2489:
1615:
1257:
1342:
3206:
For typical applications in nuclear physics, where one particle's mass is much larger than the other the reduced mass can be approximated as the smaller mass of the system. The limit of the reduced mass formula as one mass goes to infinity is the smaller mass, thus this approximation is used to ease
1930:
3085:
1042:
2304:
1024:
359:
1610:{\displaystyle \mathbf {a} _{\rm {rel}}=\mathbf {a} _{1}-\mathbf {a} _{2}={\frac {d^{2}\mathbf {x} _{1}}{dt^{2}}}-{\frac {d^{2}\mathbf {x} _{2}}{dt^{2}}}={\frac {d^{2}}{dt^{2}}}(\mathbf {x} _{1}-\mathbf {x} _{2})={\frac {d^{2}\mathbf {x} _{\rm {rel}}}{dt^{2}}}}
3357:
3554:
2831:
3362:
we find that the position of the first body with respect to the second is governed by the same differential equation as the position of a body with the reduced mass orbiting a body with a mass equal to the sum of the two masses, because
1768:
2387:
1252:{\displaystyle \mathbf {a} _{\rm {rel}}:=\mathbf {a} _{1}-\mathbf {a} _{2}=\left(1+{\frac {m_{1}}{m_{2}}}\right)\mathbf {a} _{1}={\frac {m_{2}+m_{1}}{m_{1}m_{2}}}m_{1}\mathbf {a} _{1}={\frac {\mathbf {F} _{12}}{\mu }}}
499:
3186:
3613:
1724:. Thus we have reduced our problem to a single degree of freedom, and we can conclude that particle 1 moves with respect to the position of particle 2 as a single particle of mass equal to the reduced mass,
2160:
2072:
2148:
901:
614:
146:
833:
773:
2467:
1665:
890:
2750:
2677:
426:
3436:
3221:
1702:
1334:
1297:
3472:. They orbit each other about a common centre of mass, a two body problem. To analyze the motion of the electron, a one-body problem, the reduced mass replaces the electron mass
1964:
690:
657:
2819:
3478:
549:
2602:
2575:
2548:
2521:
1991:
1742:
1722:
97:
69:
2773:
2012:
1925:{\displaystyle {\mathcal {L}}={1 \over 2}m_{1}\mathbf {\dot {r}} _{1}^{2}+{1 \over 2}m_{2}\mathbf {\dot {r}} _{2}^{2}-V(|\mathbf {r} _{1}-\mathbf {r} _{2}|)}
51:
be replaced with the reduced mass, if this is compensated by replacing the other mass with the sum of both masses. The reduced mass is frequently denoted by
3080:{\displaystyle I=m_{1}r_{1}^{2}+m_{2}r_{2}^{2}=R^{2}{\frac {m_{1}m_{2}^{2}}{(m_{1}+m_{2})^{2}}}+R^{2}{\frac {m_{1}^{2}m_{2}}{(m_{1}+m_{2})^{2}}}=\mu R^{2}.}
2315:
3693:
437:
140:, the equivalent one-body problem, with the position of one body with respect to the other as the unknown, is that of a single body of mass
3107:
3704:
Molecular
Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry (Volume 1), P.W. Atkins, Oxford University Press, 1977,
3565:
2299:{\displaystyle \mathbf {r} _{1}={\frac {m_{2}\mathbf {r} }{m_{1}+m_{2}}},\;\mathbf {r} _{2}=-{\frac {m_{1}\mathbf {r} }{m_{1}+m_{2}}}.}
1019:{\displaystyle m_{1}\mathbf {a} _{1}=-m_{2}\mathbf {a} _{2}\;\;\Rightarrow \;\;\mathbf {a} _{2}=-{m_{1} \over m_{2}}\mathbf {a} _{1}}
3709:
2025:
2083:
354:{\displaystyle \mu =m_{1}\parallel m_{2}={\cfrac {1}{{\cfrac {1}{m_{1}}}+{\cfrac {1}{m_{2}}}}}={\cfrac {m_{1}m_{2}}{m_{1}+m_{2}}},}
557:
76:
784:
724:
2398:
1623:
848:
842:, the force that particle 2 exerts on particle 1 is equal and opposite to the force that particle 1 exerts on particle 2:
35:
when the particles are interacting with each other. Reduced mass allows the two-body problem to be solved as if it were a
2683:
2610:
378:
3352:{\displaystyle V(|\mathbf {r} _{1}-\mathbf {r} _{2}|)=-{\frac {Gm_{1}m_{2}}{|\mathbf {r} _{1}-\mathbf {r} _{2}|}}\,,}
3095:
3369:
3656:
3637:
1670:
1302:
1265:
3619:
715:
3642:
1938:
2480:
Reduced mass can be used in a multitude of two-body problems, where classical mechanics is applicable.
839:
662:
3549:{\displaystyle m_{\text{e}}\rightarrow {\frac {m_{\text{e}}m_{\text{p}}}{m_{\text{e}}+m_{\text{p}}}}}
1759:
1753:
104:
2019:
is a function as it is only dependent on the absolute distance between the particles. If we define
622:
3647:
3631:
2778:
709:
115:
40:
3742:
3679:, G.L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.) 0-89573-752-3
514:
3705:
3689:
2825:
36:
32:
2580:
2553:
2526:
2499:
1969:
3724:
2488:
1727:
1707:
82:
54:
3676:
2758:
1997:
3747:
3736:
3469:
505:
28:
2472:
is the reduced mass. Thus we have reduced the two-body problem to that of one body.
2077:
and let the centre of mass coincide with our origin in this reference frame, i.e.
1262:
Note that (since the derivative is a linear operator) the relative acceleration
100:
72:
3652:
3207:
calculations, especially when the larger particle's exact mass is not known.
3200:
718:, the force exerted by a body (particle 2) on another body (particle 1) is:
2382:{\displaystyle {\mathcal {L}}={1 \over 2}\mu \mathbf {\dot {r}} ^{2}-V(r),}
364:
where the force on this mass is given by the force between the two bodies.
3447:
1758:
Alternatively, a
Lagrangian description of the two-body problem gives a
372:
The reduced mass is always less than or equal to the mass of each body:
494:{\displaystyle {\frac {1}{\mu }}={\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}}
108:
20:
3458:
3181:{\displaystyle \Delta K={\frac {1}{2}}\mu v_{\rm {rel}}^{2}(e^{2}-1)}
2487:
1620:
This simplifies the description of the system to one force (since
3608:{\displaystyle m_{\text{p}}\rightarrow m_{\text{e}}+m_{\text{p}}}
3211:
Motion of two massive bodies under their gravitational attraction
3688:
Dynamics and
Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009,
3634:- the general operation, of which reduced mass is just one case
2067:{\displaystyle \mathbf {r} =\mathbf {r} _{1}-\mathbf {r} _{2}}
2143:{\displaystyle m_{1}\mathbf {r} _{1}+m_{2}\mathbf {r} _{2}=0}
2321:
1774:
609:{\displaystyle {\mu }={\frac {m_{1}}{2}}={\frac {m_{2}}{2}}}
2824:
This holds for a rotation around the center of mass. The
828:{\displaystyle \mathbf {F} _{21}=m_{2}\mathbf {a} _{2}}
768:{\displaystyle \mathbf {F} _{12}=m_{1}\mathbf {a} _{1}}
315:
286:
255:
243:
217:
205:
197:
185:
3559:
and the proton mass becomes the sum of the two masses
2462:{\displaystyle \mu ={\frac {m_{1}m_{2}}{m_{1}+m_{2}}}}
318:
289:
258:
246:
220:
208:
200:
188:
3568:
3481:
3372:
3224:
3110:
2834:
2781:
2761:
2686:
2613:
2583:
2556:
2529:
2502:
2401:
2318:
2163:
2086:
2028:
2000:
1972:
1941:
1771:
1730:
1710:
1673:
1626:
1345:
1305:
1268:
1045:
904:
851:
787:
727:
665:
625:
560:
517:
504:
which by re-arrangement is equivalent to half of the
440:
381:
149:
85:
57:
2492:
Two point masses rotating around the center of mass.
1660:{\displaystyle \mathbf {F} _{12}=-\mathbf {F} _{21}}
885:{\displaystyle \mathbf {F} _{12}=-\mathbf {F} _{21}}
3215:In the case of the gravitational potential energy
2745:{\displaystyle r_{2}=R{\frac {m_{1}}{m_{1}+m_{2}}}}
2672:{\displaystyle r_{1}=R{\frac {m_{2}}{m_{1}+m_{2}}}}
421:{\displaystyle \mu \leq m_{1},\quad \mu \leq m_{2}}
3607:
3548:
3430:
3351:
3180:
3101:, the change in kinetic energy can be written as
3079:
2813:
2767:
2744:
2671:
2596:
2569:
2542:
2515:
2461:
2381:
2298:
2142:
2066:
2006:
1985:
1958:
1924:
1736:
1716:
1696:
1659:
1609:
1328:
1291:
1251:
1018:
884:
827:
778:The force exerted by particle 1 on particle 2 is:
767:
684:
651:
608:
543:
493:
420:
353:
91:
63:
2550:such that they are co-linear, the two distances
2484:Moment of inertia of two point masses in a line
2309:Then substituting above gives a new Lagrangian
1299:is equal to the acceleration of the separation
39:. Note, however, that the mass determining the
3199:is the relative velocity of the bodies before
8:
3431:{\displaystyle m_{1}m_{2}=(m_{1}+m_{2})\mu }
2828:around this axis can be then simplified to
2228:
961:
960:
956:
955:
431:and has the reciprocal additive property:
3599:
3586:
3573:
3567:
3537:
3524:
3512:
3502:
3495:
3486:
3480:
3416:
3403:
3387:
3377:
3371:
3345:
3337:
3331:
3326:
3316:
3311:
3305:
3297:
3287:
3277:
3263:
3257:
3252:
3242:
3237:
3231:
3223:
3163:
3150:
3138:
3137:
3120:
3109:
3068:
3049:
3039:
3026:
3011:
3001:
2996:
2989:
2983:
2967:
2957:
2944:
2929:
2924:
2914:
2907:
2901:
2888:
2883:
2873:
2860:
2855:
2845:
2833:
2805:
2792:
2780:
2760:
2733:
2720:
2709:
2703:
2691:
2685:
2660:
2647:
2636:
2630:
2618:
2612:
2588:
2582:
2561:
2555:
2534:
2528:
2507:
2501:
2450:
2437:
2425:
2415:
2408:
2400:
2355:
2344:
2343:
2329:
2320:
2319:
2317:
2284:
2271:
2260:
2254:
2247:
2235:
2230:
2216:
2203:
2192:
2186:
2179:
2170:
2165:
2162:
2128:
2123:
2116:
2103:
2098:
2091:
2085:
2058:
2053:
2043:
2038:
2029:
2027:
1999:
1977:
1971:
1950:
1944:
1943:
1940:
1914:
1908:
1903:
1893:
1888:
1882:
1867:
1862:
1851:
1850:
1843:
1829:
1820:
1815:
1804:
1803:
1796:
1782:
1773:
1772:
1770:
1729:
1709:
1681:
1680:
1675:
1672:
1651:
1646:
1633:
1628:
1625:
1598:
1576:
1575:
1570:
1563:
1556:
1544:
1539:
1529:
1524:
1511:
1497:
1491:
1479:
1464:
1459:
1452:
1445:
1433:
1418:
1413:
1406:
1399:
1390:
1385:
1375:
1370:
1353:
1352:
1347:
1344:
1313:
1312:
1307:
1304:
1276:
1275:
1270:
1267:
1238:
1233:
1230:
1221:
1216:
1209:
1196:
1186:
1174:
1161:
1154:
1145:
1140:
1126:
1116:
1110:
1090:
1085:
1075:
1070:
1053:
1052:
1047:
1044:
1010:
1005:
996:
986:
980:
968:
963:
949:
944:
937:
921:
916:
909:
903:
876:
871:
858:
853:
850:
819:
814:
807:
794:
789:
786:
759:
754:
747:
734:
729:
726:
676:
664:
643:
630:
624:
595:
589:
575:
569:
561:
559:
535:
522:
516:
483:
474:
463:
454:
441:
439:
412:
392:
380:
337:
324:
319:
305:
295:
290:
283:
264:
259:
247:
240:
226:
221:
209:
202:
201:
189:
182:
173:
160:
148:
84:
56:
3675:Encyclopaedia of Physics (2nd Edition),
3655:, a relativistic equivalent used in the
2604:to the rotation axis may be found with
1697:{\displaystyle \mathbf {x} _{\rm {rel}}}
1329:{\displaystyle \mathbf {x} _{\rm {rel}}}
1292:{\displaystyle \mathbf {a} _{\rm {rel}}}
700:The equation can be derived as follows.
3668:
114:Reduced mass is particularly useful in
47:reduced. In the computation, one mass
101:a number of other physical quantities
7:
1036:between the two bodies is given by:
3442:Non-relativistic quantum mechanics
3145:
3142:
3139:
3111:
2496:In a system with two point masses
1959:{\displaystyle {\mathbf {r} }_{i}}
1688:
1685:
1682:
1583:
1580:
1577:
1360:
1357:
1354:
1320:
1317:
1314:
1283:
1280:
1277:
1060:
1057:
1054:
14:
685:{\displaystyle \mu \approx m_{2}}
33:system with two or more particles
3618:This idea is used to set up the
3327:
3312:
3253:
3238:
2349:
2346:
2261:
2231:
2193:
2166:
2124:
2099:
2054:
2039:
2030:
1945:
1904:
1889:
1856:
1853:
1809:
1806:
1676:
1647:
1629:
1571:
1540:
1525:
1460:
1414:
1386:
1371:
1348:
1308:
1271:
1234:
1217:
1141:
1086:
1071:
1048:
1006:
964:
945:
917:
872:
854:
815:
790:
755:
730:
126:Given two bodies, one with mass
77:standard gravitational parameter
1966:is the position vector of mass
401:
3579:
3492:
3422:
3396:
3338:
3306:
3268:
3264:
3232:
3228:
3175:
3156:
3046:
3019:
2964:
2937:
2373:
2367:
1919:
1915:
1883:
1879:
1550:
1520:
957:
652:{\displaystyle m_{1}\gg m_{2}}
27:is a measure of the effective
1:
2814:{\displaystyle R=r_{1}+r_{2}}
2775:is the sum of both distances
1336:between the two particles.
544:{\displaystyle m_{1}=m_{2}}
3764:
3096:coefficient of restitution
1751:
1029:The relative acceleration
707:
511:In the special case that
3657:post-Newtonian expansion
3638:Center-of-momentum frame
2015:). The potential energy
133:and the other with mass
3622:for the hydrogen atom.
3090:Collisions of particles
16:Effective inertial mass
3609:
3550:
3432:
3353:
3182:
3094:In a collision with a
3081:
2815:
2769:
2746:
2673:
2598:
2571:
2544:
2517:
2493:
2463:
2383:
2300:
2144:
2068:
2008:
1987:
1960:
1926:
1738:
1718:
1698:
1661:
1611:
1330:
1293:
1253:
1020:
886:
829:
769:
686:
653:
610:
545:
495:
422:
355:
93:
65:
3643:Momentum conservation
3610:
3551:
3433:
3354:
3183:
3082:
2816:
2770:
2747:
2674:
2599:
2597:{\displaystyle r_{2}}
2572:
2570:{\displaystyle r_{1}}
2545:
2543:{\displaystyle m_{2}}
2518:
2516:{\displaystyle m_{1}}
2491:
2464:
2384:
2301:
2145:
2069:
2009:
1988:
1986:{\displaystyle m_{i}}
1961:
1927:
1739:
1719:
1699:
1662:
1612:
1331:
1294:
1254:
1021:
887:
830:
770:
687:
654:
611:
546:
496:
423:
356:
94:
66:
3620:Schrödinger equation
3566:
3479:
3370:
3222:
3108:
2832:
2779:
2759:
2684:
2611:
2581:
2554:
2527:
2500:
2399:
2316:
2161:
2084:
2026:
1998:
1970:
1939:
1769:
1754:Lagrangian mechanics
1748:Lagrangian mechanics
1737:{\displaystyle \mu }
1728:
1717:{\displaystyle \mu }
1708:
1671:
1624:
1343:
1303:
1266:
1043:
902:
849:
785:
725:
663:
623:
558:
515:
438:
379:
147:
92:{\displaystyle \mu }
83:
64:{\displaystyle \mu }
55:
3648:Harmonic oscillator
3632:Parallel (operator)
3155:
3006:
2934:
2893:
2865:
1872:
1825:
716:Newton's second law
710:Newtonian mechanics
704:Newtonian mechanics
317:
288:
257:
245:
219:
207:
199:
187:
116:classical mechanics
79:is also denoted by
41:gravitational force
3605:
3546:
3428:
3349:
3178:
3133:
3077:
2992:
2920:
2879:
2851:
2811:
2765:
2742:
2669:
2594:
2567:
2540:
2513:
2494:
2459:
2379:
2296:
2140:
2064:
2004:
1983:
1956:
1922:
1849:
1802:
1734:
1714:
1694:
1667:), one coordinate
1657:
1607:
1326:
1289:
1249:
1016:
882:
840:Newton's third law
825:
765:
682:
649:
606:
541:
491:
418:
351:
344:
312:
276:
271:
252:
233:
214:
194:
89:
61:
3694:978-0-470-01460-8
3602:
3589:
3576:
3544:
3540:
3527:
3515:
3505:
3489:
3343:
3128:
3056:
2974:
2826:moment of inertia
2768:{\displaystyle R}
2740:
2667:
2457:
2352:
2337:
2291:
2223:
2007:{\displaystyle i}
1859:
1837:
1812:
1790:
1605:
1518:
1486:
1440:
1247:
1203:
1132:
1002:
604:
584:
489:
469:
449:
346:
316:
287:
278:
273:
256:
244:
235:
218:
206:
198:
186:
3755:
3712:
3702:
3696:
3686:
3680:
3673:
3614:
3612:
3611:
3606:
3604:
3603:
3600:
3591:
3590:
3587:
3578:
3577:
3574:
3555:
3553:
3552:
3547:
3545:
3543:
3542:
3541:
3538:
3529:
3528:
3525:
3518:
3517:
3516:
3513:
3507:
3506:
3503:
3496:
3491:
3490:
3487:
3437:
3435:
3434:
3429:
3421:
3420:
3408:
3407:
3392:
3391:
3382:
3381:
3358:
3356:
3355:
3350:
3344:
3342:
3341:
3336:
3335:
3330:
3321:
3320:
3315:
3309:
3303:
3302:
3301:
3292:
3291:
3278:
3267:
3262:
3261:
3256:
3247:
3246:
3241:
3235:
3187:
3185:
3184:
3179:
3168:
3167:
3154:
3149:
3148:
3129:
3121:
3086:
3084:
3083:
3078:
3073:
3072:
3057:
3055:
3054:
3053:
3044:
3043:
3031:
3030:
3017:
3016:
3015:
3005:
3000:
2990:
2988:
2987:
2975:
2973:
2972:
2971:
2962:
2961:
2949:
2948:
2935:
2933:
2928:
2919:
2918:
2908:
2906:
2905:
2892:
2887:
2878:
2877:
2864:
2859:
2850:
2849:
2820:
2818:
2817:
2812:
2810:
2809:
2797:
2796:
2774:
2772:
2771:
2766:
2751:
2749:
2748:
2743:
2741:
2739:
2738:
2737:
2725:
2724:
2714:
2713:
2704:
2696:
2695:
2678:
2676:
2675:
2670:
2668:
2666:
2665:
2664:
2652:
2651:
2641:
2640:
2631:
2623:
2622:
2603:
2601:
2600:
2595:
2593:
2592:
2576:
2574:
2573:
2568:
2566:
2565:
2549:
2547:
2546:
2541:
2539:
2538:
2522:
2520:
2519:
2514:
2512:
2511:
2468:
2466:
2465:
2460:
2458:
2456:
2455:
2454:
2442:
2441:
2431:
2430:
2429:
2420:
2419:
2409:
2388:
2386:
2385:
2380:
2360:
2359:
2354:
2353:
2345:
2338:
2330:
2325:
2324:
2305:
2303:
2302:
2297:
2292:
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2010:
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1992:
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1931:
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3728:on HyperPhysics
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376:
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333:
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314:
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3719:External links
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1752:Main article:
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366:
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192:
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172:
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120:
103:). It has the
88:
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15:
13:
10:
9:
6:
4:
3:
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3710:0-19-855129-0
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3473:
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3470:hydrogen atom
3464:
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3453:
3449:
3446:Consider the
3441:
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3417:
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3409:
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1076:
1066:
1039:
1038:
1037:
1032:
1011:
997:
993:
987:
983:
977:
974:
969:
950:
938:
934:
930:
927:
922:
910:
906:
898:
897:
896:
877:
867:
864:
859:
845:
844:
843:
841:
838:According to
820:
808:
804:
800:
795:
781:
780:
779:
760:
748:
744:
740:
735:
721:
720:
719:
717:
711:
703:
701:
695:
693:
677:
673:
669:
666:
644:
640:
636:
631:
627:
601:
596:
592:
586:
581:
576:
572:
566:
562:
554:
553:
552:
536:
532:
528:
523:
519:
509:
507:
506:harmonic mean
484:
480:
476:
471:
464:
460:
456:
451:
446:
443:
434:
433:
432:
413:
409:
405:
402:
398:
393:
389:
385:
382:
375:
374:
373:
367:
365:
348:
338:
334:
330:
325:
321:
306:
302:
296:
292:
280:
265:
261:
248:
237:
227:
223:
210:
190:
179:
174:
170:
166:
161:
157:
153:
150:
143:
142:
141:
136:
129:
121:
119:
117:
112:
110:
107:of mass, and
106:
102:
86:
78:
74:
58:
50:
46:
42:
38:
34:
30:
29:inertial mass
26:
22:
3726:Reduced Mass
3725:
3700:
3684:
3671:
3617:
3558:
3462:
3451:
3445:
3361:
3214:
3205:
3193:
3191:
3098:
3093:
2823:
2754:
2495:
2479:
2476:Applications
2471:
2391:
2308:
2153:
2076:
2016:
1994:
1934:
1757:
1619:
1261:
1030:
1028:
894:
837:
777:
713:
699:
618:
510:
503:
430:
371:
363:
134:
127:
125:
113:
48:
44:
25:reduced mass
24:
18:
3677:R.G. Lerner
895:Therefore:
3737:Categories
3663:References
3653:Chirp mass
1760:Lagrangian
708:See also:
696:Derivation
368:Properties
105:dimensions
3743:Mechanics
3580:→
3493:→
3468:) in the
3426:μ
3323:−
3275:−
3249:−
3201:collision
3170:−
3131:μ
3112:Δ
3062:μ
2403:μ
2362:−
2350:˙
2340:μ
2245:−
2050:−
1900:−
1874:−
1857:˙
1810:˙
1732:μ
1712:μ
1643:−
1536:−
1443:−
1382:−
1245:μ
1082:−
978:−
958:⇒
931:−
868:−
670:≈
667:μ
637:≫
563:μ
447:μ
406:≤
403:μ
386:≤
383:μ
167:∥
151:μ
87:μ
59:μ
3626:See also
3448:electron
122:Equation
99:(as are
659:, then
109:SI unit
21:physics
3708:
3692:
3461:(mass
3459:proton
3457:) and
3450:(mass
3192:where
2755:where
2392:where
1935:where
714:Using
2154:then
31:of a
3748:Mass
3706:ISBN
3690:ISBN
2577:and
2523:and
111:kg.
3197:rel
1762:of
1034:rel
619:If
49:can
45:not
43:is
19:In
3739::
3203:.
2821:.
1744:.
1653:21
1635:12
1240:12
1067::=
878:21
860:12
796:21
736:12
692:.
551::
508:.
118:.
73:mu
23:,
3601:p
3597:m
3593:+
3588:e
3584:m
3575:p
3571:m
3539:p
3535:m
3531:+
3526:e
3522:m
3514:p
3510:m
3504:e
3500:m
3488:e
3484:m
3466:p
3463:m
3455:e
3452:m
3423:)
3418:2
3414:m
3410:+
3405:1
3401:m
3397:(
3394:=
3389:2
3385:m
3379:1
3375:m
3347:,
3339:|
3333:2
3328:r
3318:1
3313:r
3307:|
3299:2
3295:m
3289:1
3285:m
3281:G
3272:=
3269:)
3265:|
3259:2
3254:r
3244:1
3239:r
3233:|
3229:(
3226:V
3194:v
3188:,
3176:)
3173:1
3165:2
3161:e
3157:(
3152:2
3146:l
3143:e
3140:r
3135:v
3126:2
3123:1
3118:=
3115:K
3099:e
3075:.
3070:2
3066:R
3059:=
3051:2
3047:)
3041:2
3037:m
3033:+
3028:1
3024:m
3020:(
3013:2
3009:m
3003:2
2998:1
2994:m
2985:2
2981:R
2977:+
2969:2
2965:)
2959:2
2955:m
2951:+
2946:1
2942:m
2938:(
2931:2
2926:2
2922:m
2916:1
2912:m
2903:2
2899:R
2895:=
2890:2
2885:2
2881:r
2875:2
2871:m
2867:+
2862:2
2857:1
2853:r
2847:1
2843:m
2839:=
2836:I
2807:2
2803:r
2799:+
2794:1
2790:r
2786:=
2783:R
2763:R
2735:2
2731:m
2727:+
2722:1
2718:m
2711:1
2707:m
2701:R
2698:=
2693:2
2689:r
2662:2
2658:m
2654:+
2649:1
2645:m
2638:2
2634:m
2628:R
2625:=
2620:1
2616:r
2590:2
2586:r
2563:1
2559:r
2536:2
2532:m
2509:1
2505:m
2452:2
2448:m
2444:+
2439:1
2435:m
2427:2
2423:m
2417:1
2413:m
2406:=
2377:,
2374:)
2371:r
2368:(
2365:V
2357:2
2347:r
2335:2
2332:1
2327:=
2322:L
2294:.
2286:2
2282:m
2278:+
2273:1
2269:m
2262:r
2256:1
2252:m
2242:=
2237:2
2232:r
2226:,
2218:2
2214:m
2210:+
2205:1
2201:m
2194:r
2188:2
2184:m
2177:=
2172:1
2167:r
2150:,
2138:0
2135:=
2130:2
2125:r
2118:2
2114:m
2110:+
2105:1
2100:r
2093:1
2089:m
2060:2
2055:r
2045:1
2040:r
2035:=
2031:r
2017:V
2002:i
1979:i
1975:m
1952:i
1946:r
1920:)
1916:|
1910:2
1905:r
1895:1
1890:r
1884:|
1880:(
1877:V
1869:2
1864:2
1854:r
1845:2
1841:m
1835:2
1832:1
1827:+
1822:2
1817:1
1807:r
1798:1
1794:m
1788:2
1785:1
1780:=
1775:L
1689:l
1686:e
1683:r
1677:x
1648:F
1640:=
1630:F
1600:2
1596:t
1592:d
1584:l
1581:e
1578:r
1572:x
1565:2
1561:d
1554:=
1551:)
1546:2
1541:x
1531:1
1526:x
1521:(
1513:2
1509:t
1505:d
1499:2
1495:d
1489:=
1481:2
1477:t
1473:d
1466:2
1461:x
1454:2
1450:d
1435:2
1431:t
1427:d
1420:1
1415:x
1408:2
1404:d
1397:=
1392:2
1387:a
1377:1
1372:a
1367:=
1361:l
1358:e
1355:r
1349:a
1321:l
1318:e
1315:r
1309:x
1284:l
1281:e
1278:r
1272:a
1235:F
1228:=
1223:1
1218:a
1211:1
1207:m
1198:2
1194:m
1188:1
1184:m
1176:1
1172:m
1168:+
1163:2
1159:m
1152:=
1147:1
1142:a
1136:)
1128:2
1124:m
1118:1
1114:m
1108:+
1105:1
1101:(
1097:=
1092:2
1087:a
1077:1
1072:a
1061:l
1058:e
1055:r
1049:a
1031:a
1012:1
1007:a
998:2
994:m
988:1
984:m
975:=
970:2
965:a
951:2
946:a
939:2
935:m
928:=
923:1
918:a
911:1
907:m
873:F
865:=
855:F
821:2
816:a
809:2
805:m
801:=
791:F
761:1
756:a
749:1
745:m
741:=
731:F
678:2
674:m
645:2
641:m
632:1
628:m
602:2
597:2
593:m
587:=
582:2
577:1
573:m
567:=
537:2
533:m
529:=
524:1
520:m
485:2
481:m
477:1
472:+
465:1
461:m
457:1
452:=
444:1
414:2
410:m
399:,
394:1
390:m
349:,
339:2
335:m
331:+
326:1
322:m
307:2
303:m
297:1
293:m
281:=
266:2
262:m
249:1
238:+
228:1
224:m
211:1
191:1
180:=
175:2
171:m
162:1
158:m
154:=
138:2
135:m
131:1
128:m
71:(
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