Лекции по классической механике. Учебное пособие
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Лекция2.ОСНОВНЫЕПОСТУЛАТЫДИНАМИКИНЬЮТОНА
ɧɢɸ ɤ ɪɚɡɧɵɦ ɂɋɈ Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ ɡɚɤɨɧɵ ɩɪɢɪɨɞɵ ɧɟ ɞɨɥɠɧɵ ɦɟɧɹɬɶ ɫɜɨɸ ɮɨɪɦɭ ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɨɞɧɨɣ ɂɋɈ ɤ ɞɪɭɝɨɣ ɉɪɢ ɷɬɨɦ ɫɚɦɢ ɮɢɡɢɱɟɫɤɢɟ ɜɟɥɢɱɢɧɵ ɦɨɝɭɬ ɦɟɧɹɬɶɫɹ ɩɪɢ ɩɟɪɟɯɨɞɚɯ ɨɬ ɨɞɧɨɣ ɂɋɈ ɤ ɞɪɭɝɨɣ ɦɚɬɟɦɚɬɢɱɟ ɫɤɢɟ ɜɵɪɚɠɟɧɢɹ ɫɜɹɡɵɜɚɸɳɢɟ ɢɯ ɨɫɬɚɸɬɫɹ ɧɟɢɡɦɟɧɧɵɦɢ Ɂɚɤɨɧɵ ɩɪɢɪɨɞɵ ɢɦɟɸɬ ɬɚɤɢɦ ɨɛɪɚɡɨɦ ɨɞɢɧɚɤɨɜɵɣ ɜɢɞ ɜ ɪɚɡɧɵɯ ɂɋɈ
ɉɪɨɰɟɫɫɵ ɛɭɞɭɬ ɜɵɝɥɹɞɟɬɶ ɨɞɢɧɚɤɨɜɵɦ ɨɛɪɚɡɨɦ ɜ ɪɚɡɧɵɯ ɂɋɈ ɟɫɥɢ ɞɥɹ
ɧɢɯ ɪɟɚɥɢɡɨɜɚɬɶ ɨɞɢɧɚɤɨɜɵɟ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɉɪɢɧɰɢɩ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɭɫ
ɬɚɧɚɜɥɢɜɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɬɚɤɨɣ ɪɟɚɥɢɡɚɰɢɢ
Ɉɩɢɫɚɧɢɟ ɫɨɫɬɨɹɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɉɪɢɧɰɢɩ ɩɪɢɱɢɧɧɨɫɬɢ
Ɉɞɧɨɣ ɢɡ ɝɥɚɜɧɵɯ ɢ ɩɟɪɜɵɯ ɡɚɞɚɱ ɥɸɛɨɣ ɮɢɡɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɹɜɥɹɟɬɫɹ ɡɚɞɚɱɚ ɨɩɢɫɚɧɢɹ ɫɨɫɬɨɹɧɢɹ ɮɢɡɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɉɨɞ ɫɨɫɬɨɹɧɢɟɦ ɫɢɫɬɟɦɵ ɩɨɧɢɦɚɟɬɫɹ ɤɨɧɤɪɟɬɧɚɹ ɫɢɬɭɚɰɢɹ ɜ ɤɨɬɨɪɨɣ ɧɚɯɨɞɹɬɫɹ ɨɛɴɟɤɬɵ ɫɢɫɬɟɦɵ ȼ ɪɚɡɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɬɟɨɪɢɹɯ ɨɩɢɫɚɧɢɟ ɫɨɫɬɨɹɧɢɹ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɪɚɡɥɢɱ ɧɵɦɢ ɫɩɨɫɨɛɚɦɢ ȼɫɟ ɷɬɢ ɫɩɨɫɨɛɵ ɞɨɥɠɧɵ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɜɩɨɥɧɟ ɨɩɪɟɞɟɥɟɧ ɧɵɦ ɬɪɟɛɨɜɚɧɢɹɦ
Ɉɩɢɫɚɧɢɟ ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵ ɞɨɥɠɧɨ ɛɵɬɶ ɩɨ ɜɨɡɦɨɠɧɨɫɬɢ ɧɚɢɛɨɥɟɟ ɩɨɥɧɵɦ ɗɬɨ ɨɡɧɚɱɚɟɬ ɱɬɨ ɧɭɠɧɨ ɡɚɞɚɬɶ ɬɚɤɭɸ ɢɧɮɨɪɦɚɰɢɸ ɨ ɫɢɫɬɟɦɟ ɜ ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɤɨɬɨɪɚɹ ɩɨɡɜɨɥɹɥɚ ɛɵ ɨɩɪɟɞɟɥɹɬɶ ɜɫɟɜɨɡ ɦɨɠɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫɢɫɬɟɦɵ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɫɬɚɬɢɱɟɫɤɢɣ ɚɫ ɩɟɤɬ ɫɨɫɬɨɹɧɢɟ ɫɢɫɬɟɦɵ ɜ ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɞɨɥɠɧɨ ɩɪɟɞɨɩɪɟ ɞɟɥɹɬɶ ɟɟ ɫɨɫɬɨɹɧɢɟ ɜ ɩɨɫɥɟɞɭɸɳɢɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ ɞɢɧɚɦɢɱɟɫɤɢɣ ɚɫɩɟɤɬ ȼɬɨɪɨɟ ɬɪɟɛɨɜɚɧɢɟ ɜɵɪɚɠɚɟɬ ɫɨɛɨɣɩɪɢɧɰɢɩ ɩɪɢɱɢɧɧɨɫɬɢɜ ɫɚɦɨɣ ɨɛɳɟɣ ɟɝɨ ɮɨɪɦɭɥɢɪɨɜɤɟ
ɋɨɫɬɨɹɧɢɟ ɡɚɦɤɧɭɬɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ ɨɩɢɫɵɜɚɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɡɚɞɚɧɢɹ ɤɨɨɪɞɢɧɚɬ ɢ ɫɤɨɪɨɫɬɟɣ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɂɯ ɡɚ
ɞɚɧɢɟ ɜ ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɨɞɧɨɡɧɚɱɧɨ ɩɪɟɞɨɩɪɟɞɟɥɹɟɬ ɫɨɫɬɨɹɧɢɟ ɫɢɫ |
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ɬɟɦɵ ɜɨ ɜɫɟ ɩɨɫɥɟɞɭɸɳɢɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ ɬ ɟ ɷɜɨɥɸɰɢɸ ɫɢɫɬɟɦɵ |
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Ɂɞɟɫɶ ɱɟɪɟɡ r&a v&a ɨɛɨɡɧɚɱɟɧɵ ɪɚɞɢɭɫ ɜɟɤɬɨɪɵ ɢ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ
ɩɟɪɟɦɟɧɧɵɟ ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵ ɇɚɛɨɪ ɤɨɨɪɞɢɧɚɬ ɢ ɫɤɨɪɨɫɬɟɣ ɱɚɫɬɢɰ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ɨɩɪɟɞɟɥɹɟɬ ɜ ɱɚɫɬɧɨɫɬɢ ɢ ɭɫɤɨɪɟɧɢɹ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ
ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɢ ɜ ɩɨɫɥɟɞɭɸɳɢɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ |
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Ɏɭɧɤɰɢɢ f ɜ ɞɚɧɧɨɣ ɮɨɪɦɭɥɟ ɧɚɡɵɜɚɸɬɫɹ ɮɭɧɤɰɢɹɦɢ ɜɨɡɞɟɣɫɬɜɢɹ ɱɚɫ |
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ɬɢɰ ɫɢɫɬɟɦɵ ɧɚ ɱɚɫɬɢɰɭa >a = |
N @Ɉɧɢ ɫɜɹɡɵɜɚɸɬ ɦɟɠɞɭ ɫɨɛɨɣ ɤɢɧɟɦɚɬɢɱɟ |
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ɫɤɭɸ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ± ɭɫɤɨɪɟɧɢɟ
11
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
&
w&a t = d drta t
ɫ ɩɟɪɟɦɟɧɧɵɦɢ ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ȼ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ ɜɢɞ ɮɭɧɤ ɰɢɣ ɜɨɡɞɟɣɫɬɜɢɹ ɜɜɨɞɢɬɫɹ ɱɟɪɟɡ ɢɡɦɟɪɟɧɢɟ ɤɢɧɟɦɚɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɞɜɢɠɟɧɢɹ ± ɭɫɤɨɪɟɧɢɹ ɫɤɨɪɨɫɬɢ ɢ ɪɚɞɢɭɫ ɜɟɤɬɨɪɵ ɱɚɫɬɢɰ
Ɇɚɫɫɚ
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɦɤɧɭɬɭɸ ɫɢɫɬɟɦɭ ɢɡ ɞɜɭɯ ɱɚɫɬɢɰ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɞɪɭɝ ɫ ɞɪɭɝɨɦ Ɉɩɵɬ ɩɨɤɚɡɵɜɚɟɬ ɱɬɨ ɜɟɤɬɨɪɵ ɭɫɤɨɪɟɧɢɣ ɱɚɫɬɢɰ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɧɚɩɪɚɜɥɟɧɵ ɥɟɠɚɬ ɧɚ ɨɞɧɨɣ ɩɪɹɦɨɣ ɢ ɨɬɧɨɲɟɧɢɟ ɢɯ ɦɨɞɭɥɟɣ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɫɨ ɫɬɨɹɧɢɹ ɜ ɤɨɬɨɪɨɦ ɧɚɯɨɞɢɬɫɹ ɫɢɫɬɟɦɚ ɬ ɟ
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↑↓ w |
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Ɉɬɧɨɲɟɧɢɟɭɫɤɨɪɟɧɢɣɱɚɫɬɢɰ k ɧɚɡɵɜɚɟɬɫɹɢɯɨɬɧɨɫɢɬɟɥɶɧɨɣɦɚɫɫɨɣ ɤɨɬɨɪɭɸ ɨɛɨɡɧɚɱɢɦ m Ɉɱɟɜɢɞɧɨ ɱɬɨ m = 
m ȼɨɡɶɦɟɦ ɬɪɟɬɶɸ ɩɪɨɛɧɭɸ ɱɚɫɬɢɰɭ ɉɭɫɬɶ ɨɧɚ ɫɧɚɱɚɥɚ ɜɡɚɢɦɨɞɟɣɫɬɜɭɟɬ ɫ ɩɟɪɜɨɣ ɱɚɫɬɢɰɟɣ ɚ ɡɚɬɟɦ ɫɨ ɜɬɨ ɪɨɣȼ ɪɟɡɭɥɶɬɚɬɟɢɡɦɟɪɢɦɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟɨɬɧɨɫɢɬɟɥɶɧɵɟɦɚɫɫɵɢ ɧɚɣɞɟɦ ɱɬɨm m =m ɢɥɢm 
m = m Ⱦɚɧɧɨɟ ɨɬɧɨɲɟɧɢɟ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɢɞɚ ɩɪɨɛɧɨɣ ɱɚɫɬɢɰɵ Ⱦɚɥɟɟ ɥɨɝɢɱɧɨ ɨɬɧɨɫɢɬɟɥɶɧɭɸ ɦɚɫɫɭm ɩɪɟɞɫɬɚɜɢɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛ ɪɚɡɨɦ
m = m m
Ɂɞɟɫɶm m ± ɦɚɫɫɵ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɱɚɫɬɢɰ ɢɥɢ ɢɯ ɨɬɧɨɫɢɬɟɥɶɧɵɟ ɦɚɫ ɫɵ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɦɚɫɫɟ ɷɬɚɥɨɧɧɨɣ ɩɪɨɛɧɨɣ ɱɚɫɬɢɰɵ ɤɨɬɨɪɚɹ ɩɪɢɧɢɦɚɟɬɫɹ ɡɚ ɟɞɢɧɢɰɭ ɦɚɫɫɵ ɂɡ ɫɨɨɬɧɨɲɟɧɢɣ ɢ ɫɥɟɞɭɟɬ ɱɬɨ
m w& = −m w&
Ɇɚɫɫɚ ɱɚɫɬɢɰɵ ɹɜɥɹɟɬɫɹ ɟɟ ɜɧɭɬɪɟɧɧɟɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɜ ɬɨɦ ɫɦɵɫɥɟ ɱɬɨ ɨɧɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɫɨɫɬɨɹɧɢɹ ɜ ɤɨɬɨɪɨɦ ɧɚɯɨɞɢɬɫɹ ɱɚɫɬɢɰɚ
Ɇɚɫɫɚ ɱɚɫɬɢɰɵ ɨɩɪɟɞɟɥɟɧɧɚɹ ɫɩɨɫɨɛɨɦ ɨɩɢɫɚɧɧɵɦ ɜɵɲɟ ɜɨɨɛɳɟ ɝɨɜɨ ɪɹ ɧɚɡɵɜɚɟɬɫɹɢɧɟɪɬɧɨɣ ɦɚɫɫɨɣɈɧɚ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɩɨɫɨɛɧɨɫɬɶ ɱɚɫɬɢɰɵ ɫɨ ɯɪɚɧɹɬɶ ɯɚɪɚɤɬɟɪ ɫɜɨɟɝɨ ɞɜɢɠɟɧɢɹ ɩɪɢ ɜɨɡɞɟɣɫɬɜɢɢ ɧɚ ɧɟɟ ɑɟɦ ɛɨɥɶɲɟ ɦɚɫɫɚ ɱɚɫɬɢɰɵ ɬɟɦ ɫɥɨɠɧɟɟ ɢɡɦɟɧɢɬɶ ɯɚɪɚɤɬɟɪ ɟɟ ɞɜɢɠɟɧɢɹ
ɋɢɥɚ ɉɪɢɧɰɢɩ ɧɟɡɚɜɢɫɢɦɨɫɬɢ ɞɟɣɫɬɜɢɹ ɫɢɥ
ɋɢɥɨɣ ɞɟɣɫɬɜɭɸɳɟɣɧɚɱɚɫɬɢɰɭ a ɫɨɫɬɨɪɨɧɵɞɪɭɝɢɯɱɚɫɬɢɰɫɢɫɬɟɦɵ ɧɚɡɵɜɚɟɬɫɹ ɩɪɨɢɡɜɟɞɟɧɢɟ ɦɚɫɫɵ ɱɚɫɬɢɰɵ ɧɚ ɮɭɧɤɰɢɸ ɜɨɡɞɟɣɫɬɜɢɹ ɱɚɫɬɢɰ ɫɢɫ ɬɟɦɵ ɧɚ ɞɚɧɧɭɸ ɱɚɫɬɢɰɭ
& & & & & & = & & & & & & .
Fa r t r t rN t v t v t vN t t ma fa r t r t rN t v t v t vN t t
12
Лекция2.ОСНОВНЫЕПОСТУЛАТЫДИНАМИКИНЬЮТОНА
ɉɨɥɧɭɸ ɫɢɥɭ ɞɟɣɫɬɜɭɸɳɭɸ ɧɚ ɱɚɫɬɢɰɭ ɫɢɫɬɟɦɵa ɦɨɠɧɨ ɩɪɟɞɫɬɚ ɜɢɬɶ ɜ ɜɢɞɟ ɜɟɤɬɨɪɧɨɣ ɫɭɦɦɵ ɫɢɥ ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɞɚɧɧɭɸ ɱɚɫɬɢɰɭ ɫɨ ɫɬɨɪɨɧɵ ɞɪɭɝɢɯ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɬ ɟ
& N &
Fa =¦ Fab
b=
Ⱦɚɧɧɨɟɭɬɜɟɪɠɞɟɧɢɟɧɨɫɢɬɧɚɡɜɚɧɢɟɩɪɢɧɰɢɩɚɧɟɡɚɜɢɫɢɦɨɫɬɢɞɟɣɫɬɜɢɹ ɫɢɥɋɭɦɦɢɪɨɜɚɧɢɟ ɜ ɮɨɪɦɭɥɟ ɩɪɨɢɫɯɨɞɢɬ ɩɨ ɱɢɫɥɭ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɡɧɚɤ ©ɲɬɪɢɯª ɭ ɫɭɦɦɵ ɨɡɧɚɱɚɟɬ ɱɬɨ a ≠ b ɋɢɥɭ ɨɩɪɟɞɟɥɹɟɦɭɸ ɞɚɧɧɨɣ ɮɨɪɦɭɥɨɣ ɧɚɡɵɜɚɸɬɪɚɜɧɨɞɟɣɫɬɜɭɸɳɟɣɫɢɥ ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɱɚɫɬɢɰɭ ɫɢɫɬɟɦɵ
ȿɫɥɢɫɢɫɬɟɦɚɱɚɫɬɢɰ ɧɟɹɜɥɹɟɬɫɹɡɚɦɤɧɭɬɨɣɬɨɩɨɥɧɚɹɫɢɥɚ ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɱɚɫɬɢɰɭ a ɹɜɥɹɟɬɫɹ ɫɭɦɦɨɣ ɫɢɥ ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɱɚɫɬɢɰɭ ɫɨ ɫɬɨɪɨɧɵ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɢ ɱɚɫɬɢɰ ɧɟ ɜɯɨɞɹɳɢɯ ɜ ɷɬɭ ɫɢɫɬɟɦɭ ɬ ɟ
& N & & & &
Fa =¦ Fab +FaH[S =Fain +FaH[S
b=
Ɂɞɟɫɶ F&ain ± ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɱɚɫɬɢɰɭ ɫɢɫɬɟɦɵ F&aH[S ±ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹɜɧɟɲɧɢɯɫɢɥ F&ab ±ɫɢɥɚɞɟɣɫɬɜɭɸɳɚɹɧɚ
ɱɚɫɬɢɰɭ a ɫɨ ɫɬɨɪɨɧɵ ɱɚɫɬɢɰɵ b Ɋɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɜɧɟɲɧɢɯ ɫɢɥ ɫɱɢɬɚɟɬɫɹ ɢɡɜɟɫɬɧɨɣ ɡɚɞɚɧɧɨɣ
ɉɪɢɧɰɢɩɧɟɡɚɜɢɫɢɦɨɫɬɢɞɟɣɫɬɜɢɹɫɢɥɫɥɟɞɭɟɬɪɚɫɫɦɚɬɪɢɜɚɬɶɤɚɤɨɛɨɛ ɳɟɧɢɟ ɰɟɥɨɝɨ ɪɹɞɚ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɮɚɤɬɨɜ
Ɂɚɤɨɧɵ ɩɨɫɬɭɥɚɬɵ ɇɶɸɬɨɧɚ
ȼɞɚɧɧɨɦɩɭɧɤɬɟɦɵɫɮɨɪɦɭɥɢɪɭɟɦɬɪɢɨɫɧɨɜɧɵɯɩɨɫɬɭɥɚɬɚɤɥɚɫɫɢɱɟ ɫɤɨɣ ɦɟɯɚɧɢɤɢ ɇɶɸɬɨɧɚ ɤɨɬɨɪɵɟ ɧɚɡɵɜɚɸɬ ɡɚɤɨɧɚɦɢ ɇɶɸɬɨɧɚ
ɉɟɪɜɵɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ ɂɋɈ ɫɭɳɟɫɬɜɭɸɬ
ɑɬɨɛɵ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɜɬɨɪɨɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ ɩɟɪɟɩɢɲɟɦ ɮɨɪɦɭɥɭ
ɢɫɩɨɥɶɡɭɹ ɮɨɪɦɭɥɭ ɢ ɨɩɪɟɞɟɥɟɧɢɟ ɭɫɤɨɪɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɜɢɞɟ
d |
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Ɂɞɟɫɶ F&a ± ɪɚɜɧɨɞɟɣɫɬɜɭɸɳɚɹ ɜɧɭɬɪɟɧɧɢɯ ɢ ɜɧɟɲɧɢɯ ɫɢɥ ɞɟɣɫɬɜɭɸɳɢɯ
ɧɚ ɱɚɫɬɢɰɭ
& ȼɬɨɪɨɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚɉɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ Fa ɱɚɫɬɢɰɚ ɦɚɫɫɵ ma ɩɨɥɭ
ɱɚɟɬ ɭɫɤɨɪɟɧɢɟ
w&a = Fa ma
Ɍɪɟɬɢɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ ɮɨɪɦɭɥɢɪɭɟɬɫɹ ɧɚ ɨɫɧɨɜɟ ɮɨɪɦɭɥɵ
13
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
Ɍɪɟɬɢɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚȾɜɟ ɱɚɫɬɢɰɵ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɦɟɠɞɭ ɫɨɛɨɣ ɫɢ ɥɚɦɢ ɪɚɜɧɵɦɢ ɩɨ ɜɟɥɢɱɢɧɟ ɢ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɧɚɩɪɚɜɥɟɧɧɵɦɢ
F&ab = −F&ba
ɍɪɚɜɧɟɧɢɟ ɫ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɞɢɮ
ɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɢ ɧɚɡɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ ɞɜɢɠɟɧɢɹ
ɱɚɫɬɢɰɵ Ⱦɥɹ ɟɝɨ ɨɞɧɨɡɧɚɱɧɨɝɨ ɪɟɲɟɧɢɹ ɧɟɨɛɯɨɞɢɦɨ ɡɚɞɚɬɶ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ
Ɍɚɤɢɦɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɜ ɫɢɥɭ ɩɪɢɧɰɢɩɚ ɩɪɢɱɢɧɧɨɫɬɢ ɜɵɫɬɭɩɚɸɬ ɧɚ
ɱɚɥɶɧɨɟ ɩɨɥɨɠɟɧɢɟ ɱɚɫɬɢɰɵ ɢ ɟɟ ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ Ɍɚɤ ɤɚɤ ɜ ɫɢɥɭ ɨɞɧɨɪɨɞ
ɧɨɫɬɢ ɜɪɟɦɟɧɢ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɦɨɠɧɨ ɜɵɛɪɚɬɶ ɩɪɨɢɡɜɨɥɶɧɵɦ ɨɛɪɚ
ɡɨɦ ɬɨ ɞɨɫɬɚɬɨɱɧɨ ɡɚɞɚɬɶ ɩɨɥɨɠɟɧɢɟ ɢ ɫɤɨɪɨɫɬɶ ɱɚɫɬɢɰɵ ɜ ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ
ɜɪɟɦɟɧɢ Ɉɫɧɨɜɧɚɹ ɡɚɞɚɱɚ ɞɢɧɚɦɢɤɢ ɱɚɫɬɢɰɵɉɨ ɡɚɞɚɧɧɨɣ ɫɢɥɟ ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ
ɱɚɫɬɢɰɭ ɢɡɜɟɫɬɧɨɣ ɦɚɫɫɟ ɱɚɫɬɢɰɵ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɢ ɩɨɥɨɠɟɧɢɸ ɱɚɫɬɢɰɵ ɬɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵr&a = r&a t
Ⱦɚɧɧɚɹ ɡɚɞɚɱɚ ɪɟɲɚɟɬɫɹ ɨɞɧɨɡɧɚɱɧɨ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ ɭɪɚɜɧɟɧɢɹ ȼ ɷɬɨɦ ɧɚɯɨɞɢɬ ɫɜɨɟ ɨɬɪɚɠɟɧɢɟ ɩɪɢɧɰɢɩ ɩɪɢɱɢɧɧɨɫɬɢ ɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ Ɋɚɫɫɦɨɬɪɢɦ ɩɪɢɦɟɪɵ ɧɚ ɪɟɲɟɧɢɟ ɨɫɧɨɜɧɨɣ ɡɚɞɚɱɢ ɦɟɯɚɧɢɤɢ ɢ ɨɛɪɚɬɧɨɣ ɞɥɹ ɧɟɟ ɡɚɞɚɱɢ ɩɨ ɡɚɤɨɧɭ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɧɚɣɬɢ ɫɢɥɭ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɤɨɬɨ
ɪɨɣ ɩɪɨɢɫɯɨɞɢɬ ɞɜɢɠɟɧɢɟ
ɉɪɢɦɟɪɑɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ ɜ ɩɥɨɫɤɨɫɬɢ x y ɩɨ ɡɚɤɨɧɭ x = bt y = at − ct ɇɚɣɬɢ ɫɢɥɭ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɤɨɬɨɪɨɣ ɩɪɨɢɫɯɨɞɢɬ ɞɜɢɠɟɧɢɟ
ɉɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɹ ɧɚɯɨɞɢɦ ɤɚɤ ɜɬɨɪɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ ɜɪɟɦɟɧɢ ɨɬ ɤɨ ɨɪɞɢɧɚɬ ɱɚɫɬɢɰwx = wy = − c Ⱦɚɥɟɟ ɩɨ ɮɨɪɦɭɥɟ ɧɚɯɨɞɢɦ ɩɪɨɟɤɰɢɢ ɫɢ
ɥɵ Fx = m wx = Fy = mwy = − m c Ɍɚɤɢɦɨɛɪɚɡɨɦɱɚɫɬɢɰɚɫɨɜɟɪɲɚɟɬɞɜɢɠɟɧɢɟ
ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɩɨɫɬɨɹɧɧɨɣ ɫɢɥɵ ɤɨɬɨɪɚɹ ɧɚɩɪɚɜɥɟɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɧɚɩɪɚɜ ɥɟɧɢɸ ɨɫɢy
ɉɪɢɦɟɪȿɫɥɢ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɡɚɞɚɧɨ ɜ ɩɨɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯr ϕ ɪɢɫ ɬɨ ɞɥɹ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ ɢɦɟɟɦ
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v& = |
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ɭɫɤɨɪɟɧɢɟ ɱɚɫɬɢɰɵ |
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Ⱦɥɹ ɩɪɨɟɤɰɢɣ ɫɢɥ ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ ɱɚɫɬɢɰɵ r ɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɟɤ ɧɟɦɭɧɚɩɪɚɜɥɟɧɢɟɜ ɫɬɨɪɨɧɭɭɜɟɥɢɱɟɧɢɹɭɝɥɚ ϕ ɦɨɠɧɨ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɡɚɩɢɫɚɬɶ
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Лекция2.ОСНОВНЫЕПОСТУЛАТЫДИНАМИКИНЬЮТОНА
U
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[
Ɋɢɫ. 2.1.ɉɨɥɹɪɧɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɊɢɫ. 2.1. ɚɤɨɨɪɞɢɧɚɬ
ɉɪɢɦɟɪɁɚɤɨɧ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɩɨɥɹɪɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɢɦɟ ɟɬ ɜɢɞr = at ϕ = bt ɝɞɟa b ± ɩɨɫɬɨɹɧɧɵɟ ɇɚɣɬɢ ɫɢɥɭ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɤɨɬɨɪɨɣ
ɞɜɢɠɟɬɫɹ ɱɚɫɬɢɰɚ ɉɨ ɮɨɪɦɭɥɚɦ ɩɪɢɦɟɪɚ ɧɚɯɨɞɢɦ Fr = −mab t Fϕ = ma b Ɇɨɞɭɥɶ ɫɢɥɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɚɤF = Fr + Fϕ = mab
b t + a
ɉɪɢɦɟɪɇɚɣɬɢ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɨɞɧɨɪɨɞɧɨɦ ɩɨɥɟ ɫɢɥɵ ɬɹ ɠɟɫɬɢ Ɂɟɦɥɢ ɤɨɬɨɪɚɹ ɛɪɨɲɟɧɚ ɩɨɞ ɭɝɥɨɦ ɤ ɝɨɪɢɡɨɧɬɭα ɫ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ v& ɪɢɫ
ɉɪɨɟɤɰɢɢ ɫɢɥɵ ɞɟɣɫɬɜɭɸɳɟɣ |
ɧɚ |
ɱɚɫɬɢɰɭ Fx = Fy = Fz = − mg |
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ɢ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɢɦɟɸɬ ɜɢɞ |
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Ɇɚɫɫɭ ɱɚɫɬɢɰɵ ɜ ɞɚɧɧɵɯ ɭɪɚɜɧɟɧɢɹɯ ɦɨɠɧɨ ɫɨɤɪɚɬɢɬɶ ɗɬɨ ɨɛɫɬɨɹɬɟɥɶɫɬ
ɜɨ ɭɤɚɡɵɜɚɟɬ ɧɚ ɬɨ ɱɬɨ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɜ ɩɨɥɟ ɫɢɥɵ ɬɹɠɟɫɬɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɟɟ
ɦɚɫɫɵ ɉɪɨɢɧɬɟɝɪɢɪɨɜɚɜ ɞɜɚɠɞɵ ɞɚɧɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɩɨɥɭɱɢɦ
x = C t + C y = C t + C |
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ɝɞɟCk ± ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɤɨɬɨɪɵɟ ɧɚɯɨɞɢɦ ɢɡ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ
x = y = z = dx dt |
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t = = v FRVα |
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Ɉɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɚɟɦ |
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x = v t FRVα y = z = − |
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Ɋɢɫ. 2.2ɤ ɩɪɢɦɟɪɭɊɢɫ. 2.2ɤ ɩɪɢɦɟɪɭ |
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15
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
ɉɪɢɦɟɪ ɑɚɫɬɢɰɚɦɚɫɫɵ m ɩɚɞɚɟɬɜɟɪɬɢɤɚɥɶɧɨɜɧɢɡɩɨɞɞɟɣɫɬɜɢɟɦ ɫɢɥɵ ɬɹɠɟɫɬɢmgɢ ɫɢɥɵ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡɞɭɯɚ R = α v ɋɱɢɬɚɹ ɱɬɨ ɧɚɱɚɥɶɧɚɹ ɤɨɨɪɞɢɧɚɬɚ z = ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ v = ɨɫɶ z ɧɚɩɪɚɜɥɟɧɚ ɜɟɪɬɢɤɚɥɶɧɨ
ɜɧɢɡ ɧɚɣɬɢ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ
ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɢɦɟɟɬ ɜɢɞ
m |
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ɢɥɢ
dvdt = g − α v
ɉɨɫɥɟɞɧɟɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɪɟɲɚɟɬɫɹ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ
dv |
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g − α v |
ɂɧɬɟɝɪɢɪɭɟɦ ɞɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɢ ɩɨɥɭɱɚɟɦ
OQg − α v = −α v + C
ɉɨɫɬɨɹɧɧɭɸ C ɨɩɪɟɞɟɥɹɟɦ ɢɡ ɧɚɱɚɥɶɧɨɝɨ ɭɫɥɨɜɢɹ v = ȼ ɪɟɡɭɥɶɬɚɬɟ
ɧɚɯɨɞɢɦ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ v = αg ( − e−α t )
ɉɨɫɥɟɞɧɟɟ ɭɪɚɜɧɟɧɢɟ ɩɪɟɞɫɬɚɜɢɦ ɜ ɜɢɞɟ dz = αg ( − e−α t )dt
ɉɪɨɢɧɬɟɝɪɢɪɭɟɦ ɟɝɨ
z = αg t + αg e−α t + C
ɉɨɫɬɨɹɧɧɚɹC ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɧɚɱɚɥɶɧɨɝɨ ɭɫɥɨɜɢɹ z = Ɉɤɨɧɱɚɬɟɥɶɧɨ
ɩɨɥɭɱɚɟɦ
z = αg t − αg ( − e−α t )
Ɉɞɧɨɪɨɞɧɨɫɬɶ ɢɡɨɬɪɨɩɧɨɫɬɶ ɩɪɨɫɬɪɚɧɫɬɜɚ ɨɞɧɨɪɨɞɧɨɫɬɶ ɜɪɟɦɟɧɢ ɩɪɢɧ
ɰɢɩ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɧɚɤɥɚɞɵɜɚɸɬ ɪɹɞ ɨɝɪɚɧɢɱɟɧɢɣ ɧɚ ɡɚɜɢɫɢɦɨɫɬɶ ɫɢɥɵ ɜɡɚɢ
ɦɨɞɟɣɫɬɜɢɹ ɱɚɫɬɢɰ ɨɬ ɢɯ ɤɨɨɪɞɢɧɚɬ ɫɤɨɪɨɫɬɟɣ ɢ ɜɪɟɦɟɧɢ Ɋɚɫɫɦɨɬɪɢɦ ɡɚɦɤɧɭɬɭɸ ɫɢɫɬɟɦɭ ɢɡ ɞɜɭɯ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɛɟɫɫɬɪɭɤ
ɬɭɪɧɵɯ ɱɚɫɬɢɰ ɉɭɫɬɶ ɱɚɫɬɢɰɵ ɩɨɤɨɹɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɪɭɝ ɞɪɭɝɚ ȼ ɫɢɥɭ ɨɞɧɨ
ɪɨɞɧɨɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɫɢɥɚ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɞɨɥɠɧɚ ɡɚɜɢɫɟɬɶ ɨɬ ɪɚɡɧɨɫɬɢ ɪɚɞɢ ɭɫ ɜɟɤɬɨɪɨɜ ɱɚɫɬɢɰ r& = r& − r& ɬɚɤ ɤɚɤ ɢɦɟɧɧɨ ɷɬɚ ɜɟɥɢɱɢɧɚ ɹɜɥɹɟɬɫɹ ɢɧɜɚɪɢɚɧɬ ɧɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ ɩɪɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɦ ɫɞɜɢ ɝɟ ȼ ɫɢɥɭ ɢɡɨɬɪɨɩɧɨɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɫɥɟɞɭɟɬ ɱɬɨ ɟɞɢɧɫɬɜɟɧɧɵɦ ɜɵɞɟɥɟɧɧɵɦ ɧɚɩɪɚɜɥɟɧɢɟɦ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɹɜɥɹɟɬɫɹ ɧɚɩɪɚɜɥɟɧɢɟ ɨɩɪɟɞɟɥɹɟɦɨɟ ɟɞɢɧɢɱɧɵɦ
16 |
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Лекция2.ОСНОВНЫЕПОСТУЛАТЫДИНАМИКИНЬЮТОНА
ɜɟɤɬɨɪɨɦ ɧɚɩɪɚɜɥɟɧɧɵɦ ɩɨ ɩɪɹɦɨɣ ɫɨɟɞɢɧɹɸɳɟɣ ɱɚɫɬɢɰɵ r&
r Ɉɬɫɸɞɚ ɫɥɟ ɞɭɟɬ ɱɬɨ ɫɢɥɚ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɞɨɥɠɧɚ ɢɦɟɬɶ ɜɢɞ
F& = F r rr
ɉɪɟɞɩɨɥɨɠɢɦ ɬɟɩɟɪɶ ɱɬɨ ɱɚɫɬɢɰɵ ɞɜɢɠɭɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɪɭɝ ɞɪɭɝɚ ȼ ɫɢɥɭ ɩɪɢɧɰɢɩɚ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɫɢɥɚ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɞɨɥɠɧɚ ɡɚɜɢɫɟɬɶ ɨɬ ɨɬ ɧɨɫɢɬɟɥɶɧɨɣ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰv& = v& − v& ɤɨɬɨɪɚɹ ɢɧɜɚɪɢɚɧɬɧɚ ɨɬɧɨɫɢ ɬɟɥɶɧɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ Ƚɚɥɢɥɟɹ ɋɥɟɞɭɟɬ ɨɛɪɚɬɢɬɶ ɜɧɢɦɚɧɢɟ ɧɚ ɬɨ ɨɛɫɬɨɹɬɟɥɶ ɫɬɜɨ ɱɬɨ ɞɚɧɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ ɫɩɪɚɜɟɞɥɢɜɨ ɬɨɥɶɤɨ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ ɱɬɨ ɜɡɚɢ ɦɨɞɟɣɫɬɜɢɟ ɦɟɠɞɭ ɱɚɫɬɢɰɚɦɢ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɦɝɧɨɜɟɧɧɨ ɧɟɬ ɷɮɮɟɤɬɨɜ ɡɚɩɚɡ ɞɵɜɚɧɢɹ Ⱦɚɧɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ ɫɩɪɚɜɟɞɥɢɜɨ ɜ ɤɥɚɫɫɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ ɧɨ ɬɟɪɹ ɟɬ ɫɜɨɸ ɫɢɥɭ ɜ ɬɟɨɪɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ
ȼ ɫɢɥɭ ɨɞɧɨɪɨɞɧɨɫɬɢ ɜɪɟɦɟɧɢ ɫɢɥɚ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɦɟɠɞɭ ɱɚɫɬɢɰɚɦɢ ɧɟ
ɦɨɠɟɬ ɹɜɧɵɦ ɨɛɪɚɡɨɦ ɡɚɜɢɫɟɬɶ ɨɬ ɜɪɟɦɟɧɢ ɋɢɥɵ ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɵ ɜ ɜɢɞɟ ɧɚɡɵɜɚɸɬɫɹɰɟɧ
ɬɪɚɥɶɧɵɦɢ ɫɢɥɚɦɢ Ɉɱɟɜɢɞɧɨ ɱɬɨ ɞɥɹ ɬɚɤɢɯ ɫɢɥ ɜɵɩɨɥɧɹɟɬɫɹ ɬɪɟɬɢɣ ɡɚɤɨɧ ɇɶɸɬɨɧɚ
17
ɅȿɄɐɂə ɈɋɇɈȼɇɕȿ ɌȿɈɊȿɆɕ ȾɂɇȺɆɂɄɂ ɑȺɋɌɂɐɕ
Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ Ɂɚɤɨɧ ɢɧɟɪɰɢɢ
ɂɦɩɭɥɶɫɨɦ ɱɚɫɬɢɰɵ ɧɚɡɵɜɚɟɬɫɹ ɩɪɨɢɡɜɟɞɟɧɢɟ ɦɚɫɫɵ ɱɚɫɬɢɰɵ ɧɚ ɟɟ
ɫɤɨɪɨɫɬɶ |
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ɋɨɝɥɚɫɧɨ ɜɬɨɪɨɦɭ ɡɚɤɨɧɭ ɇɶɸɬɨɧɚ |
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Ɂɞɟɫɶ ɦɵ ɢɫɩɨɥɶɡɨɜɚɥɢ ɬɨɬ ɮɚɤɬ ɱɬɨ ɦɚɫɫɚ ɱɚɫɬɢɰɵ ɟɫɬɶ ɜɟɥɢɱɢɧɚ ɩɨɫɬɨ ɹɧɧɚɹɢ ɟɟɦɨɠɧɨɜɧɟɫɬɢɩɨɞɡɧɚɤɩɪɨɢɡɜɨɞɧɨɣɂɫɩɨɥɶɡɭɹɨɩɪɟɞɟɥɟɧɢɟɢɦ ɩɭɥɶɫɚ ɱɚɫɬɢɰɵ ɩɟɪɟɩɢɲɟɦ ɩɨɫɥɟɞɧɟɟ ɪɚɜɟɧɫɬɜɨ ɜ ɜɢɞɟ
F& = ddtp
Ⱦɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢ ɜɵɪɚɠɚɟɬ ɫɨɛɨɣɬɟɨɪɟɦɭ ɨɛ ɢɡɦɟɧɟɧɢɢ ɢɦɩɭɥɶɫɚɱɚɫɬɢɰɵ ɫɤɨɪɨɫɬɶɢɡɦɟɧɟɧɢɹɢɦɩɭɥɶɫɚɱɚɫɬɢɰɵɪɚɜɧɚɫɢɥɟɞɟɣɫɬ ɜɭɸɳɟɣ ɧɚ ɱɚɫɬɢɰɭ
Ɋɚɫɫɦɨɬɪɢɦ ɫɜɨɛɨɞɧɭɸ ɱɚɫɬɢɰɭ ɬ ɟ ɱɚɫɬɢɰɭ ɧɚ ɤɨɬɨɪɭɸ ɧɟ ɞɟɣɫɬɜɭɸɬ
ɫɢɥɵ ɢɥɢ ɞɟɣɫɬɜɢɟ ɫɢɥ ɧɚ ɤɨɬɨɪɭɸ ɫɤɨɦɩɟɧɫɢɪɨɜɚɧɨ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɞɥɹ ɫɜɨ ɛɨɞɧɨɣ ɱɚɫɬɢɰɵ F = ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ p& = const ɗɬɨ ɨɡɧɚɱɚɟɬ ɱɬɨ ɫɜɨɛɨɞɧɚɹ ɱɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ ɢ ɪɚɜɧɨɦɟɪɧɨ ɥɢɛɨ ɩɨɤɨɢɬɫɹɡɚɤɨɧ ɢɧɟɪɰɢɢ Ɂɚɤɨɧ ɢɧɟɪɰɢɢ ɹɜɥɹɟɬɫɹ ɫɥɟɞɫɬɜɢɟɦ ɨɞɧɨɪɨɞɧɨɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɢ ɜɪɟɦɟɧɢ ɢɡɨɬɪɨɩɧɨɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɢ ɩɪɢɧɰɢɩɚ ɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢ ɇɚ ɫɚɦɨɦ
ɞɟɥɟ ɭɫɤɨɪɟɧɢɟ ɱɚɫɬɢɰɵ
w& = f r& v* t
ɟɫɬɶ ɮɭɧɤɰɢɹ ɜɪɟɦɟɧɢ ɢ ɟɟ ɤɨɨɪɞɢɧɚɬ ɢ ɫɤɨɪɨɫɬɢ ɜ ɬɨɬ ɠɟ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ȼ ɫɢɥɭ ɨɞɧɨɪɨɞɧɨɫɬɢ ɜɪɟɦɟɧɢ ɷɬɚ ɮɭɧɤɰɢɹ ɞɥɹ ɫɜɨɛɨɞɧɨɣ ɱɚɫɬɢɰɵ ɧɟ ɞɨɥɠɧɚ ɡɚɜɢɫɟɬɶ ɨɬ ɜɪɟɦɟɧɢ ɜ ɫɢɥɭ ɨɞɧɨɪɨɞɧɨɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɨɧɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɤɨ ɨɪɞɢɧɚɬɱɚɫɬɢɰɵɉɪɢɧɰɢɩɨɬɧɨɫɢɬɟɥɶɧɨɫɬɢɬɪɟɛɭɟɬɱɬɨɛɵɜɵɩɨɥɧɹɥɨɫɶɪɚ
ɜɟɧɫɬɜɨ
f v* = f v* − V
ɉɨɫɤɨɥɶɤɭ ɜɟɤɬɨɪV ɹɜɥɹɟɬɫɹ ɩɪɨɢɡɜɨɥɶɧɵɦ ɬɨ ɞɚɧɧɚɹ ɮɭɧɤɰɢɹ ɞɨɥɠɧɚ ɛɵɬɶ ɩɨɫɬɨɹɧɧɵɦ ɜɟɤɬɨɪɨɦ ɤɨɬɨɪɵɣ ɜ ɫɢɥɭ ɢɡɨɬɪɨɩɧɨɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɞɨɥɠɟɧ ɛɵɬɶ ɪɚɜɟɧ ɧɭɥɸ
Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ
Ɇɨɦɟɧɬɨɦ ɫɢɥɵɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ ɰɟɧɬɪɚ ɧɚɡɵɜɚɟɬɫɹ ɜɟɤɬɨɪɧɨɟ ɩɪɨ ɢɡɜɟɞɟɧɢɟ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ ɱɚɫɬɢɰɵ ɧɚ ɜɟɤɬɨɪ ɫɢɥɵ
M = >r& F @
Ɇɨɦɟɧɬɨɦɢɦɩɭɥɶɫɚɨɪɛɢɬɚɥɶɧɵɦɦɨɦɟɧɬɨɦ ɱɚɫɬɢɰɵɧɚɡɵɜɚɟɬɫɹɜɟɤ ɬɨɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ ɱɚɫɬɢɰɵ ɧɚ ɟɟ ɢɦɩɭɥɶɫ
18
Лекция3.ОСНОВНЫЕТЕОРЕМЫДИНАМИКИЧАСТИЦЫ
l = >r* p&@ = m>r& v&@
ɇɚɣɞɟɦ ɫɤɨɪɨɫɬɶ ɢɡɦɟɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ |
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dl& |
d & & |
dr& & |
& dp& |
& dp& |
dt = dt >r p@ = > dt p@+ >r dt @ = >r dt @
ɝɞɟ ɩɪɢ ɩɟɪɟɯɨɞɟ ɤ ɩɨɫɥɟɞɧɟɦɭ ɪɚɜɟɧɫɬɜɭ ɢɫɩɨɥɶɡɨɜɚɧɨ ɱɬɨ ɜɟɤɬɨɪɵ ɫɤɨɪɨɫɬɢ ɢ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ ɩɚɪɚɥɥɟɥɶɧɵ ɢ ɢɯ ɜɟɤɬɨɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɪɚɜɧɨ ɧɭɥɸ ɂɫɩɨɥɶɡɭɹ ɬɟɨɪɟɦɭ ɨɛ ɢɡɦɟɧɟɧɢɢ ɢɦɩɭɥɶɫɚ ɢ ɨɩɪɟɞɟɥɟɧɢɟ ɩɪɢɯɨɞɢɦ
ɤɬɟɨɪɟɦɟ ɨɛ ɢɡɦɟɧɟɧɢɢ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ
&
ddtl = M&
ɫɤɨɪɨɫɬɶɢɡɦɟɧɟɧɢɹɦɨɦɟɧɬɚɢɦɩɭɥɶɫɚɱɚɫɬɢɰɵɪɚɜɧɚɦɨɦɟɧɬɭɫɢɥɵɩɪɢɥɨ ɠɟɧɧɨɣ ɤ ɱɚɫɬɢɰɟ
ȿɫɥɢ ɦɨɦɟɧɬ ɫɢɥɵ ɩɪɢɥɨɠɟɧɧɨɣ ɤ ɱɚɫɬɢɰɟ ɪɚɜɟɧ ɧɭɥɸ ɬɨ ɦɨɦɟɧɬ ɢɦ
ɩɭɥɶɫɚ ɱɚɫɬɢɰɵ ɟɫɬɶ ɜɟɥɢɱɢɧɚ ɩɨɫɬɨɹɧɧɚɹ ȼ ɱɚɫɬɧɨɫɬɢ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɫɜɨ
ɛɨɞɧɨɣ ɱɚɫɬɢɰɵ ɫɨɯɪɚɧɹɟɬɫɹ
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ
Ɋɚɛɨɬɚ ɦɨɳɧɨɫɬɶ Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɱɚɫɬɢɰɵ
Ʉɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɟɣɱɚɫɬɢɰɵ ɧɚɡɵɜɚɟɬɫɹ ɜɟɥɢɱɢɧɚ
T = mv
ɗɥɟɦɟɧɬɚɪɧɨɣ ɪɚɛɨɬɨɣɧɚɡɵɜɚɟɬɫɹ ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɫɢɥɵ ɞɟɣɫɬ ɜɭɸɳɟɣ ɧɚ ɱɚɫɬɢɰɭ ɧɚ ɜɟɤɬɨɪ ɷɥɟɦɟɧɬɚɪɧɨɝɨ ɩɟɪɟɦɟɳɟɧɢɹ ɱɚɫɬɢɰɵ
δA = Fdr&
Ɋɚɛɨɬɚ ɫɢɥɵ ɩɨ ɩɟɪɟɦɟɳɟɧɢɸ ɱɚɫɬɢɰɵ ɢɡ ɩɨɥɨɠɟɧɢɹ ɜ ɩɨɥɨɠɟɧɢɟ ɨɩ
ɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɫɭɦɦɚ ɷɥɟɦɟɧɬɚɪɧɵɯ ɪɚɛɨɬ ɧɚ ɞɚɧɧɨɦ ɩɟɪɟɦɟɳɟɧɢɢ ɬ ɟ
= & &
A → ³Fdr
ɉɪɟɨɛɪɚɡɭɟɦ ɮɨɪɦɭɥɭ ɞɥɹ ɪɚɛɨɬɵ ɫɢɥɵ ɫ ɭɱɟɬɨɦ ɜɬɨɪɨɝɨ ɡɚɤɨɧɚ ɇɶɸɬɨɧɚ ɢ ɨɩɪɟɞɟɥɟɧɢɹ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ
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dv& & |
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mv |
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A → = ³m |
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dr |
= ³mvdv |
= ³d |
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= ³dT = T − T |
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dt |
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ɬ ɟ
δA = dT
A → = T − T
Ⱦɚɧɧɵɟ ɮɨɪɦɭɥɵ ɜɵɪɚɠɚɸɬ ɫɨɛɨɣɬɟɨɪɟɦɭ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢɱɚɫɬɢɰɵ ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɣ ɢ ɢɧɬɟɝɪɚɥɶɧɨɣ ɮɨɪɦɟ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
Ɇɨɳɧɨɫɬɶɸɧɚɡɵɜɚɸɬ ɪɚɛɨɬɭ ɫɨɜɟɪɲɟɧɧɭɸ ɡɚ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ
.
19
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
N = δA = F&dr& = F&v&
dt dt
ɂɫɩɨɥɶɡɭɹ ɩɟɪɜɭɸ ɮɨɪɦɭɥɭ ɩɨɥɭɱɢɦ ɮɨɪɦɭɥɭ ɞɥɹ ɢɡɦɟɧɟɧɢɹ ɤɢ
ɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɱɚɫɬɢɰɵ ɜ ɜɢɞɟ
dTdt = F&v&
ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɫɢɥɚ ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ
Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ
ɋɢɥɚ ɧɚɡɵɜɚɟɬɫɹɩɨɬɟɧɰɢɚɥɶɧɨɣɟɫɥɢ ɪɚɛɨɬɚ ɷɬɨɣ ɫɢɥɵ ɩɨ ɩɟɪɟɦɟɳɟɧɢɸ ɱɚɫɬɢɰɵ ɢɡ ɩɨɥɨɠɟɧɢɹ ɜ ɩɨɥɨɠɟɧɢɟ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɢɞɚ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟ ɧɢɹ ɱɚɫɬɢɰɵ ɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɥɶɤɨ ɧɚɱɚɥɶɧɵɦ ɢ ɤɨɧɟɱɧɵɦ ɩɨɥɨɠɟɧɢɹɦɢ ɱɚɫ ɬɢɰɵ Ɋɚɛɨɬɚ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɫɢɥɵ ɩɨ ɡɚɦɤɧɭɬɨɦɭ ɤɨɧɬɭɪɭ ɪɚɜɧɚ ɧɭɥɸ ɬɟɨɪɟɦɚ ɨ ɰɢɪɤɭɥɹɰɢɢ
³Fdr&=
L
ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɫɢɥɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɢ ɦɨɠɟɬ
ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ ɜɢɞɟ
F* = − ∂∂Ur& = −gradU
Ɂɞɟɫɶ U r& t ±ɫɤɚɥɹɪɧɚɹɮɭɧɤɰɢɹɤɨɨɪɞɢɧɚɬɱɚɫɬɢɰɵɢ ɜɪɟɦɟɧɢɧɚɡɵ ɜɚɟɦɚɹɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɟɣɂɡ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɫɥɟɞɭɟɬ ɱɬɨ ɨɧɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɧɟɤɨɬɨɪɨɣ ɩɨɫɬɨɹɧɧɨɣ ɜ ɤɨɬɨɪɭɸ ɜ ɤɚɱɟɫɬɜɟ ɩɚɪɚɦɟɬɪɚ ɦɨɠɟɬ ɜɯɨɞɢɬɶ ɜɪɟɦɹ ȼɵɛɨɪ ɷɬɨɣ ɩɨɫɬɨɹɧɧɨɣ ɧɚɡɵɜɚɟɬɫɹ ɧɨɪɦɢɪɨɜɤɨɣ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ
ȼɵɱɢɫɥɢɦɷɥɟɦɟɧɬɚɪɧɭɸɪɚɛɨɬɭɩɨɬɟɧɰɢɚɥɶɧɨɣɫɢɥɵɩɨɩɟɪɟɦɟɳɟɧɢɸ
ɱɚɫɬɢɰɵ
δA = F&dr& = − ∂∂Ur& dr& = − ∂∂Ur& dr& − ∂∂Ut dt + ∂∂Ut dt = −dU + ∂∂Ut
δA = −dU + ∂∂Ut
Ɏɨɪɦɭɥɚ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɬɟɨɪɟɦɭ ɨɛ ɢɡɦɟɧɟɧɢɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ
ɷɧɟɪɝɢɢ ɱɚɫɬɢɰɵ ɉɪɢɦɟɪȼɵɱɢɫɥɢɦ ɪɚɛɨɬɭ ɫɢɥɵ ɬɹɠɟɫɬɢF = mg&
A = ³F&dr& |
= ³(Fx dx + Fy dy + Fz dz)= ³Fz dz = −³mg dz = −mg(z − z )= mgh |
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ɝɞɟ h = z − z ± ɜɟɥɢɱɢɧɚ ɜɟɪɬɢɤɚɥɶɧɨɝɨ ɫɦɟɳɟɧɢɹ ɱɚɫɬɢɰɵ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɪɚ ɛɨɬɚ ɫɢɥɵ ɬɹɠɟɫɬɢ ɪɚɜɧɚ ɩɪɨɢɡɜɟɞɟɧɢɸ ɫɢɥɵ ɧɚ ɜɟɪɬɢɤɚɥɶɧɨɟ ɫɦɟɳɟɧɢɟ ɱɚɫɬɢ ɰɵ Ɋɚɛɨɬɚ ɫɢɥɵ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɢɞɚ ɬɪɚɟɤɬɨɪɢɢ ɩɨ ɤɨɬɨɪɨɣ ɩɟɪɟɦɟɳɚɟɬɫɹ ɱɚɫ ɬɢɰɚ ɬ ɟ ɫɢɥɚ ɬɹɠɟɫɬɢ ɹɜɥɹɟɬɫɹ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɫɢɥɨɣ ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɜ ɩɨɥɟ ɫɢɥɵ ɬɹɠɟɫɬɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟU = mgz + const
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