Лекции по классической механике. Учебное пособие
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Лекция3.ОСНОВНЫЕТЕОРЕМЫДИНАМИКИЧАСТИЦЫ
ɉɪɢɦɟɪȼɵɱɢɫɥɢɦ ɪɚɛɨɬɭ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢF = −kx
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A = ³F dx = −³kx dx = − |
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ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɞɜɢɠɭɳɟɣɫɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɭɩɪɭɝɨɣ ɫɢɥɵ
U= k x + const
ɉɪɢɦɟɪɊɚɛɨɬɚ ɫɢɥɵ ɧɶɸɬɨɧɨɜɚ ɬɹɝɨɬɟɧɢɹF& = −G |
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A = ³F dr |
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ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɦɚɫɫɵm ɜ ɩɨɥɟ ɫɢɥɵ ɬɹɝɨɬɟɧɢɹ
U = −G Mrm + const
Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɩɨɥɧɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ
ɉɭɫɬɶ ɫɢɥɚ ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɱɚɫɬɢɰɭ ɹɜɥɹɟɬɫɹ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɧɚɪɹɞɭ ɫ ɬɟɨɪɟɦɨɣ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɩɪɚɜɟɞ ɥɢɜɚɬɟɨɪɟɦɚɨɛɢɡɦɟɧɟɧɢɢɩɨɬɟɧɰɢɚɥɶɧɨɣɷɧɟɪɝɢɢ ɉɪɢɪɚɜɧɢɜɚɹɞɪɭɝ ɤ ɞɪɭɝɭ ɩɪɚɜɵɟ ɱɚɫɬɢ ɩɟɪɜɨɝɨ ɪɚɜɟɧɫɬɜɚ ɢ ɪɚɜɟɧɫɬɜɚ ɩɨɥɭɱɢɦ
d T + U = ∂∂Ut dt
ɋɭɦɦɚ ɤɢɧɟɬɢɱɟɫɤɨɣ ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɣ ɱɚɫɬɢɰɵ ɧɚɡɵɜɚɟɬɫɹɩɨɥ ɧɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɟɣ ɱɚɫɬɢɰɵ
T + U = E
ɋ ɭɱɟɬɨɦ ɞɚɧɧɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ ɮɨɪɦɭɥɚ ɩɪɢɨɛɪɟɬɚɟɬ ɜɢɞ
dE = ∂∂Ut dt
ɢɥɢ
dEdt = ∂∂Ut
ɬ ɟ ɩɨɥɧɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɩɨ ɜɪɟɦɟɧɢ ɨɬ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɱɚɫɬɢɰɵ ɪɚɜɧɚ ɱɚɫɬɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɩɨ ɜɪɟɦɟɧɢ ɨɬ ɟɟ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢɬɟɨɪɟɦɚ ɨɛ ɢɡ ɦɟɧɟɧɢɢ ɩɨɥɧɨɣ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɱɚɫɬɢɰɵ
ȿɫɥɢ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɧɟ ɡɚɜɢɫɢɬ ɹɜɧɨ ɨɬ ɜɪɟɦɟɧɢ ɬɨ ɩɨɥ ɧɚɹɦɟɯɚɧɢɱɟɫɤɚɹɷɧɟɪɝɢɹɱɚɫɬɢɰɵɫɨɯɪɚɧɹɟɬɫɹ ɡɚɤɨɧɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟ ɫɤɨɣ ɷɧɟɪɝɢɢ ɱɚɫɬɢɰɵȼ ɱɚɫɬɧɨɫɬɢ ɭ ɫɜɨɛɨɞɧɨɣ ɱɚɫɬɢɰɵ ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪ ɝɢɹ ɫɨɜɩɚɞɚɟɬ ɫ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɟɣ ɢ ɫɨɯɪɚɧɹɟɬɫɹ
ɉɪɢɦɟɪɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɜ ɩɨɥɟ ɫɢɥɵ ɬɹɠɟɫɬɢ Ɂɟɦɥɢ U = mgh ɝɞɟh± ɜɵɫɨɬɚ ɱɚɫɬɢɰɵ ɧɚɞ ɩɨɜɟɪɯɧɨɫɬɶɸ Ɂɟɦɥɢ ɉɨɥɧɚɹ ɦɟɯɚɧɢɱɟ
ɫɤɚɹ ɷɧɟɪɝɢɹ E = |
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+ mgh ȼ ɩɪɨɰɟɫɫɟ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɟɟ ɤɢɧɟɬɢɱɟɫɤɚɹ |
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М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
ɢ ɩɨɬɟɧɰɢɚɥɶɧɚɹɷɧɟɪɝɢɢɦɟɧɹɸɬɫɹ ɫɨɜɪɟɦɟɧɟɦ ɩɨɥɧɚɹ ɷɧɟɪɝɢɹ ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ
Ɂɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɢ ɢɯ ɫɜɹɡɶ ɫɨ ɫɜɨɣɫɬɜɚɦɢ ɫɢɦɦɟɬɪɢɢ
ɩɪɨɫɬɪɚɧɫɬɜɚ ɢ ɜɪɟɦɟɧɢ
ȼɵɩɢɲɟɦ ɮɨɪɦɭɥɵ ɞɥɹ ɢɡɦɟɧɟɧɢɹ ɫɨ ɜɪɟɦɟɧɟɦ ɢɦɩɭɥɶɫɚ ɨɪɛɢɬɚɥɶɧɨɝɨ
ɦɨɦɟɧɬɚ ɢ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɱɚɫɬɢɰɵ |
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ȼ ɫɥɭɱɚɟ ɤɨɝɞɚ ɫɢɥɚ ɹɜɥɹɟɬɫɹ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɞɚɧɧɵɟ ɮɨɪɦɭɥɵ ɩɪɢɨɛɪɟ |
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ɬɚɸɬ ɜɢɞ |
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= − |
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= −>r |
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ɂɡ ɞɚɧɧɵɯ ɬɟɨɪɟɦ ɜɵɬɟɤɚɸɬ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɡɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɤɨɬɨ
ɪɵɟ ɫɜɹɡɚɧɵ ɨɩɪɟɞɟɥɟɧɧɵɦ ɨɛɪɚɡɨɦ ɫɨ ɫɜɨɣɫɬɜɚɦɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɢ ɜɪɟɦɟɧɢ
ɂɡ ɩɟɪɜɨɣ ɮɨɪɦɭɥɵ ɫɥɟɞɭɟɬ ɱɬɨ ɢɦɩɭɥɶɫ ɱɚɫɬɢɰɵ ɫɨɯɪɚɧɹɟɬɫɹ ɟɫ
ɥɢ ɫɢɥɚ ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɱɚɫɬɢɰɭ ɪɚɜɧɚ ɧɭɥɸ ȿɫɥɢ ɪɚɜɧɚ ɧɭɥɸ ɩɪɨɟɤɰɢɹ ɫɢɥɵ
ɧɚ ɧɟɤɨɬɨɪɨɟ ɧɚɩɪɚɜɥɟɧɢɟ ɬɨ ɫɨɯɪɚɧɹɟɬɫɹ ɩɪɨɟɤɰɢɹ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ ɧɚ ɞɚɧ
ɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ ɂɡ ɜɬɨɪɨɣ ɮɨɪɦɭɥɵ ɜɵɬɟɤɚɟɬ ɱɬɨ ɨɪɛɢɬɚɥɶɧɵɣ ɦɨɦɟɧɬ ɱɚɫ
ɬɢɰɵ ɫɨɯɪɚɧɹɟɬɫɹ ɟɫɥɢ ɫɢɥɚ ɪɚɜɧɚ ɧɭɥɸ ɢɥɢ ɜɟɤɬɨɪ ɫɢɥɵ ɩɚɪɚɥɥɟɥɟɧ ɪɚɞɢɭɫ
ɜɟɤɬɨɪɭ ɱɚɫɬɢɰɵ ɦɨɦɟɧɬ ɫɢɥɵ ɪɚɜɟɧ ɧɭɥɸ ȿɫɥɢ ɪɚɜɧɚ ɧɭɥɸ ɩɪɨɟɤɰɢɹ ɦɨɦɟɧ
ɬɚ ɫɢɥɵ ɧɚ ɧɟɤɨɬɨɪɨɟ ɧɚɩɪɚɜɥɟɧɢɟ ɬɨ ɫɨɯɪɚɧɹɟɬɫɹ ɩɪɨɟɤɰɢɹ ɨɪɛɢɬɚɥɶɧɨɝɨ ɦɨ
ɦɟɧɬɚ ɧɚ ɷɬɨ ɧɚɩɪɚɜɥɟɧɢɟ ɇɚɤɨɧɟɰ ɟɫɥɢ ɜɟɤɬɨɪ ɫɢɥɵ ɩɟɪɩɟɧɞɢɤɭɥɹɪɟɧ ɜɟɤɬɨɪɭ
ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ ɬɨ ɫɨɯɪɚɧɹɟɬɫɹ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɉɪɢɦɟɪɑɚɫɬɢɰɚ ɜ ɩɨɥɟ ɫɢɥɵ ɬɹɠɟɫɬɢ ɋɢɥɚ ɬɹɠɟɫɬɢ ɧɚɩɪɚɜɥɟɧɚ ɩɨ
ɨɫɢz ɉɪɨɟɤɰɢɢ ɫɢɥɵ ɧɚ ɨɫɢ x y ɪɚɜɧɵ ɧɭɥɸ ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɩɪɨɟɤɰɢɢ ɢɦ ɩɭɥɶɫɚ ɱɚɫɬɢɰɵ px py ɫɨɯɪɚɧɹɸɬɫɹ Ɇɨɦɟɧɬ ɫɢɥɵ ɬɹɠɟɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢz ɪɚɜɟɧ ɧɭɥɸ ɉɨɷɬɨɦɭ ɩɪɨɟɤɰɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵlz ɫɨɯɪɚɧɹɟɬɫɹ
ɉɪɢɦɟɪɑɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɦɚɝɧɢɬɧɨɣ ɫɨɫɬɚɜɥɹɸɳɟɣ ɫɢɥɵ Ʌɨɪɟɧɰɚ F& = qc [v& B&(r&)]ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɫɢɥɚ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚ ɫɤɨɪɨɫɬɢ
ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɢ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɫɨɯɪɚɧɹɟɬɫɹ Ɂɚɦɟɬɢɦ ɱɬɨ
ɦɚɝɧɢɬɧɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ ɫɢɥɵ Ʌɨɪɟɧɰɚ ɩɪɢɧɚɞɥɟɠɢɬ ɤ ɱɢɫɥɭ ɝɢɪɨɫɤɨɩɢɱɟɫɤɢɯ
ɫɢɥ ɤɨɬɨɪɵɟ ɥɢɧɟɣɧɨ ɡɚɜɢɫɹɬ ɨɬ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰɵ ɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ ɫɤɨɪɨ
ɫɬɢ Ɋɚɛɨɬɚ ɝɢɪɨɫɤɨɩɢɱɟɫɤɨɣ ɫɢɥɵ ɪɚɜɧɚ ɧɭɥɸ
Ʉɨɝɞɚ ɫɢɥɚ ɹɜɥɹɟɬɫɹ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɬɨ ɢɦɩɭɥɶɫ ɫɨɯɪɚɧɹɟɬɫɹ ɜ ɬɨɦ ɫɥɭ ɱɚɟ ɟɫɥɢ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɧɟ ɦɟɧɹɟɬɫɹ ɩɪɢ ɫɞɜɢɝɚɯ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ȿɫɥɢ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɧɟ ɦɟɧɹɟɬɫɹ ɩɪɢ ɫɞɜɢɝɟ ɜɞɨɥɶ ɧɟ ɤɨɬɨɪɨɝɨ ɧɚɩɪɚɜɥɟɧɢɹ ɬɨ ɫɨɯɪɚɧɹɟɬɫɹ ɩɪɨɟɤɰɢɹ ɢɦɩɭɥɶɫɚ ɧɚ ɷɬɨ ɧɚɩɪɚɜɥɟɧɢɟ Ɉɪɛɢɬɚɥɶɧɵɣ ɦɨɦɟɧɬ ɫɨɯɪɚɧɹɟɬɫɹ ɬɨɝɞɚ ɤɨɝɞɚ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɧɟ ɦɟ ɧɹɟɬɫɹ ɩɪɢ ɩɨɜɨɪɨɬɚɯ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ȼ ɬɨɦ ɫɥɭɱɚɟ ɤɨɝɞɚ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɧɟ ɦɟɧɹɟɬɫɹ ɩɪɢ ɩɨɜɨɪɨɬɟ ɜɨɤɪɭɝ ɧɟɤɨɬɨɪɨɝɨ ɧɚɩɪɚɜɥɟɧɢɹ ɬɨ ɫɨɯɪɚ ɧɹɟɬɫɹ ɩɪɨɟɤɰɢɹ ɨɪɛɢɬɚɥɶɧɨɝɨ ɢɦɩɭɥɶɫɚ ɧɚ ɷɬɨ ɧɚɩɪɚɜɥɟɧɢɟ Ɇɟɯɚɧɢɱɟɫɤɚɹ
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Лекция3.ОСНОВНЫЕТЕОРЕМЫДИНАМИКИЧАСТИЦЫ
ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɫɨɯɪɚɧɹɟɬɫɹ ɤɨɝɞɚ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɧɟ ɡɚɜɢɫɢɬ ɹɜɧɨ ɨɬ
ɜɪɟɦɟɧɢ ɧɟ ɦɟɧɹɟɬɫɹ ɩɪɢ ɜɪɟɦɟɧɧɵɯ ɫɞɜɢɝɚɯ Ⱦɚɧɧɵɟ ɭɬɜɟɪɠɞɟɧɢɹ ɫɥɟɞɭɸɬ
ɢɡ ɮɨɪɦɭɥ ɉɪɢɦɟɪ Ⱦɜɢɠɟɧɢɟ ɱɚɫɬɢɰɵ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɰɟɧɬɪɚɥɶɧɨɣ ɫɢɥɵ
F& = F(r)rr ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ
ɦɨɞɭɥɹ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ ɱɚɫɬɢɰɵU =U r Ɉɧɚ ɧɟ ɦɟɧɹɟɬɫɹ ɩɪɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧ ɧɵɯ ɩɨɜɨɪɨɬɚɯ ɜɨɤɪɭɝ ɫɢɥɨɜɨɝɨ ɰɟɧɬɪɚ ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɱɚɫ ɬɢɰɵ ɜ ɰɟɧɬɪɚɥɶɧɨɦ ɩɨɥɟ ɫɨɯɪɚɧɹɟɬɫɹ
Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɫɜɹɡɚɧ ɫ ɨɞɧɨɪɨɞɧɨɫɬɶɸ ɩɪɨɫɬɪɚɧɫɬɜɚ ɨɪɛɢ
ɬɚɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ± ɫ ɢɡɨɬɪɨɩɧɨɫɬɶɸ ɩɪɨɫɬɪɚɧɫɬɜɚ ɷɧɟɪɝɢɢ ± ɫ ɨɞɧɨɪɨɞɧɨɫɬɶɸ ɜɪɟɦɟɧɢ ȿɫɥɢ ɬɟɪɹɟɬɫɹ ɨɞɧɨ ɢɡ ɫɜɨɣɫɬɜ ɫɢɦɦɟɬɪɢɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɢ ɜɪɟɦɟɧɢ ɧɚɩɪɢɦɟɪ ɢɡ ɡɚ ɧɚɥɢɱɢɹ ɜɧɟɲɧɢɯ ɩɨɥɟɣ ɬɨ ɬɟɪɹɟɬɫɹ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɡɚ ɤɨɧ ɫɨɯɪɚɧɟɧɢɹ
23
ɅȿɄɐɂə ɁȺɄɈɇɕ ɂ ɌȿɈɊȿɆɕ ȾɂɇȺɆɂɄɂ ɋɂɋɌȿɆɕ ɑȺɋɌɂɐ
ȼɧɟɲɧɢɟ ɢ ɜɧɭɬɪɟɧɧɢɟ ɫɢɥɵ ɐɟɧɬɪ ɢɧɟɪɰɢɢ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ
ɉɭɫɬɶ ɢɦɟɟɬɫɹ ɫɢɫɬɟɦɚ ɫɨɫɬɨɹɳɚɹ ɢɡ N ɱɚɫɬɢɰ ȼɫɟ ɫɢɥɵ ɞɟɣɫɬɜɭɸɳɢɟ |
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ɧɚ ɱɚɫɬɢɰɵ ɫɢɫɬɟɦɵ ɦɨɠɧɨ ɪɚɡɞɟɥɢɬɶ ɧɚ ɜɧɟɲɧɢɟ ± F&ex ɢ ɜɧɭɬɪɟɧɧɢɟ ± F&in |
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ɫɢɥɚ ɞɟɣɫɬɜɭɸɳɚɹ ɫɨ ɫɬɨɪɨɧɵ ɱɚɫɬɢɰɵbɫɢɫɬɟɦɵ ɧɚ ɱɚɫɬɢɰɭa |
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ȼɧɭɬɪɟɧɧɢɟ ɫɢɥɵ ɩɨɞɱɢɧɹɸɬɫɹ ɬɪɟɬɶɟɦɭ ɡɚɤɨɧɭ ɇɶɸɬɨɧɚ ɬ ɟ |
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F&in = −F&in |
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Ɍɟɨɪɟɦɚɋɭɦɦɚ ɜɫɟɯ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɱɚɫɬɢɰɵ ɫɢɫɬɟ |
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ɦɵ ɪɚɜɧɚ ɧɭɥɸ |
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Fabin = |
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a= b= |
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ɝɞɟ ɲɬɪɢɯ ɭ ɜɬɨɪɨɣ ɫɭɦɦɵ ɨɡɧɚɱɚɟɬ ɱɬɨa ≠ b |
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Ɍɟɨɪɟɦɚɋɭɦɦɚ ɦɨɦɟɧɬɨɜ ɜɫɟɯ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ ɪɚɜɧɚ ɧɭɥɸ |
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N & |
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a= |
a= b= |
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ɋɥɟɞɭɟɬɩɨɦɧɢɬɶɱɬɨɜɧɭɬɪɟɧɧɢɟɫɢɥɵɩɪɢɥɨɠɟɧɵɤ ɪɚɡɧɵɦɱɚɫɬɢɰɚɦ ɢ ɧɟ ɭɪɚɜɧɨɜɟɲɢɜɚɸɬ ɞɪɭɝ ɞɪɭɝɚ ɬ ɟ ɨɧɢ ɫɩɨɫɨɛɧɵ ɜɵɡɵɜɚɬɶ ɞɜɢɠɟɧɢɟ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɪɭɝ ɞɪɭɝɚ
ɐɟɧɬɪɨɦ ɢɧɟɪɰɢɢɫɢɫɬɟɦɵ ɧɚɡɵɜɚɟɬɫɹ ɬɨɱɤɚ ɫ ɪɚɞɢɭɫ ɜɟɤɬɨɪɨɦ
& |
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¦ma ra |
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ɝɞɟm = ¦ma ± ɦɚɫɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ
a =
Ɍɟɨɪɟɦɚ ɨ ɞɜɢɠɟɧɢɢ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ
Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɤɚɠɞɨɣ ɱɚɫɬɢɰɵ ɫɢɫɬɟɦɵ ma d r&= F&ain + F&aex a= → N
dt
ɢ ɩɪɨɫɭɦɦɢɪɭɟɦ ɢɯ ɩɨ ɱɢɫɥɭ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ |
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= m d |
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¦ma d ra |
= m d ¦mara |
RC = ¦F&ain + ¦F&aex = ¦F&aex = F&ex |
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ɝɞɟ F&ex ± ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɜɧɟɲɧɢɯ ɫɢɥ ɢ ɢɫɩɨɥɶɡɨɜɚɧɚ ɬɟɨɪɟɦɚ ɨ ɫɭɦɦɟ ɜɧɭɬ ɪɟɧɧɢɯ ɫɢɥ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ
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d R&C |
= m |
dV&C |
&ex |
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ɝɞɟV&C = |
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± ɫɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ Ⱦɚɧɧɚɹ ɮɨɪɦɭɥɚ ɦɚɬɟɦɚɬɢɱɟɫɤɢ |
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a = |
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ɜɵɪɚɠɚɟɬɬɟɨɪɟɦɭɨ ɞɜɢɠɟɧɢɢɰɟɧɬɪɚɢɧɟɪɰɢɢ ɰɟɧɬɪɢɧɟɪɰɢɢɫɢɫɬɟɦɵɞɜɢ
24
Лекция4.ЗАКОНЫИТЕОРЕМЫДИНАМИКИСИСТЕМЫЧАСТИЦ
ɠɟɬɫɹ ɤɚɤ ɬɨɱɤɚ ɜ ɤɨɬɨɪɨɣ ɫɨɫɪɟɞɨɬɨɱɟɧɚ ɜɫɹ ɦɚɫɫɚ ɫɢɫɬɟɦɵ ɢ ɤ ɤɨɬɨɪɨɣ ɩɪɢ
ɥɨɠɟɧ ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɜɧɟɲɧɢɯ ɫɢɥ ȿɫɥɢ ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɜɧɟɲɧɢɯ ɫɢɥ ɪɚɜɟɧ
ɧɭɥɸ ɬɨ ɰɟɧɬɪ ɢɧɟɪɰɢɢ ɫɢɫɬɟɦɵ ɞɜɢɠɟɬɫɹ ɩɪɹɦɨɥɢɧɟɣɧɨ ɢ ɪɚɜɧɨɦɟɪɧɨ ɥɢɛɨ
ɩɨɤɨɢɬɫɹ
ɂɦɩɭɥɶɫ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ
Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ
ɂɦɩɭɥɶɫɨɦ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰɧɚɡɵɜɚɟɬɫɹ ɫɭɦɦɚɪɧɵɣ ɢɦɩɭɥɶɫ ɜɫɟɯ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ ɬ ɟ
P& |
N |
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= ¦p&a |
= ¦mav&a |
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ɂɫɩɨɥɶɡɭɹ ɨɩɪɟɞɟɥɟɧɢɟ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɫɢɫɬɟɦɵ ɢ ɟɝɨ ɫɤɨɪɨɫɬɢ ɜɵ
ɪɚɠɟɧɢɟ ɞɥɹ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɦɨɠɧɨ ɩɪɟɨɛɪɚɡɨɜɚɬɶ ɤ ɜɢɞɭ
P& = mV&C ɝɞɟ m ± ɦɚɫɫɚ ɫɢɫɬɟɦɵ ɢV&C ± ɫɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ
ɩɨ ɜɪɟɦɟɧɢ ɢ ɢɫɩɨɥɶɡɭɹ ɩɨɥɭɱɢɦɬɟɨɪɟɦɭ ɨɛ ɢɡɦɟɧɟɧɢɢ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟ ɦɵ ɱɚɫɬɢɰ &
ddPt = F&ex
ɬ ɟ ɫɤɨɪɨɫɬɶ ɢɡɦɟɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɪɚɜɧɚ ɝɥɚɜɧɨɦɭ ɜɟɤɬɨɪɭ ɜɧɟɲɧɢɯ ɫɢɥ ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɱɚɫɬɢɰɵ ɫɢɫɬɟɦɵ
ȿɫɥɢ ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɜɧɟɲɧɢɯ ɫɢɥ ɪɚɜɟɧ ɧɭɥɸ ɬɨ ɢɦɩɭɥɶɫ ɫɢɫɬɟɦɵ ɱɚɫ ɬɢɰ ɫɨɯɪɚɧɹɟɬɫɹɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ
Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ
Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ
Ɇɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɨɪɛɢɬɚɥɶɧɵɣ ɦɨɦɟɧɬ ɱɚɫɬɢɰ ɪɚɜɟɧ ɫɭɦɦɟ ɦɨɦɟɧɬɨɜ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ
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L& |
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= ¦l&a = ¦ma>r&a v&a @ |
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Ɂɚɩɢɲɟɦ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɞɥɹ ɤɚɠɞɨɣ ɱɚɫɬɢɰɵ |
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dl&a |
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ɢ ɩɪɨɫɭɦɦɢɪɭɟɦ ɞɚɧɧɵɟN ɭɪɚɜɧɟɧɢɣ |
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N & |
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dla |
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¦la = ¦Main |
+¦Maex = ¦Maex = M ex |
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ɝɞɟ M& ex ± ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɦɨɦɟɧɬɚ ɜɧɟɲɧɢɯ ɫɢɥ ɢ ɢɫɩɨɥɶɡɨɜɚɧɚ ɬɟɨɪɟɦɚ ɨ ɫɭɦɦɟ ɦɨɦɟɧɬɨɜ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ ɋ ɭɱɟɬɨɦ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɥɭɱɚɟɦ
25
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
&
ddLt = M& ex
Ⱦɚɧɧɚɹ ɮɨɪɦɭɥɚ ɜɵɪɚɠɚɟɬ ɫɨɛɨɣ ɬɟɨɪɟɦɭ ɨɛ ɢɡɦɟɧɟɧɢɢ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶ
ɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɫɤɨɪɨɫɬɶ ɢɡɦɟɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɪɚɜ ɧɚ ɝɥɚɜɧɨɦɭ ɜɟɤɬɨɪɭ ɦɨɦɟɧɬɚ ɜɧɟɲɧɢɯ ɫɢɥ ȿɫɥɢ ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɦɨɦɟɧɬɚ ɜɧɟɲɧɢɯ ɫɢɥ ɪɚɜɟɧ ɧɭɥɸ ɬɨ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɫɨɯɪɚɧɹɟɬɫɹɡɚ ɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ
ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ
Ɋɚɫɫɦɨɬɪɢɦ ɤɚɤ ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬɨɞɧɨɣɂɋɈɤ ɞɪɭɝɨɣɉɭɫɬɶɧɟɲɬɪɢɯɨɜɚɧɧɚɹɂɋɈɹɜɥɹɟɬɫɹɧɟɩɨɞɜɢɠɧɨɣ ɚ ɲɬɪɢɯɨɜɚɧɧɚɹ ɂɋɈ ɫɜɹɡɚɧɚ ɫ ɰɟɧɬɪɨɦ ɢɧɟɪɰɢɢ ɫɢɫɬɟɦɵ ɢ ɟɟ ɨɫɢ ɤɨɨɪɞɢɧɚɬ ɩɚɪɚɥɥɟɥɶɧɵ ɨɫɹɦ ɧɟɩɨɞɜɢɠɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɪɢɫ
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UD UDF |
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x[
Ɋɢɫ. 4.1.Ʉ ɜɵɜɨɞɭ ɮɨɪɦɭɥɵ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ
Ⱦɥɹɢɦɩɭɥɶɫɚɫɢɫɬɟɦɵɱɚɫɬɢɰɜ ɞɚɧɧɨɦɫɥɭɱɚɟɩɪɨɜɟɞɹɫɨɨɬɜɟɬɫɬɜɭɸ ɳɢɟɩɪɟɨɛɪɚɡɨɜɚɧɢɹɢ ɢɫɩɨɥɶɡɭɹɬɨɬɮɚɤɬɱɬɨɨɬɧɨɫɢɬɟɥɶɧɨɰɟɧɬɪɚɢɧɟɪɰɢɢ ɢɦɩɭɥɶɫ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɢ ɜɟɤɬɨɪ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɪɚɜɧɵ ɧɭɥɸ ɢɦɟɟɦ
& |
N |
& & |
N |
& & |
& & |
& & |
N |
& & |
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= ¦ma>ra va @ = ¦ma>rac + RC vac + VC @ = m >RC Vɋ @+ ¦ma>rac vac @ = Lɩɨɫɬ |
+ Lɨɬɧ |
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± ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟ
ɦɵ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɤɚɤ ɟɞɢɧɨɝɨ ɰɟɥɨɝɨ ɢ ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ
Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɢɡ ɞɜɭɯ ɱɚɫɬɢɰ ɜ ɫɢɫɬɟɦɟ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɂɦɟɟɦ
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m r& + m r& = r& − r& = r& |
26 |
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Лекция4.ЗАКОНЫИТЕОРЕМЫДИНАМИКИСИСТЕМЫЧАСТИЦ
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L&ɩɨɫɬ = |
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L&ɨɬɧ = m >r& v&@+ m >r& v& @= μ >r& − r& v& − v& @ |
ɝɞɟμ = |
m m |
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± ɩɪɢɜɟɞɟɧɧɚɹ ɦɚɫɫɚ |
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m + m |
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ȼ ɤɜɚɧɬɨɜɨɣ ɦɟɯɚɧɢɤɟ ɩɨɤɚɡɵɜɚɟɬɫɹ ɱɬɨ ɱɚɫɬɢɰɵ ɦɨɝɭɬ ɨɛɥɚɞɚɬɶ ɫɨɛɫɬ ɜɟɧɧɵɦ ɜɧɭɬɪɟɧɧɢɦ ɦɨɦɟɧɬɨɦ ɢɦɩɭɥɶɫɚ ɤɨɬɨɪɵɣ ɧɚɡɵɜɚɟɬɫɹɫɩɢɧɨɦɋɨɛ ɫɬɜɟɧɧɵɣ ɦɨɦɟɧɬ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɩɪɢ ɷɬɨɦ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɭɦ ɦɭ ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɦɨɦɟɧɬɚ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟɦɵ ɢ ɫɨɛɫɬɜɟɧɧɵɯ ɦɨɦɟɧɬɨɜ ɢɦ ɩɭɥɶɫɚ ɱɚɫɬɢɰ ɢɯ ɫɩɢɧɨɜ
L&ɫɨɛɫɬɜ = L&ɨɬɧ + ¦N s&a
a =
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ
Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ Ɍɟɨɪɟɦɚ Ʉɟɧɢɝɚ
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵɱɚɫɬɢɰ ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ ɫɭɦɦɵ ɤɢɧɟɬɢɱɟ ɫɤɢɯ ɷɧɟɪɝɢɣ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ
N |
N |
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T = ¦Ta |
= ¦ |
mava |
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Ɂɚɩɢɲɟɦɬɟɨɪɟɦɭɨɛɢɡɦɟɧɟɧɢɢɤɢɧɟɬɢɱɟɫɤɨɣɷɧɟɪɝɢɢɤɚɠɞɨɣɱɚɫɬɢɰɵ ɫɢɫɬɟɦɵ
dTa = δAain +δAaex
ɫɤɥɚɞɵɜɚɟɦ ɞɚɧɧɵɟNɭɪɚɜɧɟɧɢɣ
N N N
¦dTa = dT = ¦δAain + ¦δAaex = δAin +δAex
a= |
a = |
a = |
Ⱦɚɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣɬɟɨɪɟɦɭ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɢɧɟɬɢɱɟ ɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵɋɥɟɞɭɟɬ ɡɚɦɟɬɢɬɶ ɱɬɨ ɢɡɦɟɧɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɡɚɜɢɫɢɬ ɨɬ ɪɚɛɨɬɵ ɤɚɤ ɜɧɟɲɧɢɯ ɬɚɤ ɢ ɜɧɭɬɪɟɧɧɢɯ ɫɢɥ
Ɋɚɫɫɦɨɬɪɢɦ ɤɚɤ ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɩɪɢ ɩɟɪɟ
ɯɨɞɟ ɨɬ ɨɞɧɨɣ ɂɋɈ ɤ ɞɪɭɝɨɣ ɪɢɫ
N |
m v |
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& & & & |
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= ¦ |
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vac +VC vac +VC = |
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VC + ¦ |
a ac |
= Tɩɨɫɬ +Tɨɬɧ |
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ɝɞɟ m ± ɦɚɫɫɚ ɫɢɫɬɟɦɵ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɪɚɜɧɚ ɫɭɦɦɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɤɚɤ ɟɞɢɧɨɝɨ ɰɟɥɨɝɨ ɜ ɟɟ ɩɨɫɬɭɩɚ ɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɢ ɢ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɜ ɟɟ ɞɜɢɠɟɧɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢɬɟɨɪɟɦɚ Ʉɟɧɢɝɚ
ȼ ɱɚɫɬɧɨɫɬɢ ɞɥɹ ɡɚɦɤɧɭɬɨɣ ɫɢɫɬɟɦɵ ɢɡ ɞɜɭɯ ɱɚɫɬɢɰ ɜ ɫɢɫɬɟɦɟ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ
Tɩɨɫɬ = Tɨɬɧ = μ v& −v&
ɝɞɟμ ± ɩɪɢɜɟɞɟɧɧɚɹ ɦɚɫɫɚ ɫɢɫɬɟɦɵ
27
М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ
ɉɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ
ɋɨɛɫɬɜɟɧɧɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ
Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭa = → N ɱɚɫɬɢɰ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɢ ɫ ɱɚɫɬɢɰɚɦɢ ɤɨɬɨɪɵɟ ɧɟ ɜɯɨɞɹɬ ɜ ɞɚɧɧɭɸ ɫɢɫɬɟɦɭ ɋɢɥɵ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɦɟ ɠɞɭ ɱɚɫɬɢɰɚɦɢ ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɩɨɬɟɧɰɢɚɥɶɧɵɦɢ ɢ ɫɬɚɰɢɨɧɚɪɧɵɦɢ ȼ ɷɬɨɦ ɫɥɭ ɱɚɟ ɞɥɹ ɤɚɠɞɨɣ ɱɚɫɬɢɰɵ ɫɢɫɬɟɦɵ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
δAa = − dU ain − dU aex
ɝɞɟ Uain ± ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰɵ ɜ ɩɨɥɟ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ U aex ± ɜɨ
ɜɧɟɲɧɟɦ ɩɨɥɟ Ⱦɥɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɷɥɟɦɟɧɬɚɪɧɚɹ ɪɚɛɨɬɚ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ
N N
δA = ¦δAa = −¦ dUain + dUaex = −dU
a = a =
ɝɞɟU ± ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɜɨ ɜɧɟɲɧɟɦ ɩɨɥɟ ɬɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ Ɉɧɚ ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ ɩɨ ɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɜ ɨɬɫɭɬɫɬɜɢɟ ɜɧɟɲɧɟɝɨ ɜɨɡɞɟɣɫɬɜɢɹ ɧɚ ɫɢɫɬɟɦɭ
N
U in = ¦Uain a =
ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɨɛɭɫɥɨɜɥɟɧɧɨɣ ɜɧɟɲɧɢɦ ɫɢɥɨɜɵɦ ɩɨɥɟɦ
N
U ex = ¦Uaex a =
ɗɧɟɪɝɢɹUin ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ ɷɧɟɪɝɢɣ ɩɚɪɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɱɚɫɬɢɰ ɫɢɫ ɬɟɦɵ ɩɪɢɧɰɢɩ ɫɭɩɟɪɩɨɡɢɰɢɢ
N |
N N |
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N |
U in = ¦Uain = ¦¦ Uabin |
= |
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¦Uabin r&a − r&b |
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a= |
a= b= |
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a b |
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ɝɞɟ ɦɧɨɠɢɬɟɥɶ ò ɤɨɦɩɟɧɫɢɪɭɟɬ ɜɤɥɚɞ ɨɞɢɧɚɤɨɜɵɯ ɱɥɟɧɨɜ a = b ȼ ɫɥɭɱɚɟ ɰɟɧ ɬɪɚɥɶɧɵɯ ɫɢɥ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɡɚɜɢɫɢɬ ɨɬ ɦɨɞɭɥɹ ɪɚɡɧɨɫɬɢ ɪɚɞɢɭɫ ɜɟɤɬɨɪɨɜ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ
ɋɨɛɫɬɜɟɧɧɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɦɚɫɫ ɫɢɫɬɟɦɵ
Tɨɬɧ = ¦N mav&ac a=
ɢɡ ɷɧɟɪɝɢɢUin ɢ ɫɨɛɫɬɜɟɧɧɵɯ ɜɧɭɬɪɟɧɧɢɯ ɷɧɟɪɝɢɣ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ
N
¦Ea a =
ɤɨɬɨɪɵɟ ɨɩɪɟɞɟɥɹɸɬɫɹ ɮɨɪɦɭɥɨɣ ɗɣɧɲɬɟɣɧɚEa = ma c Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɫɨɛɫɬ
ɜɟɧɧɚɹ ɷɧɟɪɝɢɹ ɱɚɫɬɢɰ ɫɢɫɬɟɦɵ
N
Eɫɨɛɫɬ = Tɨɬɧ + U in + ¦Ea
a =
28 |
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Лекция4.ЗАКОНЫИТЕОРЕМЫДИНАМИКИСИСТЕМЫЧАСТИЦ
ȿɫɥɢ ɱɚɫɬɢɰɵ ɫɢɫɬɟɦɵ ɧɟ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɬɨ ɜɬɨɪɨɟ ɫɥɚ
ɝɚɟɦɨɟ ɜ ɩɨɫɥɟɞɧɟɣ ɮɨɪɦɭɥɟ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ
Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ
ɋɪɚɜɧɢɜɚɹ ɦɟɠɞɭ ɫɨɛɨɣ ɮɨɪɦɭɥɵ ɢ ɞɥɹ ɢɡɦɟɧɟɧɢɹ ɤɢɧɟɬɢɱɟ
ɫɤɨɣ ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɣ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ d T +U = T +U = E = const
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɟɫɥɢ ɜɫɟ ɫɢɥɵ ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɱɚɫɬɢɰɵ ɫɢɫɬɟɦɵ ɹɜɥɹ
ɸɬɫɹ ɩɨɬɟɧɰɢɚɥɶɧɵɦɢ ɢ ɫɬɚɰɢɨɧɚɪɧɵɦɢ ɬɨ ɩɨɥɧɚɹ ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫ
ɬɟɦɵ ɱɚɫɬɢɰ ɫɨɯɪɚɧɹɟɬɫɹ
ȼ ɬɨɦ ɫɥɭɱɚɟ ɟɫɥɢ ɩɨɬɟɧɰɢɚɥɶɧɵɟ ɜɧɟɲɧɢɟ ɫɢɥɵ ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɱɚɫɬɢ
ɰɵ ɫɢɫɬɟɦɵ ɧɟ ɹɜɥɹɸɬɫɹ ɫɬɚɰɢɨɧɚɪɧɵɦɢ ɫɩɪɚɜɟɞɥɢɜɚ ɬɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ
ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ
dEdt = ∂∂Ut
Ɉɬɫɸɞɚ ɜɢɞɧɨ ɱɬɨ ɢɡɦɟɧɟɧɢɟ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɨɩɪɟɞɟɥɹ ɟɬɫɹ ɜɪɟɦɟɧɧɨɣ ɡɚɜɢɫɢɦɨɫɬɶɸ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ
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ɇȿɄɈɌɈɊɕȿ ɁȺȾȺɑɂ ȾɂɇȺɆɂɄɂ ɇɖɘɌɈɇȺ Ⱦȼɂɀȿɇɂȿ
ȼ ɐȿɇɌɊȺɅɖɇɈ ɋɂɆɆȿɌɊɂɑɇɈɆ ɉɈɅȿ
ɅȿɄɐɂə ɈȾɇɈɆȿɊɇɈȿ Ⱦȼɂɀȿɇɂȿ ɁȺȾȺɑȺ Ⱦȼɍɏ ɌȿɅ
Ⱦȼɂɀȿɇɂȿ ɑȺɋɌɂɐɕ ȼ ɐȿɇɌɊȺɅɖɇɈ ɋɂɆɆȿɌɊɂɑɇɈɆ ɉɈɅȿ
Ɉɞɧɨɦɟɪɧɨɟ ɞɜɢɠɟɧɢɟ
Ⱦɜɢɠɟɧɢɟ ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɨɞɧɨɦɟɪɧɵɦ ɟɫɥɢ ɩɨɥɨɠɟɧɢɟ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɨɞɧɨɝɨ ɩɚɪɚɦɟɬɪɚ ȼ ɤɚɱɟɫɬɜɟ ɬɚ ɤɨɝɨ ɩɚɪɚɦɟɬɪɚ ɦɨɝɭɬ ɜɵɫɬɭɩɚɬɶ ɤɨɨɪɞɢɧɚɬɚ ɭɝɨɥ ɩɨɜɨɪɨɬɚ ɢ ɞɪ Ⱦɚɧɧɵɣ ɩɚɪɚ ɦɟɬɪ ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶx
Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɢɦɟɟɬ ɜɢɞ
T= m §¨dx ·¸
© dt ¹
ɝɞɟ m ± ɤɨɷɮɮɢɰɢɟɧɬ ɢɧɟɪɰɢɢ ɜ ɫɥɭɱɚɟ ɟɫɥɢ ɩɚɪɚɦɟɬɪɨɦ ɹɜɥɹɟɬɫɹ ɤɨɨɪɞɢɧɚ ɬɚ ± ɦɚɫɫɚ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɟɫɬɶ ɮɭɧɤɰɢɹ ɩɚɪɚɦɟɬɪɚ ɢ ɜɪɟɦɟɧɢ ±U x t ɋɨɝɥɚɫɧɨ ɬɟɨɪɟɦɟ ɨɛ ɢɡɦɟɧɟɧɢɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢE = T +U
dEdt = ∂∂Ut
ȿɫɥɢ ɩɨɬɟɧɰɢɚɥɶɧɚɹ ɷɧɟɪɝɢɹ ɹɜɧɨ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɪɟɦɟɧɢ ɬɨ ∂U ∂t =
ɢ ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɨɯɪɚɧɹɟɬɫɹ Ɍɚɤ ɤɚɤ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɧɟ ɨɬɪɢɰɚ
ɬɟɥɶɧɚ ɬɨ
T = E − U x ≥
Ⱦɚɧɧɨɟ ɧɟɪɚɜɟɧɫɬɜɨ ɧɚɤɥɚɞɵɜɚɟɬ ɨɝɪɚɧɢɱɟɧɢɹ ɧɚ ɞɨɩɭɫɬɢɦɭɸ ɨɛɥɚɫɬɶ ɞɜɢɠɟɧɢɹ ɉɭɫɬɶ ɮɭɧɤɰɢɹU x ɢɦɟɟɬ ɜɢɞ ɩɨɤɚɡɚɧɧɵɣ ɧɚ ɪɢɫ
U x
E
x x x x [
[ [ [ [ [
Ɋɢɫ. 5.1.Ⱦɨɩɭɫɬɢɦɵɟ ɨɛɥɚɫɬɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɩɨɬɟɧɰɢɚɥɶɧɨɦ ɩɨɥɟ
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