:
2440:
2435:
2430:
2426:
2421:
2416:
2410:
2404:
2395:
2388:
2379:
2370:
2347:
2342:
2338:
2334:
2329:
2325:
2318:
2315:
2312:
2309:
2302:
2288:
2280:
2275:
2271:
2267:
2262:
2258:
2249:
2246:
2239:
2238:
2237:
2218:
2213:
2209:
2203:
2200:
2197:
2194:
2190:
2185:
2180:
2176:
2170:
2167:
2164:
2161:
2154:
2140:
2133:
2129:
2123:
2119:
2113:
2110:
2107:
2104:
2097:
2079:
2072:
2068:
2062:
2057:
2050:
2046:
2038:
2035:
2028:
2027:
2026:
2012:
2008:
2004:
2001:
1998:
1995:
1991:
1987:
1984:
1981:
1978:
1972:
1966:
1957:
1944:
1940:
1936:
1930:
1927:
1920:
1916:
1913:
1906:
1903:
1896:
1892:
1886:
1882:
1878:
1872:
1869:
1862:
1858:
1855:
1850:
1846:
1842:
1836:
1830:
1808:
1803:
1799:
1793:
1788:
1784:
1761:
1757:
1734:
1730:
1726:
1723:
1717:
1708:
1705:
1680:
1676:
1672:
1667:
1663:
1640:
1636:
1613:
1609:
1605:
1599:
1593:
1573:
1570:
1567:
1545:
1541:
1518:
1514:
1493:
1487:
1482:
1477:
1471:
1468:
1448:
1444:
1440:
1437:
1433:
1429:
1426:
1421:
1417:
1413:
1409:
1405:
1402:
1398:
1394:
1391:
1386:
1382:
1378:
1372:
1366:
1358:
1354:
1349:
1336:
1333:
1328:
1325:
1320:
1317:
1309:
1305:
1294:
1289:
1270:
1268:
1263:
1258:
1254:
1246:
1245:inertial mass
1226:
1223:
1220:
1217:
1214:
1206:
1202:
1191:
1186:
1173:
1170:
1153:
1144:
1142:
1138:
1134:
1130:
1126:
1122:
1118:
1108:
1106:
1101:
1099:
1095:
1091:
1086:
1084:
1079:
1073:
1067:
1059:
1058:
1057:
1054:
1052:
1047:
1042:
1038:
1033:
1028:
1024:
1019:
1002:
994:
991:
988:
974:
968:
966:
962:
958:
954:
950:
946:
941:
939:
935:
931:
921:
919:
914:
912:
908:
904:
900:
896:
892:
888:
884:
880:
876:
872:
867:
865:
861:
858:or any other
857:
853:
849:
845:
841:
837:
834:
830:
824:
820:
816:
804:
799:
797:
792:
790:
785:
784:
782:
781:
775:
765:
762:
757:
751:
750:
749:
748:
740:
737:
735:
732:
730:
727:
725:
722:
720:
717:
715:
712:
710:
707:
705:
702:
700:
697:
695:
692:
690:
687:
685:
682:
680:
677:
675:
672:
670:
667:
665:
662:
660:
657:
655:
652:
650:
647:
645:
642:
640:
637:
635:
632:
630:
627:
625:
622:
621:
614:
613:
606:
602:
598:
594:
591:
590:
586:
583:
581:
578:
576:
573:
571:
568:
564:
561:
560:
559:
556:
554:
551:
549:
546:
544:
541:
540:
536:
531:
530:
522:
519:
517:
514:
510:
507:
505:
502:
501:
500:
497:
495:
492:
490:
487:
485:
480:
477:
473:
470:
469:
465:
462:
459:
455:
451:
450:
446:
443:
441:
438:
436:
433:
431:
426:
424:
421:
419:
416:
414:
411:
410:
403:
402:
391:
388:
386:
383:
381:
378:
376:
373:
371:
368:
366:
363:
362:
360:
359:
354:
351:
350:
345:
344:
338:
337:
329:
326:
324:
321:
319:
316:
314:
311:
309:
306:
304:
301:
299:
296:
294:
290:
288:
285:
283:
279:
277:
274:
271:
267:
263:
261:
258:
256:
253:
251:
248:
246:
243:
239:
236:
234:
231:
230:
229:
226:
224:
221:
219:
216:
214:
211:
209:
206:
205:
198:
197:
189:
186:
184:
181:
179:
176:
174:
171:
169:
166:
164:
161:
159:
156:
154:
151:
150:
143:
142:
136:
133:
131:
128:
126:
123:
122:
120:
119:
115:
96:
93:
83:
77:
63:
62:
59:
56:
55:
51:
50:
44:
40:
36:
32:
28:
23:
19:
4351:HyperPhysics
4322:
4303:
4284:
4265:
4247:
4235:
4226:
4219:
4212:
4208:
4198:
4192:
4182:
4162:
4026:
4021:
4002:Circle group
3931:
3914:
3867:
3863:
3856:
3846:
3790:
3784:
3741:
3699:is given by
3675:
3668:
3654:oscillations
3585:
3574:
3568:
3562:
3552:
3546:
3535:
3484:closed space
3478:
3472:
3469:
3457:
3450:An undamped
3039:
3033:
3010:
3004:
3001:
2989:
2940:
2930:
2916:
2857:
2853:
2850:
2754:
2751:
2744:
2623:
2616:
2612:
2459:acceleration
2448:
2438:
2428:
2424:
2414:
2408:
2402:
2393:
2386:
2377:
2368:
2365:
2235:
1958:
1351:Solving the
1350:
1271:
1261:
1253:displacement
1145:
1124:
1114:
1102:
1087:
1071:
1063:
1055:
1051:displacement
1045:
1031:
1017:
976:is given by
972:
969:
959:a restoring
942:
934:acceleration
927:
924:Introduction
915:
868:
844:proportional
826:
822:
812:
603: /
599: /
597:displacement
595: /
515:
456: /
418:Displacement
356:
347:
341:Formulations
328:Virtual work
268: /
208:Acceleration
201:Fundamentals
37:. (Here the
18:
3972:Isochronous
3928:Scotch yoke
3922:Scotch yoke
2992:isochronous
2914:and, since
1272:Therefore,
1257:equilibrium
1133:Hooke's law
1105:phase space
1066:accelerates
965:Hooke's law
949:equilibrium
938:fixed point
887:Hooke's law
860:dissipation
848:oscillation
739:von Neumann
406:Core topics
27:phase space
4370:Categories
4240:References
3860:is small,
3538:projection
1749:, so that
1628:, so that
891:sinusoidal
674:d'Alembert
654:Maupertuis
617:Scientists
499:Rigid body
173:Kinematics
4381:Pendulums
4112:φ
4108:
4073:φ
4063:ω
4055:
3900:α
3891:θ
3879:−
3829:α
3820:θ
3817:
3802:−
3716:π
3607:π
3503:π
3354:φ
3351:−
3345:ω
3339:
3222:φ
3219:−
3213:ω
3207:
3163:φ
3160:−
3154:ω
3148:
3119:ω
2961:π
2884:π
2818:ω
2785:ω
2781:−
2721:φ
2718:−
2712:ω
2706:
2694:ω
2687:−
2584:−
2568:ω
2538:φ
2535:−
2529:ω
2523:
2517:ω
2511:−
2420:amplitude
2319:
2310:φ
2201:φ
2198:
2168:φ
2165:
2111:φ
2108:
2005:φ
2002:−
1996:ω
1988:
1917:
1859:
1809:ω
1727:ω
1709:˙
1469:ω
1438:ω
1430:
1403:ω
1395:
1321:−
1255:from the
1218:−
1043:·m), and
992:−
903:net force
815:mechanics
719:Liouville
601:frequency
521:Vibration
238:potential
163:Continuum
158:Celestial
135:Textbooks
4115:′
4076:′
3945:See also
3650:pendulum
3442:Examples
3240:and the
2559:Speed:
2455:velocity
2451:calculus
1125:constant
1111:Dynamics
1094:periodic
1083:velocity
1078:momentum
930:particle
895:resonant
856:friction
852:sinusoid
833:periodic
774:Category
699:Hamilton
684:Lagrange
679:Clairaut
644:Horrocks
605:velocity
575:Pendulum
563:reactive
535:Rotation
504:dynamics
454:Inertial
440:Friction
323:Velocity
298:Momentum
178:Kinetics
168:Dynamics
146:Branches
130:Timeline
43:position
39:velocity
35:periodic
4179:
4167:
3854:. When
3850:is the
3669:In the
3560:of the
3470:A mass
3024:
3012:
2935:
2921:
2432:is the
2418:is the
1251:is its
1243:is the
1049:is the
1035:is the
1025:units:
961:elastic
883:elastic
819:physics
734:Koopman
694:Poisson
689:Laplace
634:Huygens
629:Galileo
474: (
413:Damping
266:Inertia
260:Impulse
233:kinetic
183:Statics
153:Applied
125:History
4329:
4310:
4291:
4272:
4165:= sin(
4159:since
4097:where
3844:where
3558:origin
3028:, the
2998:Energy
2938:where
2851:Since
2809:where
2453:, the
2436:, and
2025:where
1461:where
1239:where
1141:spring
1135:for a
1090:energy
1015:where
957:exerts
879:spring
864:energy
836:motion
772:
724:Appell
709:Cauchy
704:Jacobi
649:Halley
639:Newton
624:Kepler
476:linear
472:Motion
318:Torque
293:Moment
228:Energy
218:Couple
29:. The
4349:from
4215:) = 1
4014:Notes
3661:Hertz
3008:with
2444:phase
1139:on a
1123:with
953:force
877:on a
729:Gibbs
714:Routh
669:Euler
308:Speed
303:Space
245:Force
31:orbit
4327:ISBN
4308:ISBN
4289:ISBN
4270:ISBN
4207:cos(
4161:cos
3862:sin
3043:is
2457:and
2375:and
1533:and
1137:mass
1131:and
875:mass
817:and
313:Time
276:Mass
41:and
4223:max
4202:max
4105:tan
4052:sin
3814:sin
3540:of
3330:cos
3244:is
3198:sin
3139:sin
2856:= 2
2703:cos
2520:sin
2427:= 2
2391:is
2389:= 0
2316:arg
2195:cos
2162:sin
2105:tan
1985:cos
1914:sin
1856:cos
1427:sin
1392:cos
1115:In
1074:= 0
1029:),
866:).
862:of
828:SHM
813:In
33:is
4372::
4225:=
4211:±
4209:ωt
4181:−
3866:≈
3648:A
3578:.
3563:xy
3462:.
2919:=
2861:,
2858:πf
2745:Aω
2617:ωA
2615:=
2446:.
2429:πf
1359::
1143:.
1100:.
1023:SI
967:.
920:.
913:.
821:,
4335:.
4316:.
4297:.
4278:.
4255:.
4230:.
4227:A
4220:x
4213:φ
4199:x
4187:.
4185:)
4183:θ
4176:2
4173:/
4170:π
4163:θ
4146:,
4139:2
4135:c
4129:1
4125:c
4119:=
4085:,
4081:)
4069:+
4066:t
4059:(
4049:A
4046:=
4043:)
4040:t
4037:(
4034:x
3915:θ
3897:I
3894:=
3888:l
3885:g
3882:m
3868:θ
3864:θ
3857:θ
3847:I
3832:,
3826:I
3823:=
3811:l
3808:g
3805:m
3791:α
3770:g
3750:g
3725:g
3722:l
3713:2
3710:=
3707:T
3687:g
3676:l
3658:6
3616:g
3613:l
3604:2
3601:=
3598:T
3575:ω
3569:r
3553:r
3547:ω
3512:k
3509:m
3500:2
3497:=
3494:T
3479:k
3473:m
3427:.
3422:2
3418:A
3414:k
3408:2
3405:1
3399:=
3396:U
3393:+
3390:K
3387:=
3384:E
3360:.
3357:)
3348:t
3342:(
3334:2
3324:2
3320:A
3316:k
3310:2
3307:1
3301:=
3298:)
3295:t
3292:(
3287:2
3283:x
3279:k
3273:2
3270:1
3264:=
3261:)
3258:t
3255:(
3252:U
3228:,
3225:)
3216:t
3210:(
3202:2
3192:2
3188:A
3184:k
3178:2
3175:1
3169:=
3166:)
3157:t
3151:(
3143:2
3133:2
3129:A
3123:2
3115:m
3109:2
3106:1
3100:=
3097:)
3094:t
3091:(
3086:2
3082:v
3078:m
3072:2
3069:1
3063:=
3060:)
3057:t
3054:(
3051:K
3040:t
3034:K
3021:m
3018:/
3015:k
3005:ω
2976:.
2970:k
2967:m
2958:2
2955:=
2952:T
2941:T
2931:f
2927:/
2924:1
2917:T
2902:,
2896:m
2893:k
2881:2
2877:1
2872:=
2869:f
2854:ω
2835:m
2832:k
2827:=
2822:2
2797:.
2794:x
2789:2
2778:=
2775:)
2772:x
2769:(
2766:a
2755:m
2727:.
2724:)
2715:t
2709:(
2698:2
2690:A
2684:=
2676:2
2672:t
2667:d
2661:x
2656:2
2651:d
2643:=
2640:)
2637:t
2634:(
2631:a
2613:v
2592:2
2588:x
2579:2
2575:A
2544:,
2541:)
2532:t
2526:(
2514:A
2508:=
2502:t
2498:d
2492:x
2488:d
2481:=
2478:)
2475:t
2472:(
2469:v
2439:φ
2425:ω
2415:A
2409:ω
2406:2
2403:c
2397:1
2394:c
2387:t
2381:2
2378:c
2372:1
2369:c
2351:)
2348:i
2343:2
2339:c
2335:+
2330:1
2326:c
2322:(
2313:=
2289:,
2285:|
2281:i
2276:2
2272:c
2268:+
2263:1
2259:c
2254:|
2250:=
2247:A
2219:A
2214:1
2210:c
2204:=
2191:,
2186:A
2181:2
2177:c
2171:=
2141:,
2134:1
2130:c
2124:2
2120:c
2114:=
2080:2
2073:2
2069:c
2063:+
2058:2
2051:1
2047:c
2039:=
2036:A
2013:,
2009:)
1999:t
1992:(
1982:A
1979:=
1976:)
1973:t
1970:(
1967:x
1945:.
1941:)
1937:t
1931:m
1928:k
1921:(
1907:m
1904:k
1897:0
1893:v
1887:+
1883:)
1879:t
1873:m
1870:k
1863:(
1851:0
1847:x
1843:=
1840:)
1837:t
1834:(
1831:x
1804:0
1800:v
1794:=
1789:2
1785:c
1762:2
1758:c
1735:2
1731:c
1724:=
1721:)
1718:0
1715:(
1706:x
1681:0
1677:x
1673:=
1668:1
1664:c
1641:1
1637:c
1614:1
1610:c
1606:=
1603:)
1600:0
1597:(
1594:x
1574:0
1571:=
1568:t
1546:2
1542:c
1519:1
1515:c
1494:.
1488:m
1483:/
1478:k
1472:=
1449:,
1445:)
1441:t
1434:(
1422:2
1418:c
1414:+
1410:)
1406:t
1399:(
1387:1
1383:c
1379:=
1376:)
1373:t
1370:(
1367:x
1337:,
1334:x
1329:m
1326:k
1318:=
1310:2
1306:t
1301:d
1295:x
1290:2
1285:d
1262:k
1249:x
1241:m
1227:,
1224:x
1221:k
1215:=
1207:2
1203:t
1198:d
1192:x
1187:2
1182:d
1174:m
1171:=
1165:t
1162:e
1159:n
1154:F
1072:x
1046:x
1041:N
1039:(
1032:k
1027:N
1018:F
1003:,
999:x
995:k
989:=
985:F
973:F
802:e
795:t
788:v
478:)
97:t
94:d
88:p
84:d
78:=
73:F
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