Лекции по классической механике. Учебное пособие

81

ɅȿɄɐɂə ȾɂɇȺɆɂɄȺ ɑȺɋɌɂɐɕ ȼ ɋɌɈ

Ɂɚɤɨɧ ɢɧɟɪɰɢɢ Ɇɚɫɫɚ ɱɚɫɬɢɰɵ ɢɦɩɭɥɶɫ

ȼɵɩɢɲɟɦ ɭɫɤɨɪɟɧɢɟ ɱɚɫɬɢɰɵ ɨɩɪɟɞɟɥɟɧɢɟ ɤɨɬɨɪɨɝɨ ɛɵɥɨ ɞɚɧɨ ɜ ɩɪɟɞɵɞɭɳɟɣ ɥɟɤɰɢɢ

Ɋɚɫɫɦɨɬɪɢɦ ɪɟɥɹɬɢɜɢɫɬɫɤɭɸ ɢɡɨɥɢɪɨɜɚɧɧɭɸ ɱɚɫɬɢɰɭ ȿɟ ɭɫɤɨɪɟɧɢɟ ɜ ɫɢɥɭɩɪɢɧɰɢɩɚɩɪɢɱɢɧɧɨɫɬɢɞɨɥɠɧɨɨɩɪɟɞɟɥɹɬɶɫɹɱɟɪɟɡɟɟɪɚɞɢɭɫ ɜɟɤɬɨɪ ɢ ɫɤɨɪɨɫɬɶ ȼ ɫɢɥɭ ɨɞɧɨɪɨɞɧɨɫɬɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɢ ɜɪɟɦɟɧɢ ɫɥɟɞɭɟɬ ɢɫɤɥɸɱɢɬɶ ɡɚɜɢɫɢɦɨɫɬɶ ɨɬ ɪɚɞɢɭɫ ɜɟɤɬɨɪɚ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɤɨɦɩɨɧɟɧɬɵ ɭɫɤɨɪɟɧɢɹ ɞɨɥɠɧɵ ɡɚɜɢɫɟɬɶ ɬɨɥɶɤɨ ɨɬ ɤɨɦɩɨɧɟɧɬ ɫɤɨɪɨɫɬɢ ɤɜɚɞɪɚɬ ɤɨɬɨɪɨɣ ɟɫɬɶ ɜɟɥɢ ɱɢɧɚ ɩɨɫɬɨɹɧɧɚɹ ɗɬɨ ɩɪɢɜɨɞɢɬ ɤ ɬɨɦɭ ɱɬɨ ɤɨɦɩɨɧɟɧɬɵ ɭɫɤɨɪɟɧɢɹ ɢ ɫɤɨɪɨɫɬɢ ɞɨɥɠɧɵ ɛɵɬɶ ɫɜɹɡɚɧɵ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɫɨɨɬɧɨɲɟɧɢɟɦ

Wμ = const •U μ

ɉɨɫɤɨɥɶɤɭ ɫɤɨɪɨɫɬɶ ɢ ɭɫɤɨɪɟɧɢɟ ɨɪɬɨɝɨɧɚɥɶɧɵ ɞɪɭɝ ɞɪɭɝɭ ɬɨ ɤɨɧ ɫɬɚɧɬɚ ɜ ɩɨɫɥɟɞɧɟɣ ɮɨɪɦɭɥɟ ɞɨɥɠɧɚ ɛɵɬɶ ɪɚɜɧɚ ɧɭɥɸ ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɞɥɹ ɢɡɨ ɥɢɪɨɜɚɧɧɨɣ ɱɚɫɬɢɰɵ ɜ ɥɸɛɨɣ ɂɋɈ

Wμ =

ɂɡ ɮɨɪɦɭɥɵ ɫɥɟɞɭɟɬ ɱɬɨ w& =

ɍɫɤɨɪɟɧɢɟ ɢɡɨɥɢɪɨɜɚɧɧɨɣ ɱɚɫɬɢɰɵ ɪɚɜɧɨ ɧɭɥɸ

Ɇɚɫɫɚ ɱɚɫɬɢɰɵ ɛɵɥɚ ɨɩɪɟɞɟɥɟɧɚ ɱɟɪɟɡ ɨɬɧɨɲɟɧɢɹ ɭɫɤɨɪɟɧɢɣ ɜɡɚɢɦɨɞɟɣ

ɫɬɜɭɸɳɢɯ ɱɚɫɬɢɰ ɜ ɧɟɪɟɥɹɬɢɜɢɫɬɫɤɨɣ ɦɟɯɚɧɢɤɟ ɗɬɚ ɜɟɥɢɱɢɧɚ ɹɜɥɹɟɬɫɹ ɫɤɚɥɹɪ

ɧɨɣ ɜɧɭɬɪɟɧɧɟɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɱɚɫɬɢɰɵ ɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɟɟ ɫɨɫɬɨɹɧɢɹ ɜ ɬɨɦ ɱɢɫɥɟ ɢ ɨɬ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɉɨ ɞɚɧɧɨɦɭ ɨɩɪɟɞɟɥɟɧɢɸɦɚɫɫɚ ɟɫɬɶ ɢɧɜɚɪɢɚɧɬɧɚɹ ɜɟɥɢɱɢɧɚ ± ɫɤɚɥɹɪȼ ɫɢɥɭ ɷɬɨɝɨ ɜ ɪɟɥɹɬɢɜɢɫɬɫɤɨɣ ɮɢɡɢɤɟ ɫɥɟ ɞɭɟɬ ɩɨɧɢɦɚɬɶ ɩɨɞ ɦɚɫɫɨɣ ɬɭ ɠɟ ɜɟɥɢɱɢɧɭ ɱɬɨ ɢ ɜ ɧɟɪɟɥɹɬɢɜɢɫɬɫɤɨɣ

ɢɦɩɭɥɶɫɨɦ ɱɚɫɬɢɰɵɧɚɡɵɜɚɟɬɫɹ ɩɪɨɢɡɜɟɞɟɧɢɟ ɟɟ ɦɚɫɫɵ ɧɚ ɫɤɨɪɨɫɬɶ

Лекция15.ДИНАМИКАЧАСТИЦЫВСТО

ɧɚɡɵɜɚɟɬɫɹ ɪɟɥɹɬɢɜɢɫɬɫɤɢɦ ɢɦɩɭɥɶɫɨɦ ȼ ɧɟɪɟɥɹɬɢɜɢɫɬɫɤɨɦ ɫɥɭɱɚɟv << c ɪɟɥɹ ɬɢɜɢɫɬɫɤɢɣ ɢɦɩɭɥɶɫ ɩɟɪɟɯɨɞɢɬ ɜ ɨɛɵɱɧɵɣ ɢɦɩɭɥɶɫ ɱɚɫɬɢɰɵ p& = mv&

Ɉɫɧɨɜɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɢɧɚɦɢɤɢ ɱɚɫɬɢɰɵ

Ɉɫɧɨɜɧɨɟɭɪɚɜɧɟɧɢɟɞɢɧɚɦɢɤɢɧɟɪɟɥɹɬɢɜɢɫɬɫɤɨɣɱɚɫɬɢɰɵɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ

ddpt = F&

Ɉɛɨɛɳɟɧɢɟɦ ɭɪɚɜɧɟɧɢɹ ɧɚ ɪɟɥɹɬɢɜɢɫɬɫɤɢɣ ɫɥɭɱɚɣ ɹɜɥɹɟɬɫɹ ɭɪɚɜɧɟɧɢɟ

dPμ = fμ dt

ɝɞɟ Pμ ± ɤɨɦɩɨɧɟɧɬɵ ɢɦɩɭɥɶɫɚ ɱɚɫɬɢɰɵ fμ ± ɤɨɦɩɨɧɟɧɬɵ ɧɟɤɨɬɨɪɨɝɨ

ɜɟɤɬɨɪɚ ɤɨɬɨɪɵɣ ɧɚɡɵɜɚɟɬɫɹ ɫɢɥɚ ɢɥɢ ɫɢɥɚ Ɇɢɧɤɨɜɫɤɨɝɨ Ⱦɚɧɧɨɟ ɭɪɚɜɧɟɧɢɟ

ɢɦɟɟɬ ɪɟɥɹɬɢɜɢɫɬɫɤɭɸ ɤɨɜɚɪɢɚɧɬɧɭɸ ɮɨɪɦɭ ɬɚɤ ɤɚɤ ɜ ɩɪɚɜɭɸ ɢ ɥɟɜɭɸ ɟɝɨ ɱɚɫ

ɬɢ ɜɯɨɞɹɬ ɜɟɤɬɨɪɵ ɤɨɬɨɪɵɟ ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɨɞɧɨɣ ɂɋɈ ɤ ɞɪɭɝɨɣ ɢɡɦɟɧɹɸɬɫɹ

ɩɨ ɨɞɧɨɦɭ ɢ ɬɨɦɭ ɠɟ ɡɚɤɨɧɭ

ȼɵɩɢɲɟɦ ɩɟɪɜɵɟ ɬɪɢ ɤɨɦɩɨɧɟɧɬɵ ɭɪɚɜɧɟɧɢɹ

ɉɪɢv << c ɭɪɚɜɧɟɧɢɟ ɞɨɥɠɧɨ ɩɟɪɟɯɨɞɢɬɶ ɜ ɭɪɚɜɧɟɧɢɟ ɉɨɷɬɨɦɭ ɬɪɢ ɩɟɪɜɵɟ ɤɨɦɩɨɧɟɧɬɵ ɫɢɥɵ ɫɜɹɡɚɧɵ ɫ ɨɛɵɱɧɨɣ ɫɢɥɨɣF& ɫɨɨɬɧɨɲɟɧɢɟɦ

ɍɪɚɜɧɟɧɢɟ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ

83

Лекция15.ДИНАМИКАЧАСТИЦЫВСТО

ȼ ɩɪɚɜɨɣ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ ɫɬɨɢɬ ɦɨɳɧɨɫɬɶ ɫɢɥɵ ɋɥɟɞɨɜɚɬɟɥɶɧɨ

ɜ ɥɟɜɨɣ ɱɚɫɬɢ ɫɬɨɢɬ ɢɡɦɟɧɟɧɢɟ ɷɧɟɪɝɢɢ ɱɚɫɬɢɰɵ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɨɥɧɚɹ ɷɧɟɪɝɢɹ ɪɟɥɹɬɢɜɢɫɬɫɤɨɣ ɱɚɫɬɢɰɵ

− v

c

ɂɦɩɭɥɶɫ ɷɧɟɪɝɢɹ ɢ ɦɚɫɫɚ ɪɟɥɹɬɢɜɢɫɬɫɤɨɣ ɱɚɫɬɢɰɵ

ɂɡ ɨɩɪɟɞɟɥɟɧɢɣ ɢ ɜɵɬɟɤɚɟɬ ɱɬɨ ɢɦɩɭɥɶɫ ɱɚɫɬɢɰɵ ɢɦɟɟɬ

ɜɢɞ

ɜɟɤɬɨɪ ɧɚɡɵɜɚɟɬɫɹɜɟɤɬɨɪɨɦɷɧɟɪɝɢɢ ɢɦɩɭɥɶɫɚɄɨɦɩɨɧɟɧɬɵ ɷɧɟɪɝɢɢ ɢɦɩɭɥɶɫɚ ɩɪɟɨɛɪɚɡɭɸɬɫɹ ɜ ɫɨɝɥɚɫɢɢ ɫ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹɦɢ Ʌɨɪɟɧɰɚ Ʉɜɚɞɪɚɬ ɟɝɨ ɤɚɤ ɤɜɚɞɪɚɬ ɜɟɤɬɨɪɚ ɟɫɬɶ ɢɧɜɚɪɢɚɧɬ

P = P& − E = −m c c

Ɉɬɫɸɞɚɩɨɥɭɱɚɟɦɫɜɹɡɶɦɟɠɞɭɦɚɫɫɨɣɱɚɫɬɢɰɵɟɟɪɟɥɹɬɢɜɢɫɬɫɤɢɦɢɦ ɩɭɥɶɫɨɦ ɢ ɩɨɥɧɨɣ ɷɧɟɪɝɢɟɣ

P& c + m c = E

ɋɌɈɩɪɟɞɫɤɚɡɵɜɚɟɬɫɭɳɟɫɬɜɨɜɚɧɢɟɛɟɡɦɚɫɫɨɜɵɯɱɚɫɬɢɰɌɚɤɢɟɱɚɫɬɢɰɵ ɨɛɥɚɞɚɸɬ ɧɭɥɟɜɨɣ ɦɚɫɫɨɣ ɞɜɢɠɭɬɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ ɪɚɜɧɨɣ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ ɷɧɟɪɝɢɹ ɢ ɢɦɩɭɥɶɫ ɱɚɫɬɢɰɵ ɫɜɹɡɚɧɵ ɫɨɨɬɧɨɲɟɧɢɟɦ E = P c ɉɪɢɦɟɪɨɦ ɬɚɤɨɣ ɱɚɫɬɢɰɵ ɹɜɥɹɟɬɫɹ ɮɨɬɨɧ

Ɋɚɫɫɦɨɬɪɢɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɨɥɧɨɣ ɷɧɟɪɝɢɢ ɱɚɫɬɢɰɵ ɉɪɢ ɦɚɥɵɯ ɫɤɨɪɨɫɬɹɯv << c ɨɧɨ ɩɟɪɟɯɨɞɢɬ ɜ

ȼɬɨɪɨɟɫɥɚɝɚɟɦɨɟɜ ɮɨɪɦɭɥɟ ɫɨɜɩɚɞɚɟɬɫ ɤɢɧɟɬɢɱɟɫɤɨɣɷɧɟɪɝɢɟɣ ɱɚɫɬɢɰɵ ɉɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ

E = mc

ɟɫɬɶ ɷɧɟɪɝɢɹ ɩɨɤɨɹɳɟɣɫɹ ɱɚɫɬɢɰɵ ɷɧɟɪɝɢɹ ɩɨɤɨɹ

85

М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ

Ɏɨɪɦɭɥɚ ɧɚɡɵɜɚɟɬɫɹ ɮɨɪɦɭɥɨɣ ɗɣɧɲɬɟɣɧɚ Ɉɧɚ ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɫɜɹɡɶ ɦɟɠɞɭ ɦɚɫɫɨɣ ɱɚɫɬɢɰɵ ɢ ɟɟ ɷɧɟɪɝɢɟɣ ɩɨɤɨɹ Ⱦɚɧɧɵɟ ɜɟɥɢɱɢɧɵ ɫɜɹɡɚɧɵ ɩɨɫɪɟɞɫɬɜɨɦ ɮɭɧɞɚɦɟɧɬɚɥɶɧɨɣ ɩɨɫɬɨɹɧɧɨɣ ± ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ ɜ ɜɚɤɭɭɦɟ

ɗɧɟɪɝɢɹ E ɟɫɬɶ ɩɨɥɧɚɹ ɷɧɟɪɝɢɹ ɞɜɢɠɭɳɟɣɫɹ ɪɟɥɹɬɢɜɢɫɬɫɤɨɣ ɱɚɫ ɬɢɰɵɷɧɟɪɝɢɹ E ɟɫɬɶɷɧɟɪɝɢɹɩɨɤɨɹɱɚɫɬɢɰɵɊɚɡɧɨɫɬɶɷɬɢɯɷɧɟɪɝɢɣ ɟɫɬɶ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɪɟɥɹɬɢɜɢɫɬɫɤɨɣ ɱɚɫɬɢɰɵ ɫɜɹɡɚɧɧɚɹ ɫ ɞɜɢɠɟɧɢɟɦ ɱɚɫɬɢɰɵ

Ɏɭɧɤɰɢɢ Ʌɚɝɪɚɧɠɚ ɢ Ƚɚɦɢɥɶɬɨɧɚ ɫɜɨɛɨɞɧɨɣ ɪɟɥɹɬɢɜɢɫɬɫɤɨɣ

ɱɚɫɬɢɰɵ

Ɋɚɫɫɦɨɬɪɢɦ ɪɟɥɹɬɢɜɢɫɬɫɤɭɸ ɱɚɫɬɢɰɭ ɫ ɢɦɩɭɥɶɫɨɦ

P& = mv&

− v

c

ɢ ɷɧɟɪɝɢɟɣ

E = mc

− v

c

Ɏɭɧɤɰɢɹ ɅɚɝɪɚɧɠɚLɞɨɥɠɧɚ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɭɪɚɜɧɟɧɢɹɦ

ɝɞɟi = x y z

ɍɪɚɜɧɟɧɢɹɦ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɮɭɧɤɰɢɹ

L = −mc − v c

ɉɪɢ ɦɚɥɵɯ ɫɤɨɪɨɫɬɹɯv << c ɮɭɧɤɰɢɹ ɩɟɪɟɯɨɞɢɬ ɜ

L −mc + mv

ɋ ɬɨɱɧɨɫɬɶɸ ɞɨ ɩɨɫɬɨɹɧɧɨɣ(− mc )ɮɭɧɤɰɢɹ ɫɨɜɩɚɞɚɟɬ ɫ ɮɭɧɤɰɢɟɣ

Ʌɚɝɪɚɧɠɚ ɧɟɪɟɥɹɬɢɜɢɫɬɫɤɨɣ ɱɚɫɬɢɰɵ ɗɬɚ ɩɨɫɬɨɹɧɧɚɹ ɧɟ ɦɟɧɹɟɬ ɭɪɚɜɧɟɧɢɣ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ

ɂɫɩɨɥɶɡɭɹ ɮɨɪɦɭɥɭ ɜɵɪɚɠɚɟɦ ɮɭɧɤɰɢɸ Ƚɚɦɢɥɶɬɨɧɚ ɱɟɪɟɡ ɨɛɨɛ

ɳɟɧɧɵɣ ɢɦɩɭɥɶɫ

H = P& c +m c

87

ɅȿɄɐɂə ɋɂɋɌȿɆɕ ɑȺɋɌɂɐ ȼ ɋɌɈ

ɋɢɫɬɟɦɚ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɱɚɫɬɢɰ

Ɋɚɫɫɦɨɬɪɢɦ ɫɢɫɬɟɦɭ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɱɚɫɬɢɰ ɗɧɟɪɝɢɹ ɢ ɢɦɩɭɥɶɫ ɜ ɫɢɫɬɟɦɟ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɱɚɫɬɢɰ ɹɜɥɹɸɬɫɹ ɚɞɞɢɬɢɜɧɵɦɢ ɜɟɥɢɱɢɧɚɦɢ ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɷɧɟɪɝɢɹ ɢ ɢɦɩɭɥɶɫ ɫɢɫɬɟɦɵ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɱɚɫɬɢɰ

Ɂɞɟɫɶ N ± ɱɢɫɥɨ ɱɚɫɬɢɰ ɜ ɫɢɫɬɟɦɟ a ± ɢɧɞɟɤɫ ɱɚɫɬɢɰɵ ɋɤɨɪɨɫɬɢ ɜɫɟɯ ɱɚɫɬɢɰ ɩɨɫɬɨɹɧɧɵ ɉɨɷɬɨɦɭ ɩɨɥɧɚɹ ɷɧɟɪɝɢɹ ɢ ɢɦɩɭɥɶɫ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɩɨɫɬɨ ɹɧɧɵ ɜɨ ɜɪɟɦɟɧɢ E = const P = const ȼɜɟɞɟɦ ɜɟɤɬɨɪ ɷɧɟɪɝɢɢ ɢɦɩɭɥɶɫɚ ɫɢɫɬɟ

ɦɵ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɱɚɫɬɢɰ

N

Pμ = ¦Pμ(a )

a =

ɝɞɟPμ(a)± ɷɧɟɪɝɢɹ ɢɦɩɭɥɶɫ ɨɬɞɟɥɶɧɨɣ ɱɚɫɬɢɰɵ

Ʉɜɚɞɪɚɬ ɜɟɤɬɨɪɚ ɟɫɬɶ ɢɧɜɚɪɢɚɧɬ ɉɨ ɚɧɚɥɨɝɢɢ ɫ ɱɚɫɬɢɰɟɣ ɜɜɟɞɟɦ ɦɚɫɫɭ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰM ɬɚɤ ɱɬɨɛɵ ɜɵɩɨɥɧɹɥɨɫɶ ɪɚɜɟɧɫɬɜɨ

ȼɜɟɞɟɦ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ K (C )ɫɜɹɡɚɧɧɭɸ ɫ ɰɟɧɬɪɨɦ ɢɧɟɪɰɢɢ ɫɢɫɬɟɦɵ ɨɬ

ɫɱɟɬɚ ɂɦɩɭɥɶɫ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɜ ɫɢɫɬɟɦɟK (C )ɪɚɜɟɧ ɧɭɥɸ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɫɢɫ ɬɟɦɵ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ P(C ) = ɉɨ ɨɬɧɨɲɟɧɢɸ ɤ ɩɪɨɢɡɜɨɥɶɧɨɣ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɟ ɨɬɫɱɟɬɚ K ɫɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɞɜɢɠɟɧɢɟ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɤɚɤ ɰɟɥɨɝɨ ɢ &

V&(C ) = PEc

Ɂɚɩɢɲɟɦ ɫɨɨɬɧɨɲɟɧɢɟ ɜ ɫɢɫɬɟɦɟ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ

(P&=C ) − (Ec(C ))= − M c

E(C ) = M c

Ɂɞɟɫɶ ɦɵ ɢɫɩɨɥɶɡɨɜɚɥɢ ɫɜɨɣɫɬɜɨ ɢɧɜɚɪɢɚɧɬɧɨɫɬɢ ɜɵɪɚɠɟɧɢɹ ɂɫ

ɩɨɥɶɡɭɹ ɩɟɪɜɭɸ ɮɨɪɦɭɥɭ ɧɚɣɞɟɦ

ɂɡ ɮɨɪɦɭɥ ɢ ɧɚɣɞɟɦ ɦɚɫɫɭ ɫɢɫɬɟɦɵ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ

ɱɚɫɬɢɰ

Ɂɞɟɫɶ ɫɤɨɪɨɫɬɢ ɱɚɫɬɢɰ ɨɬɧɟɫɟɧɵ ɤ ɫɢɫɬɟɦɟ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɂɡ ɮɨɪɦɭɥɵ

ɫɥɟɞɭɟɬ ɱɬɨ ɦɚɫɫɚ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɧɟ ɪɚɜɧɚ ɫɭɦɦɟ ɦɚɫɫ ɨɬɞɟɥɶɧɵɯ ɱɚɫ

ɬɢɰ ɫɢɫɬɟɦɵ Ɉɧɚ ɡɚɜɢɫɢɬ ɨɬ ɫɤɨɪɨɫɬɟɣ ɱɚɫɬɢɰ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɫɢɫɬɟɦɟ ɰɟɧɬɪɚ

ɢɧɟɪɰɢɢ

Ɇɚɫɫɭ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɩɪɟɞɫɬɚɜɢɦ ɜ ɜɢɞɟ

N

M = ¦ma + m

a =

ɝɞɟ m ± ɦɚɫɫɚ ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ

Ʉɢɧɟɦɚɬɢɤɚ ɹɞɟɪɧɵɯ ɪɟɚɤɰɢɣ

Ɋɟɡɭɥɶɬɚɬɵ ɩɪɟɞɵɞɭɳɟɝɨ ɩɚɪɚɝɪɚɮɚ ɩɨɡɜɨɥɹɸɬ ɪɚɫɫɦɨɬɪɟɬɶ ɤɢɧɟɦɚɬɢɤɭ

ɹɞɟɪɧɵɯ ɪɟɚɤɰɢɣ ɢ ɪɟɚɤɰɢɣ ɪɚɫɩɚɞɚ ɱɚɫɬɢɰ ɉɪɨɰɟɫɫ ɹɞɟɪɧɨɣ ɪɟɚɤɰɢɢ ɪɚɫɫɦɚɬ ɪɢɜɚɟɬɫɹ ɜ ɬɪɢ ɫɬɚɞɢɢ ɉɟɪɜɚɹ ɱɚɫɬɢɰɵ ɧɚɯɨɞɹɬɫɹ ɧɚ ɛɨɥɶɲɢɯ ɪɚɫɫɬɨɹɧɢɹɯ ɞɪɭɝ ɨɬ ɞɪɭɝɚ ɧɟ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɦɟɠɞɭ ɫɨɛɨɣ ɢ ɞɜɢɠɭɬɫɹ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɫɤɨɪɨɫɬɹɦɢ ȼɬɨɪɚɹ ɱɚɫɬɢɰɵ ɫɛɥɢɠɚɸɬɫɹ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɞɪɭɝ ɫ ɞɪɭɝɨɦ Ɍɪɟɬɶɹ ɱɚɫɬɢɰɵ ɩɨɫɥɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɪɚɡɥɟɬɚɸɬɫɹ ɧɚ ɛɨɥɶɲɢɟ ɪɚɫɫɬɨɹɧɢɹ ɧɟ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɢ ɞɜɢɠɭɬɫɹ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɫɤɨɪɨɫɬɹɦɢ Ɉ ɜɡɚɢɦɨɞɟɣɫɬɜɢɢ ɱɚɫɬɢɰ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɝɨɜɨɪɹɬ ɤɚɤ ɨ ɫɬɨɥɤɧɨɜɟɧɢɢ ɱɚɫɬɢɰ ɂɦɩɭɥɶɫ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɞɨ ɫɬɨɥɤɧɨɜɟɧɢɹ ɪɚɜɟɧ ɢɦɩɭɥɶɫɭ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɩɨ ɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɗɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɱɚɫɬɢɰ ɞɨ ɫɬɨɥɤɧɨɜɟɧɢɹ ɪɚɜɧɚ ɷɧɟɪɝɢɢ ɫɢɫ ɬɟɦɵ ɱɚɫɬɢɰ ɩɨɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ Ɂɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɢɦɩɭɥɶɫɚ ɢ ɷɧɟɪɝɢɢ ɦɨɠ ɧɨɪɚɫɫɦɚɬɪɢɜɚɬɶɤɚɤɡɚɤɨɧɫɨɯɪɚɧɟɧɢɹɢɦɩɭɥɶɫɚ ɷɧɟɪɝɢɢɫɢɫɬɟɦɵɱɚɫɬɢɰ Ɂɚɦɟɬɢɦɱɬɨɱɢɫɥɨɱɚɫɬɢɰɞɨɫɬɨɥɤɧɨɜɟɧɢɹɜɨɨɛɳɟɝɨɜɨɪɹɧɟɪɚɜɧɨɱɢɫɥɭ ɱɚɫɬɢɰ ɩɨɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ

Ɋɟɚɤɰɢɢ ɪɚɫɩɚɞɚ ɱɚɫɬɢɰɊɚɫɫɦɨɬɪɢɦ ɪɚɫɩɚɞ ɱɚɫɬɢɰɵ

M → m + m

ȼ ɫɢɫɬɟɦɟ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɢɦɩɭɥɶɫ ɞɨ ɪɚɫɩɚɞɚ ɪɚɜɟɧ ɧɭɥɸ ɢ

I = P c − E = − E = − M c

ɉɨɫɥɟ ɪɚɫɩɚɞɚ ɱɚɫɬɢɰɵ ɫ ɦɚɫɫɚɦɢm m ɢɦɟɸɬ ɢɦɩɭɥɶɫɵ P P ɢ ɷɧɟɪɝɢɢ

ɝɞɟT T ± ɤɢɧɟɬɢɱɟɫɤɢɟ ɷɧɟɪɝɢɢ ɱɚɫɬɢɰ ɂɡ ɮɨɪɦɭɥ ɢ ɩɨɥɭɱɚɟɦ

89

М. А. Михайлов. ЛЕКЦИИ ПО КЛАССИЧЕСКОЙ МЕХАНИКЕ

M c = (m + m )c + T + T

ɂɡ ɛɚɥɚɧɫɚ ɷɧɟɪɝɢɢ ɫɥɟɞɭɟɬ ɧɟɨɛɯɨɞɢɦɨɟ ɭɫɥɨɜɢɟ ɪɚɫɩɚɞɚ ɱɚɫ

ɬɢɰɵ

M > m + m

Ɇɚɫɫɚɪɚɫɩɚɞɚɸɳɟɣɫɹɱɚɫɬɢɰɵɞɨɥɠɧɚɛɵɬɶɛɨɥɶɲɟɫɭɦɦɵɦɚɫɫɨɛɪɚ ɡɭɸɳɢɯɫɹ ɱɚɫɬɢɰ

ɉɨɫɤɨɥɶɤɭ

ɗɧɟɪɝɢɟɣ ɪɚɫɩɚɞɚ ɧɚɡɵɜɚɟɬɫɹ ɜɟɥɢɱɢɧɚ

Q = T + T = (M − m − m )c

ɂɫɩɨɥɶɡɭɹ ɨɩɪɟɞɟɥɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɪɚɫɩɚɞɚɸɳɢɯɫɹ ɱɚɫɬɢɰ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɜ ɫɢɫɬɟɦɟ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɷɧɟɪɝɢɢ ɨɛɪɚɡɭɸɳɢɯɫɹ ɱɚɫɬɢɰ

ɩɨɥɧɨɫɬɶɸ ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɯ ɦɚɫɫɚɦɢ ɢ ɦɚɫɫɨɣ ɪɚɫɩɚɞɚɸɳɟɣɫɹ ɱɚɫɬɢɰɵ ȿɫɥɢ m >> m ɬɨT << T Q ± ɛɨɥɶɲɭɸ ɱɚɫɬɶ ɷɧɟɪɝɢɢ ɪɚɫɩɚɞɚ ɭɧɨɫɢɬ ɥɟɝɤɚɹ ɱɚɫɬɢ ɰɚ Ɍɚɤɚɹ ɫɢɬɭɚɰɢɹ ɯɚɪɚɤɬɟɪɧɚ ɞɥɹ ɚɥɶɮɚ ɪɚɫɩɚɞɚ

ɗɧɟɪɝɟɬɢɱɟɫɤɢɣ ɜɵɯɨɞ ɹɞɟɪɧɨɣ ɪɟɚɤɰɢɢ ɉɨɪɨɝ ɷɧɞɨɬɟɪɦɢɱɟɫɤɨɣ ɪɟɚɤ ɰɢɢ Ɋɚɫɫɦɨɬɪɢɦ ɪɟɚɤɰɢɸ ɬɢɩɚ

a + b → c + d

ȼɟɥɢɱɢɧɚ

Q = [(ma + mb )− (mc + md )]c

ɧɚɡɵɜɚɟɬɫɹ ɷɧɟɪɝɟɬɢɱɟɫɤɢɦ ɜɵɯɨɞɨɦ ɪɟɚɤɰɢɢ ȿɫɥɢ Q > ɪɟɚɤɰɢɹ ɧɚɡɵɜɚɟɬɫɹ ɷɤɡɨɬɟɪɦɢɱɟɫɤɨɣ ɟɫɥɢ Q < ɪɟɚɤɰɢɹ ɧɚɡɵɜɚɟɬɫɹ ɷɧɞɨɬɟɪɦɢɱɟɫɤɨɣ ɉɪɢ ɭɩɪɭ ɝɨɦ ɫɬɨɥɤɧɨɜɟɧɢɢQ =

ɉɭɫɬɶ Q < Ɋɟɚɤɰɢɸ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɪɟɚɤɰɢɸ ɪɚɫɩɚɞɚ ɫɢɫɬɟɦɵ ɧɟɜɡɚɢɦɨɞɟɣɫɬɜɭɸɳɢɯ ɱɚɫɬɢɰa ɢbɉɭɫɬɶM ± ɦɚɫɫɚ ɬɚɤɨɣ ɫɢɫɬɟɦɵ Ɍɨɝɞɚ ɧɟɨɛɯɨɞɢɦɨɟ ɭɫɥɨɜɢɟ ɪɚɫɩɚɞɚ

M > mc + md