Лекции по классической механике. Учебное пособие
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Лекция5.ОДНОМЕРНОЕДВИЖЕНИЕ.ЗАДАЧАДВУХТЕЛ.ДВИЖЕНИЕЧАСТИЦЫВЦЕНТРАЛЬНО-СИММЕТРИЧНОМПОЛЕ
Ⱦɜɢɠɟɧɢɟɜɨɡɦɨɠɧɨɟɫɥɢ x ≤ x x ≤ x ≤ x x ≥ x ȿɫɥɢ x ≤ x ≤ x ɞɜɢɠɟ
ɧɢɟɨɝɪɚɧɢɱɟɧɨɜ ɩɪɨɫɬɪɚɧɫɬɜɟɢ ɧɚɡɵɜɚɟɬɫɹɮɢɧɢɬɧɵɦȿɫɥɢɱɚɫɬɢɰɚɦɨɠɟɬ ɭɣɬɢɧɚɛɟɫɤɨɧɟɱɧɨɫɬɶɞɜɢɠɟɧɢɟɧɚɡɵɜɚɟɬɫɹɢɧɮɢɧɢɬɧɵɦȾɜɢɠɟɧɢɟɧɟɜɨɡ ɦɨɠɧɨ ɟɫɥɢx ≤ x ≤ x x ≤ x ≤ x
ɂɫɩɨɥɶɡɭɹ ɨɩɪɟɞɟɥɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɪɚɜɟɧɫɬɜɨ ɦɨɠ
ɧɨ ɡɚɩɢɫɚɬɶ ɬɚɤ
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Ɉɬɫɸɞɚ ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɣ ɮɨɪɦɟ
dx = ± |
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Ʉɨɧɤɪɟɬɧɵɣ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ x = x t ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɢɞɨɦ ɮɭɧɤɰɢɢ ɩɨɬɟɧ ɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢU x ɉɟɪɢɨɞ ɨɞɧɨɦɟɪɧɨɝɨ ɮɢɧɢɬɧɨɝɨ ɩɟɪɢɨɞɢɱɟɫɤɨɝɨ ɞɜɢ ɠɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
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Ɂɚɞɚɱɚ ɞɜɭɯ ɬɟɥ
Ɂɚɞɚɱɟɣ ɞɜɭɯ ɬɟɥ ɧɚɡɵɜɚɟɬɫɹ ɡɚɞɚɱɚ ɨ ɞɜɢɠɟɧɢɢ ɞɜɭɯ ɱɚɫɬɢɰ ɫɨɫɬɚɜɥɹɸ
ɳɢɯ ɡɚɦɤɧɭɬɭɸ ɫɢɫɬɟɦɭ ɋɱɢɬɚɟɦ ɱɬɨ ɱɚɫɬɢɰɵ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɦɟɠɞɭ ɫɨɛɨɣ
ɩɨɫɪɟɞɫɬɜɨɦ ɩɨɬɟɧɰɢɚɥɶɧɵɯ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɢɥ
ȼɧɭɬɪɟɧɧɢɟ ɫɢɥɵ ɧɟ ɜɥɢɹɸɬ ɧɚ ɞɜɢɠɟɧɢɟ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɫɢɫɬɟɦɵ ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɫɢɫɬɟɦɵ ɞɜɢɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ ɢ ɪɚɜɧɨɦɟɪɧɨ ɥɢɛɨ ɩɨɤɨɢɬɫɹ ȼ ɞɚɥɶɧɟɣɲɟɦ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɜ ɫɢɫɬɟɦɟ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɪɢɫ
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Ɋɢɫ. 5.2.Ɂɚɦɤɧɭɬɚɹ ɫɢɫɬɟɦɚ ɢɡ ɞɜɭɯ ɱɚɫɬɢɰ
ɍɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɢɦɟɸɬ ɜɢɞ |
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М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
ɂɫɩɨɥɶɡɭɹ ɬɪɟɬɢɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ ɫɨɝɥɚɫɧɨ ɤɨɬɨɪɨɦɭ F r& = − F r& ɢ
r& = r& − r& |
m r& + m r& = ɩɨɥɭɱɢɦ ɱɬɨ ɞɜɚ ɭɪɚɜɧɟɧɢɹ ɫɢɫɬɟɦɵ ɷɤɜɢɜɚɥɟɧɬ |
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ɧɵ ɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ |
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± ɩɪɢɜɟɞɟɧɧɚɹ ɦɚɫɫɚ ɫɢɫɬɟɦɵ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɡɚɞɚɱɚ ɞɜɭɯ ɬɟɥ |
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ɫɜɨɞɢɬɫɹ ɤ ɡɚɞɚɱɟ ɨ ɧɚɯɨɠɞɟɧɢɢ ɡɚɤɨɧɚ ɞɜɢɠɟɧɢɹ ɢɡɨɛɪɚɠɚɸɳɟɣ ɬɨɱɤɢ ɫ ɩɪɢɜɟɞɟɧɧɨɣɦɚɫɫɨɣɫɢɫɬɟɦɵɜ ɩɨɥɟɰɟɧɬɪɚɥɶɧɨɣɫɢɥɵɊɟɲɚɹɭɪɚɜɧɟɧɢɟɧɚɯɨɞɢɦ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɢɡɨɛɪɚɠɚɸɳɟɣ ɬɨɱɤɢ r& = r& t Ⱦɚɥɟɟ ɩɨ ɮɨɪɦɭɥɚɦ
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ɧɚɯɨɞɢɦɡɚɤɨɧɵɞɜɢɠɟɧɢɹɱɚɫɬɢɰɫɢɫɬɟɦɵɂɡ ɫɥɟɞɭɟɬɱɬɨɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɢɡɨɛɪɚɠɚɸɳɟɣ ɬɨɱɤɢ ɢ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɹɜɥɹɸɬɫɹ ɩɨɞɨɛɧɵɦɢ
ɑɚɫɬɢɰɚ ɜ ɰɟɧɬɪɚɥɶɧɨ ɫɢɦɦɟɬɪɢɱɧɨɦ ɩɨɥɟ ɂɧɬɟɝɪɚɥɵ ɞɜɢɠɟɧɢɹ
Ɋɚɫɫɦɨɬɪɢɦ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɰɟɧɬɪɚɥɶɧɨɣ ɫɢɥɵ
F& r& = F r rr
ɋɢɥɚ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧɚ ɱɟɪɟɡ ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪɝɢɸ ɱɚɫɬɢɰɵ
F& r& = − ∂U∂r&r
ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɧɟ ɡɚɜɢɫɢɬ ɹɜɧɵɦ ɨɛɪɚɡɨɦ ɨɬ ɜɪɟɦɟɧɢ
ɗɬɨ ɨɡɧɚɱɚɟɬ ɱɬɨ ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɫɨɯɪɚɧɹɟɬɫɹ ɩɟɪɜɵɣ ɢɧɬɟɝɪɚɥ
ɞɜɢɠɟɧɢɹ
E = m v + U r = const
ɉɪɢ ɞɜɢɠɟɧɢɢ ɱɚɫɬɢɰɵ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ ɫɨɯɪɚɧɹɟɬɫɹ ɨɪɛɢɬɚɥɶ
ɧɵɣ ɦɨɦɟɧɬ ɜɬɨɪɨɣ ɢɧɬɟɝɪɚɥ ɞɜɢɠɟɧɢɹ
l = m>r& v&@ = const
ɂɡ ɷɬɨɝɨ ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ ɫɥɟɞɭɟɬ ɱɬɨ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɩɪɨɢɫɯɨɞɢɬ
ɜ ɧɟɤɨɬɨɪɨɣ ɩɥɨɫɤɨɫɬɢ ɬɪɚɟɤɬɨɪɢɟɣ ɞɜɢɠɟɧɢɹ ɹɜɥɹɟɬɫɹ ɩɥɨɫɤɚɹ ɤɪɢɜɚɹ ɨɛɨɛ ɳɟɧɢɟ ɩɟɪɜɨɝɨ ɡɚɤɨɧɚ Ʉɟɩɥɟɪɚ ɂɡ ɜɵɬɟɤɚɟɬ ɱɬɨ ɫɟɤɬɨɪɧɚɹ ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɩɨɫɬɨɹɧɧɚ ɜɬɨɪɨɣ ɡɚɤɨɧ Ʉɟɩɥɟɪɚ
σ& = >r& v&@ = const
Ɍɚɤ ɤɚɤ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɹɜɥɹɟɬɫɹ ɩɥɨɫɤɢɦ ɬɨ ɞɥɹ ɟɝɨ ɨɩɢɫɚɧɢɹ ɰɟɥɟɫɨ
ɨɛɪɚɡɧɨ ɩɟɪɟɣɬɢ ɜ ɩɨɥɹɪɧɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɪɢɫ ɋɤɨɪɨɫɬɶ ɱɚɫɬɢɰɵ
ɜ ɩɨɥɹɪɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɡɚɩɢɫɵɜɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
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Лекция5.ОДНОМЕРНОЕДВИЖЕНИЕ.ЗАДАЧАДВУХТЕЛ.ДВИЖЕНИЕЧАСТИЦЫВЦЕНТРАЛЬНО-СИММЕТРИЧНОМПОЛЕ
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ɝɞɟ e&r e&ϕ ± ɨɪɬɵ ɩɨɥɹɪɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ Ɉɪɛɢɬɚɥɶɧɵɣ ɦɨɦɟɧɬ ɱɚɫɬɢɰɵ ɜ ɩɨɥɹɪɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɡɚɩɢɫɵɜɚɟɬɫɹ ɬɚɤ
l = mr ddtϕ
ɋ ɭɱɟɬɨɦ ɮɨɪɦɭɥ ɢ ɢɧɬɟɝɪɚɥɵ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɩɨɥɹɪɧɨɣ
ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɢɦɟɸɬ ɜɢɞ
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HΜ M |
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Ɋɢɫ. 5.3.ɉɨɥɹɪɧɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ |
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ɗɮɮɟɤɬɢɜɧɵɣ ɩɨɬɟɧɰɢɚɥ |
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Ɋɚɫɤɪɵɜ ɫɤɨɛɤɢ ɜ ɮɨɪɦɭɥɟ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ |
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m §dr · |
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© dt ¹ |
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ɝɞɟ |
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Uɷɮɮ r = |
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ɟɫɬɶɷɮɮɟɤɬɢɜɧɵɣɩɨɬɟɧɰɢɚɥɗɮɮɟɤɬɢɜɧɵɣɩɨɬɟɧɰɢɚɥɦɨɠɟɬɛɵɬɶɩɨɥɨɠɢ ɬɟɥɶɧɵɦ ɨɬɪɢɰɚɬɟɥɶɧɵɦ ɢ ɪɚɜɧɵɦ ɧɭɥɸ ɉɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɜ ɩɪɟɞɫɬɚɜ ɥɹɟɬ ɫɨɛɨɣ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɪɚɞɢɚɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɩɟɪɜɨɟ ɫɥɚ ɝɚɟɦɨɟ ɜ ± ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɨɛɪɚɳɟɧɢɹ ɱɚɫɬɢɰɵ ɜɨɤɪɭɝ ɫɢɥɨɜɨɝɨ ɰɟɧɬɪɚ
ɂɡ ɭɪɚɜɧɟɧɢɹ ɩɨɥɭɱɚɟɦ
33
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
dr |
= ± dt |
m (E − Uɷɮɮ r ) |
ɤɨɬɨɪɨɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɪɚɞɢɚɥɶɧɨɝɨ ɞɜɢɠɟ ɧɢɹɱɚɫɬɢɰɵɂɧɬɟɝɪɢɪɭɟɦ ɢ ɧɚɯɨɞɢɦ ɡɚɜɢɫɢɦɨɫɬɶɪɚɫɫɬɨɹɧɢɹɱɚɫɬɢɰɵ ɞɨ ɫɢɥɨɜɨɝɨ ɰɟɧɬɪɚ ɤɚɤ ɮɭɧɤɰɢɸ ɜɪɟɦɟɧɢ ɡɚɤɨɧ ɪɚɞɢɚɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢ ɰɵr = r t Ⱦɚɥɟɟ ɢɡ ɧɚɯɨɞɢɦ
dϕ = l dt m r t
ɂɧɬɟɝɪɢɪɭɹ ɩɨɫɥɟɞɧɟɟ ɭɪɚɜɧɟɧɢɟ ɩɨɥɭɱɢɦ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɭɝɥɚ ɩɨɜɨɪɨ ɬɚ ɱɚɫɬɢɰɵ ɜɨɤɪɭɝ ɫɢɥɨɜɨɝɨ ɰɟɧɬɪɚ ϕ = ϕ t Ɋɚɡɞɟɥɢɜ ɭɪɚɜɧɟɧɢɟ ɧɚ ɧɚɣɞɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ
dϕ = ± |
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− Uɷɮɮ r ) |
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ɂɧɬɟɝɪɢɪɭɹ ɧɚɣɞɟɦ ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɱɚɫɬɢɰɵr = r ϕ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɡɚɞɚɱɚ ɨ ɞɜɢɠɟɧɢɢ ɱɚɫɬɢɰɵ ɜ ɰɟɧɬɪɚɥɶɧɨɦ ɩɨɥɟ ɩɨɥɧɨɫɬɶɸ ɪɟɲɟɧɚ
Ɍɪɚɟɤɬɨɪɢɹ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɰɟɧɬɪɚɥɶɧɨɦ ɩɨɥɟ ɟɫɬɶ ɩɥɨɫɤɚɹ ɤɪɢɜɚɹ Ɉɧɚ ɫɢɦɦɟɬɪɢɱɧɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɹɦɨɣ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɫɢɥɨɜɨɣ ɰɟɧɬɪ ɢ ɛɥɢɠɚɣɲɭɸ ɧɚɢɛɨɥɟɟ ɭɞɚɥɟɧɧɭɸ ɬɨɱɤɭ ɬɪɚɟɤɬɨɪɢɢ ɤ ɫɢɥɨɜɨɦɭ ɰɟɧɬɪɭ ɟɫɥɢ ɬɚɤɚɹ ɢɦɟɟɬɫɹ ɧɚ ɷɬɨ ɭɤɚɡɵɜɚɸɬ ɡɧɚɤɢ± ɜ ɮɨɪɦɭɥɟ
ɍɪɚɜɧɟɧɢɟ Ȼɢɧɟ
Ɏɨɪɦɭɥɵ ± ɩɨɡɜɨɥɹɸɬ ɩɨɥɧɨɫɬɶɸ ɪɟɲɢɬɶ ɡɚɞɚɱɭ ɨ ɞɜɢɠɟɧɢɢ ɱɚɫɬɢɰɵ ɜ ɰɟɧɬɪɚɥɶɧɨ ɫɢɦɦɟɬɪɢɱɧɨɦ ɩɨɥɟ ɟɫɥɢ ɢɡɜɟɫɬɧɚ ɫɢɥɚ ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɱɚɫɬɢɰɭ ɢɥɢ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɜ ɞɚɧɧɨɦ ɫɢɥɨɜɨɦ ɩɨɥɟ ɑɚɫɬɨ ɬɪɟɛɭɟɬɫɹ ɪɟɲɢɬɶ ɨɛɪɚɬɧɭɸ ɡɚɞɚɱɭ ɩɨ ɢɡɜɟɫɬɧɨɣ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢ ɰɵɜ ɰɟɧɬɪɚɥɶɧɨɦɩɨɥɟɧɚɣɬɢɫɢɥɭɞɟɣɫɬɜɭɸɳɭɸɧɚɱɚɫɬɢɰɭȼ ɷɬɨɦɫɥɭɱɚɟ ɭɞɨɛɧɨ ɩɨɥɶɡɨɜɚɬɶɫɹ ɭɪɚɜɧɟɧɢɟɦ Ȼɢɧɟ ɤɨɬɨɪɨɟ ɦɵ ɫɟɣɱɚɫ ɩɨɥɭɱɢɦ
Ɂɚɩɢɲɟɦ ɮɨɪɦɭɥɭ ɞɥɹ ɭɫɤɨɪɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɩɨɥɹɪɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ
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ªd r |
§dϕ · |
º |
& ª |
d ϕ |
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dr dϕ º& |
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= « |
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ɉɪɢ ɞɜɢɠɟɧɢɢ ɱɚɫɬɢɰɵ ɜ ɰɟɧɬɪɚɥɶɧɨɦ ɩɨɥɟ ɭɫɤɨɪɟɧɢɟ ɱɚɫɬɢɰɵ ɧɚɩɪɚɜ
ɥɟɧɨ ɜɞɨɥɶ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ ɱɚɫɬɢɰɵ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɞɟɣɫɬɜɭɸɳɟɣ ɫɢɥɵ ɉɨ ɷɬɨɦɭ ɫɨɫɬɚɜɥɹɸɳɚɹ ɭɫɤɨɪɟɧɢɹaϕ ɪɚɜɧɚ ɧɭɥɸ ȼɬɨɪɨɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ ɞɥɹ ɱɚɫ
ɬɢɰɵ ɜ ɰɟɧɬɪɚɥɶɧɨɦ ɩɨɥɟ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
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Лекция5.ОДНОМЕРНОЕДВИЖЕНИЕ.ЗАДАЧАДВУХТЕЛ.ДВИЖЕНИЕЧАСТИЦЫВЦЕНТРАЛЬНО-СИММЕТРИЧНОМПОЛЕ
ɉɪɨɢɡɜɨɞɧɚɹ |
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ɑɬɨɛɵ ɧɚɣɬɢ ɫɢɥɭ ɩɨ ɮɨɪɦɭɥɟ ɧɟɨɛɯɨɞɢɦɨ ɜɵɱɢɫɥɢɬɶ dt Ⱦɥɹ |
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ɷɬɨɝɨ ɜɵɱɢɫɥɹɟɦ |
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ɢ ɞɚɥɟɟ ɧɚɯɨɞɢɦ |
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ɉɨɞɫɬɚɜɢɜ ɮɨɪɦɭɥɵ ɢ ɜ ɮɨɪɦɭɥɭ ɩɪɨɜɟɞɹ ɚɥɝɟɛɪɚɢɱɟ |
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ɫɤɢɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ Ȼɢɧɟ |
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Ⱦɚɧɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɦɨɠɟɬ ɫɥɭɠɢɬɶ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɩɨ ɡɚɞɚɧɧɨɣ ɫɢɥɟ ɢɥɢ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɫɢɥɵ ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɱɚɫɬɢɰɭ ɩɨ ɡɚɞɚɧɧɨɣ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ
ɉɪɢɦɟɪ Ɍɪɚɟɤɬɨɪɢɹ ɱɚɫɬɢɰɵ |
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r = R FRVϕ ɇɚɣɬɢ ɫɢɥɭF r ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ ɱɚɫɬɢɰɭ |
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ɇɚɯɨɞɢɦ |
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ɉɨɞɫɬɚɜɢɜ ɩɨɫɥɟɞɧɸɸ ɮɨɪɦɭɥɭ ɜ ɧɚɣɞɟɦ ɫɢɥɭ |
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ɝɞɟl = mr |
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± ɢɧɬɟɝɪɚɥ ɞɜɢɠɟɧɢɹ |
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35
ɅȿɄɐɂə ɋɌɈɅɄɇɈȼȿɇɂə ɑȺɋɌɂɐ
ɍɩɪɭɝɢɟ ɢ ɧɟɭɩɪɭɝɢɟ ɫɬɨɥɤɧɨɜɟɧɢɹ
ɉɪɨɰɟɫɫɨɦ ɫɬɨɥɤɧɨɜɟɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɩɪɨɰɟɫɫ ɩɪɢ ɤɨɬɨɪɨɦ ɱɚɫɬɢɰɵ ɜ ɯɨɞɟɢɯɫɛɥɢɠɟɧɢɹɢ ɞɚɥɶɧɟɣɲɟɝɨɪɚɡɥɟɬɚɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬɞɪɭɝ ɫ ɞɪɭɝɨɦ ɉɪɢ ɷɬɨɦ ɱɚɫɬɢɰɵ ɦɨɝɭɬ ɩɪɢɯɨɞɢɬɶ ɜ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɟ ɫɨɩɪɢɤɨɫɧɨɜɟɧɢɟ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɢɥɢ ɨɫɬɚɜɚɬɶɫɹ ɧɚ ɤɨɧɟɱɧɨɦ ɪɚɫɫɬɨɹɧɢɢ ɞɪɭɝ ɨɬ ɞɪɭɝɚ
ɉɪɨɰɟɫɫ ɫɬɨɥɤɧɨɜɟɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɭɩɪɭɝɢɦ ɟɫɥɢ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɞɨ ɫɬɨɥɤɧɨɜɟɧɢɹ ɪɚɜɧɚ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɩɨɫɥɟ ɢɯ ɫɬɨɥɤɧɨɜɟɧɢɹ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɧɟ ɦɟɧɹɟɬɫɹ ɜ ɩɪɨɰɟɫɫɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ȿɫɥɢ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɜ ɩɪɨɰɟɫɫɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɦɟɧɹɟɬɫɹ ɬɨ ɫɬɨɥɤɧɨɜɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹɧɟɭɩɪɭɝɢɦȿɫ ɥɢ ɜ ɩɪɨɰɟɫɫɟ ɧɟɭɩɪɭɝɨɝɨ ɫɬɨɥɤɧɨɜɟɧɢɹ ɦɟɧɹɟɬɫɹ ɩɪɢɪɨɞɚ ɱɚɫɬɢɰ ɭɧɢɱɬɨɠɟɧɢɟ ɱɚɫɬɢɰ ɪɨɠɞɟɧɢɟ ɧɨɜɵɯ ɱɚɫɬɢɰ ɬɨ ɬɚɤɢɟ ɩɪɨɰɟɫɫɵ ɧɚɡɵɜɚɸɬɫɹɹɞɟɪɧɵɦɢ ɪɟ ɚɤɰɢɹɦɢɂɦɩɭɥɶɫ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɜ ɫɥɭɱɚɟ ɭɩɪɭɝɢɯ ɢ ɧɟɭɩɪɭɝɢɯ ɩɪɨɰɟɫɫɨɜ ɫɨ ɯɪɚɧɹɟɬɫɹ
ȼ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ ɡɚɞɚɱɢ ɨ ɫɬɨɥɤɧɨɜɟɧɢɢ ɱɚɫɬɢɰ ɫɬɚɜɹɬɫɹ ɫɥɟ ɞɭɸɳɢɦ ɨɛɪɚɡɨɦ ɋɱɢɬɚɸɬɫɹ ɡɚɞɚɧɧɵɦɢ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɞɨ ɢɯ ɫɬɨɥɤɧɨɜɟ ɧɢɹ ɱɚɫɬɢɰɵ ɩɪɢ ɷɬɨɦ ɧɟ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɞɪɭɝ ɫ ɞɪɭɝɨɦ Ɂɚɬɟɦ ɨɫɭɳɟɫɬɜɥɹɟɬ ɫɹ ɩɪɨɰɟɫɫ ɫɬɨɥɤɧɨɜɟɧɢɹ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ Ⱦɚɥɟɟ ɱɚɫɬɢɰɵ ɪɚɡɥɟɬɚɸɬɫɹ ɧɚ ɛɨɥɶɲɢɟɪɚɫɫɬɨɹɧɢɹɢ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬɞɪɭɝɫ ɞɪɭɝɨɦɌɪɟɛɭɟɬɫɹɨɩɪɟɞɟ ɥɢɬɶ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɩɨɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ Ɋɟɲɟɧɢɟ ɞɚɧɧɨɣ ɡɚɞɚɱɢ ɨɫɭɳɟɫɬɜɥɹ ɟɬɫɹ ɧɚ ɨɫɧɨɜɚɧɢɢ ɡɚɤɨɧɨɜ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɢ ɢɦɩɭɥɶɫɚ ɉɪɢ ɷɬɨɦ ɫɚɦ ɩɪɨ ɰɟɫɫɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɧɟɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɇɚɪɹɞɭɫɨ ɫɤɨɪɨɫɬɹɦɢɱɚɫɬɢɰ ɞɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɡɚɞɚɟɬɫɹ ɡɚɤɨɧ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɱɚɫɬɢɰ ±U (r)ɉɪɢ ɷɬɨɦ ɢɫɫɥɟ ɞɭɟɬɫɹ ɢ ɫɚɦ ɩɪɨɰɟɫɫ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ Ⱦɚɧɧɚɹ ɡɚɞɚɱɚ ɧɚɡɵɜɚɟɬɫɹ ɡɚɞɚɱɟɣ ɨ ɪɚɫɫɟɹɧɢɢ ɱɚɫɬɢɰ
ɍɩɪɭɝɢɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɱɚɫɬɢɰ
ɍɩɪɭɝɨɟ ɫɬɨɥɤɧɨɜɟɧɢɟ ɱɚɫɬɢɰ ɭɞɨɛɧɨ ɩɪɨɜɨɞɢɬɶ ɩɚɪɚɥɥɟɥɶɧɨ ɨɬɧɨɫɢ
ɬɟɥɶɧɨ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ ɫɜɹɡɚɧɧɨɣ ɫ ɥɚɛɨɪɚɬɨɪɢɟɣ Ʌ ɫɢɫɬɟɦɚ ɢ ɜ ɫɢɫɬɟɦɟ
ɨɬɫɱɟɬɚ ɫɜɹɡɚɧɧɨɣ ɫ ɰɟɧɬɪɨɦ ɢɧɟɪɰɢɢ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɐ ɫɢɫɬɟɦɚ ɉɪɢ ɢɫ
ɫɥɟɞɨɜɚɧɢɢ ɫɬɨɥɤɧɨɜɟɧɢɹ ɜ Ʌ ɫɢɫɬɟɦɟ ɨɛɵɱɧɨ ɫɱɢɬɚɟɬɫɹ ɱɬɨ ɨɞɧɚ ɢɡ ɱɚɫɬɢɰ
ɩɨɤɨɢɬɫɹ
Ɋɚɫɫɦɨɬɪɢɦɭɩɪɭɝɨɟ ɫɬɨɥɤɧɨɜɟɧɢɟɞɜɭɯ ɱɚɫɬɢɰ ɫ ɦɚɫɫɚɦɢ m m ɉɭɫɬɶ ɜɬɨɪɚɹ ɱɚɫɬɢɰɚ ɩɨɤɨɢɬɫɹ v = ɋɤɨɪɨɫɬɶ ɧɚɥɟɬɚɸɳɟɣ ɱɚɫɬɢɰɵ ɡɚɞɨɥɝɨ ɞɨ ɫɬɨɥɤɧɨɜɟɧɢɹ v& Ɍɨɱɧɨɟ ɜɡɚɢɦɧɨɟ ɪɚɫɩɨɥɨɠɟɧɢɹ ɱɚɫɬɢɰ ɫɱɢɬɚɟɬɫɹ ɧɟɢɡɜɟɫɬ ɧɵɦ Ⱦɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɫɥɟɞɭɟɬ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɡɚɤɨɧɚɦɢ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪ ɝɢɢɢ ɢɦɩɭɥɶɫɚ E = const P = const ɉɨɫɤɨɥɶɤɭɜɡɚɢɦɧɨɟɪɚɫɩɨɥɨɠɟɧɢɟɱɚɫɬɢɰ ɫɱɢɬɚɟɬɫɹ ɧɟɢɡɜɟɫɬɧɵɦ ɬɨ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɜ ɞɚɧɧɨɦ ɫɥɭ ɱɚɟ ɧɟ ɢɫɩɨɥɶɡɭɟɬɫɹ Ɉ ɫɨɯɪɚɧɹɸɳɢɯɫɹ ɜɟɥɢɱɢɧɚɯ ɜ ɩɪɨɰɟɫɫɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɝɨ ɜɨɪɹɬ ɤɚɤ ɨɛɢɧɬɟɝɪɚɥɚɯ ɞɜɢɠɟɧɢɹɌɚɤɢɦ ɨɛɪɚɡɨɦ ɦɵ ɢɦɟɟɦ ɲɟɫɬɶ ɧɟɢɡɜɟɫɬ ɧɵɯ ɜɟɥɢɱɢɧ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɩɨɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ v′ v′ ɢ ɱɟɬɵɪɟ ɢɧɬɟɝɪɚɥɚ ɞɜɢɠɟɧɢɹ Ɂɚɞɚɱɚ ɹɜɥɹɟɬɫɹ ɧɟɨɩɪɟɞɟɥɟɧɧɨɣ ɢ ɞɥɹ ɟɟ ɪɟɲɟɧɢɹ ɨɛɵɱɧɨ ɞɚɥɟɟ ɩɟ
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Лекция6.СТОЛКНОВЕНИЯЧАСТИЦ
ɪɟɯɨɞɹɬ ɜ ɐ ɫɢɫɬɟɦɭ ɤɨɬɨɪɚɹ ɫɜɹɡɚɧɚ ɫ ɰɟɧɬɪɨɦ ɢɧɟɪɰɢɢ ɞɜɢɠɭɳɢɦɫɹ ɫɨ ɫɤɨ
ɪɨɫɬɶɸ
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Ⱦɥɹ ɢɦɩɭɥɶɫɨɜ ɱɚɫɬɢɰ ɜ ɐ ɫɢɫɬɟɦɟ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
p& = − p& p&′ = − p&′
ɂɧɞɟɤɫɚɦɢ© ªɩɨɦɟɱɟɧɵɜɟɥɢɱɢɧɵɨɩɪɟɞɟɥɹɟɦɵɟɜ ɐ ɫɢɫɬɟɦɟɒɬɪɢ ɯɚɦɢ ɩɨɦɟɱɟɧɵ ɜɟɥɢɱɢɧɵ ɨɩɪɟɞɟɥɹɟɦɵɟ ɩɨɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɱɚɫɬɢɰ Ɂɚɤɨɧ ɫɨ ɯɪɚɧɟɧɢɹɢɦɩɭɥɶɫɚɩɪɢɜɨɞɢɬɤ ɬɨɦɭɱɬɨɜ ɪɟɡɭɥɶɬɚɬɟɩɪɨɰɟɫɫɚɫɬɨɥɤɧɨɜɟɧɢɹ ɞɜɭɯ ɱɚɫɬɢɰ ɩɪɨɢɫɯɨɞɢɬ ɩɨɜɨɪɨɬ ɫɤɨɪɨɫɬɟɣ ɱɚɫɬɢɰ ɩɪɢ ɷɬɨɦ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɨɫɬɚɸɬɫɹ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɦɢ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɪɢɫ
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Ɋɢɫ. 6.1.Ⱦɢɚɝɪɚɦɦɚ ɫɤɨɪɨɫɬɟɣ ɜ ɐ ɫɢɫɬɟɦɟ |
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ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɹ p = p = p p′ = p′ = p′ Ɂɚɩɢɲɟɦ ɷɧɟɪɝɢɸ ɫɢɫɬɟɦɵ |
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ɱɚɫɬɢɰ ɞɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɢ ɩɨɫɥɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɱɚɫɬɢɰ |
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± ɩɪɢɜɟɞɟɧɧɚɹ ɦɚɫɫɚ ɂɡ ɪɚɜɟɧɫɬɜɚ ɷɧɟɪɝɢɣ ɫɥɟɞɭɟɬ ɱɬɨ |
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɢɡ ɡɚɤɨɧɚ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɫɥɟɞɭɟɬ ɱɬɨ ɫɤɨɪɨɫɬɢ ɱɚɫ
ɬɢɰ ɨɫɬɚɸɬɫɹ ɧɟɢɡɦɟɧɧɵɦɢ ɩɨ ɫɜɨɟɦɭ ɚɛɫɨɥɸɬɧɨɦɭ ɡɧɚɱɟɧɢɸ ɋɤɨɪɨɫɬɢ ɱɚɫ
ɬɢɰ ɞɨ ɫɬɨɥɤɧɨɜɟɧɢɹ ɢ ɩɨɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɜ ɐ ɫɢɫɬɟɦɟ ɫɜɹɡɚɧɵ ɫɥɟɞɭɸɳɢɦɢ
ɫɨɨɬɧɨɲɟɧɢɹɦɢ
v&′ = v n& v&′ = −v n&
ɝɞɟ n± ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ ɪɢɫ ɇɚɪɹɞɭ ɫ ɞɚɧɧɵɦ ɜɟɤɬɨɪɨɦ ɱɚɫɬɨ ɜɜɨɞɹɬ ɭɝɨɥ ɪɚɫɫɟɹɧɢɹχ ɪɢɫ
37
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
ɂɫɩɨɥɶɡɭɹ ɡɚɤɨɧ ɫɥɨɠɟɧɢɹ ɫɤɨɪɨɫɬɟɣv&a = v&a +V&C ɧɚɣɞɟɦ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɞɨ ɫɬɨɥɤɧɨɜɟɧɢɹ ɜ ɐ ɫɢɫɬɟɦɟ
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ɉɨ ɮɨɪɦɭɥɚɦ ɞɚɥɟɟ ɧɚɯɨɞɢɦ |
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Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɪɢɦɟɧɹɹ ɡɚɤɨɧ ɫɥɨɠɟɧɢɹ ɫɤɨɪɨɫɬɟɣ ɧɚɣɞɟɦ ɫɤɨɪɨɫɬɢ |
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ɱɚɫɬɢɰ ɩɨɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɜ Ʌ ɫɢɫɬɟɦɟ |
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Ɋɚɫɫɦɨɬɪɢɦ ɩɨɫɬɪɨɟɧɢɟ ɞɢɚɝɪɚɦɦɵ ɫɬɨɥɤɧɨɜɟɧɢɹ ɋɬɪɨɢɦ ɢɡ ɩɪɨɢɡɜɨɥɶ |
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ɧɨɣ ɬɨɱɤɢ ɨɤɪɭɠɧɨɫɬɶ ɪɚɞɢɭɫɚ R |
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ɢ ɩɨɥɭɨɤɪɭɠɧɨɫɬɶ ɪɚɞɢɭɫɚ |
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Ɂɚɬɟɦ ɩɪɨɜɨɞɢɦ ɩɨɫɬɪɨɟɧɢɹ ɩɨɤɚɡɚɧɧɵɟ ɧɚ ɪɢɫ ɞɥɹ ɫɥɭ |
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ɱɚɹm |
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Ɋɢɫ. 6.2.Ⱦɢɚɝɪɚɦɦɚ ɫɬɨɥɤɧɨɜɟɧɢɣ
ɇɚ ɞɢɚɝɪɚɦɦɟ ɩɨɥɨɠɟɧɢɹ ɬɨɱɟɤ ɢA ɮɢɤɫɢɪɨɜɚɧɵ ɬɨɱɤɢ B ɢC ɦɨɝɭɬ ɢɦɟɬɶ ɥɸɛɵɟ ɩɨɥɨɠɟɧɢɹ ɧɚ ɨɤɪɭɠɧɨɫɬɹɯ ɪɚɞɢɭɫɨɜ R R ɍɝɨɥθ ɧɚɡɵɜɚɟɬɫɹ ɭɝɥɨɦ ɪɚɫɫɟɹɧɢɹ ɜ Ʌ ɫɢɫɬɟɦɟ ɭɝɨɥθ ɧɚɡɵɜɚɟɬɫɹ ɭɝɥɨɦ ɨɬɞɚɱɢ
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Лекция6.СТОЛКНОВЕНИЯЧАСТИЦ
ɉɭɫɬɶm < m ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɭɝɨɥ ɪɚɫɫɟɹɧɢɹθ ɦɟɧɹɟɬɫɹ ɨɬ ɡɧɚɱɟɧɢɹ ɞɨ π ɤɚɤ ɢ ɭɝɨɥ χ ɋɜɹɡɶ ɦɟɠɞɭ ɭɝɥɚɦɢ θ θ ɢχ ɧɚɯɨɞɢɦ ɢɡ ɬɪɟɭɝɨɥɶɧɢɤɨɜ
DB DAB |
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ɉɪɢ©ɥɨɛɨɜɨɦªɭɞɚɪɟ (χ = π ) ɭɝɨɥɪɚɫɫɟɹɧɢɹ θ = π ɭɝɨɥɨɬɞɚɱɢ θ = ɉɪɢ ©ɥɨɛɨɜɨɦª ɭɞɚɪɟ ɧɚɥɟɬɚɸɳɚɹ ɱɚɫɬɢɰɚ ɛɭɞɟɬ ɞɜɢɝɚɬɶɫɹ ɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɚ ɩɨɤɨɹɳɚɹɫɹ ɞɨ ɫɬɨɥɤɧɨɜɟɧɢɹ ɱɚɫɬɢɰɚ ɛɭɞɟɬ ɞɜɢɝɚɬɶɫɹ ©ɜɩɟɪɟɞª ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɧɚɥɟɬɚɸɳɟɣ ɱɚɫɬɢɰɵ ɍɝɨɥ ɪɚɡɥɟɬɚ
θ + θ > π
ɉɪɢ ©ɥɨɛɨɜɨɦª ɭɞɚɪɟ ɨɧ ɪɚɜɟɧ ɫɜɨɟɦɭ ɦɚɤɫɢɦɚɥɶɧɨɦɭ ɡɧɚɱɟɧɢɸπ ɂɡɮɨɪɦɭɥ ɦɨɠɧɨɭɫɬɚɧɨɜɢɬɶɡɚɜɢɫɢɦɨɫɬɶɭɝɥɚ χ ɨɬɭɝɥɨɜ θ θ
ɉɪɢ ɡɧɚɱɟɧɢɢm < m ɷɬɚ ɫɜɹɡɶ ɹɜɥɹɟɬɫɹ ɨɞɧɨɡɧɚɱɧɨɣ ɢ ɜɵɪɚɠɚɟɬɫɹ ɮɨɪɦɭɥɚɦɢ
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χ (θ )= DUFFRV«FRVθ |
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VLQ θ » |
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χ (θ )= π − θ
ɉɪɢ ɡɧɚɱɟɧɢɢ m = m ɱɚɫɬɢɰɵ ɪɚɡɥɟɬɚɸɬɫɹ ɩɨɞ ɩɪɹɦɵɦ ɭɝɥɨɦ ɩɪɢ ɡɧɚɱɟɧɢɹɯχ ≠ π ɉɪɢ ©ɥɨɛɨɜɨɦª ɭɞɚɪɟ ɧɚɥɟɬɚɸɳɚɹ ɱɚɫɬɢɰɚ ɨɫɬɚɧɚɜɥɢɜɚɟɬ ɫɹ ɚ ɩɨɤɨɹɳɚɹɫɹ ɱɚɫɬɢɰɚ ɩɨɫɥɟ ɫɨɭɞɚɪɟɧɢɹ ɞɜɢɠɟɬɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ ɩɟɪɜɨɣ ɱɚɫɬɢɰɵ
ɉɪɢ ɡɧɚɱɟɧɢɢ m > m ɭɝɨɥ ɪɚɫɫɟɹɧɢɹθ ɬɹɠɟɥɨɣ ɱɚɫɬɢɰɵ ɧɟ ɩɪɟɜɵɲɚɟɬ ɤɪɢɬɢɱɟɫɤɨɝɨ ɡɧɚɱɟɧɢɹ
(θ )kr = DUFVLQ m m
Ɏɨɪɦɭɥɵ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɫɨɯɪɚɧɹɸɬɫɹ ɚ ɫɜɹɡɶχ (θ )ɧɟ ɹɜɥɹɟɬɫɹ ɨɞ ɧɨɡɧɚɱɧɨɣ
39
ɅȿɄɐɂə ɁȺȾȺɑȺ ɄȿɉɅȿɊȺ ɊȺɋɋȿəɇɂȿ ɑȺɋɌɂɐ
Ɂɚɞɚɱɚ Ʉɟɩɥɟɪɚ
Ɂɚɞɚɱɟɣ Ʉɟɩɥɟɪɚ ɧɚɡɵɜɚɟɬɫɹ ɡɚɞɚɱɚ ɨ ɞɜɢɠɟɧɢɢ ɱɚɫɬɢɰɵ ɜ ɩɨɥɟ
U r = − αr α >
ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɜ ɬɚɤɨɦ ɩɨɥɟ ɨɬɪɢɰɚɬɟɥɶɧɚ ɢ ɨɛɪɚɬɧɨ ɩɪɨ
ɩɨɪɰɢɨɧɚɥɶɧɚ ɪɚɫɫɬɨɹɧɢɸ ɱɚɫɬɢɰɵ ɞɨ ɫɢɥɨɜɨɝɨ ɰɟɧɬɪɚ ɫɢɥɚ ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɱɚɫ
ɬɢɰɭ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɤɜɚɞɪɚɬɭ ɪɚɫɫɬɨɹɧɢɹ ɱɚɫɬɢɰɵ ɞɨ ɫɢɥɨɜɨɝɨ ɰɟɧɬɪɚ
Ʉ ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɫɜɨɞɹɬɫɹ ɡɚɞɚɱɢ ɨ ɞɜɢɠɟɧɢɢ ɱɚɫɬɢɰɵ ɜ ɝɪɚɜɢɬɚɰɢɨɧɧɨɦ ɩɨɥɟ ɬɹɝɨɬɟɧɢɹ α = G M m ɢ ɜ ɤɭɥɨɧɨɜɫɤɨɦ ɩɨɥɟ ɩɪɢɬɹɠɟɧɢɹ α = Ze
ɗɮɮɟɤɬɢɜɧɵɣɩɨɬɟɧɰɢɚɥ ɜ ɩɨɥɟɜɢɞɚ ɢɦɟɟɬɫɥɟɞɭɸɳɢɣ ɜɢɞ ɪɢɫ
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8ɷɮɮ U
U ɷɮɮ r
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Uɦɢɧ U U U
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E
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Ɋɢɫ. 7.1.ɗɮɮɟɤɬɢɜɧɵɣ ɩɨɬɟɧɰɢɚɥ
Ʉɚɱɟɫɬɜɟɧɧɨɟ ɢɫɫɥɟɞɨɜɚɧɢɟ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɩɨɬɟɧɰɢɚɥɶɧɨɦ ɩɨɥɟ ɧɚ ɨɫ
ɧɨɜɟ ɞɚɧɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɦɨɠɟɬ ɛɵɬɶ ɩɪɨɜɟɞɟɧɨ ɫɩɨɫɨɛɨɦ ɤɨɬɨɪɵɣ ɪɚɫɫɦɚɬɪɢɜɚɥ ɫɹ ɜ ɩɭɧɤɬɟ ɉɪɢ ɡɧɚɱɟɧɢɢ ɩɨɥɧɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢE = E ɞɜɢɠɟɧɢɟ ɱɚɫ
ɬɢɰɵ ɩɪɨɢɫɯɨɞɢɬ ɩɨ ɨɤɪɭɠɧɨɫɬɢ ɪɚɞɢɭɫɚr ɉɪɢ ɷɧɟɪɝɢɢ ɱɚɫɬɢɰɵ E < E < ɞɜɢ
ɠɟɧɢɟɱɚɫɬɢɰɵɹɜɥɹɟɬɫɹɮɢɧɢɬɧɵɦɨɝɪɚɧɢɱɟɧɨɜ ɩɪɨɫɬɪɚɧɫɬɜɟȼ ɫɥɭɱɚɟ ɱɚɫɬɢɰɚ ɦɨɠɟɬ ɭɣɬɢ ɧɚ ɛɟɫɤɨɧɟɱɧɨɫɬɶ ɨɬ ɫɢɥɨɜɨɝɨ ɰɟɧɬɪɚ ɞɜɢɠɟɧɢɟ ɢɧɮɢɧɢɬɧɨɟ
ɍɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɩɨɥɭɱɚɟɬɫɹ ɩɨɞɫɬɚɧɨɜɤɨɣ ɞɚɧ ɧɨɝɨ ɜɵɪɚɠɟɧɢɹ ɜ ɨɛɳɟɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɢ ɩɨɫɥɟɞɭɸɳɢɦ ɟɝɨ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ ɉɪɟɞɟɥɵ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɢɧɬɟɝɪɢɪɨ ɜɚɧɢɹ rɦɢɧ ɢr ɝɞɟ rɦɢɧ ± ɛɥɢɠɚɣɲɟɟ ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɫɢɥɨɜɨɝɨ ɰɟɧɬɪɚ ɞɨ ɬɨɱɤɢ
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