method_dynamics
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ɝɞɟ a(t) |
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ɉɪɢ k§p a(t) ɹɜɥɹɟɬɫɹ |
ɦɟɞɥɟɧɧɨ |
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ɢɡɦɟɧɹɸɳɟɣɫɹ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɮɭɧɤɰɢɟɣ ɫ ɩɟɪɢɨɞɨɦ |
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Ʉɚɤ ɫɥɟɞɭɟɬ ɢɡ ɭɪɚɜɧɟɧɢɹ (5), ɞɜɢɠɟɧɢɟ ɩɪɨɢɫɯɨɞɢɬ ɫ ɤɪɭɝɨɜɨɣ ɱɚɫɬɨɬɨɣ ɪ, ɩɨɷɬɨɦɭ ɩɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ ɪɚɜɟɧ
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(7) |
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Ɍɚɤ ɤɚɤ k § p, ɬɨ Ɍɚ >> T, ɩɪɢɱɟɦ |
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ɉɨɞɫɬɚɜɥɹɹ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɩɨɥɭɱɢɦ |
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x a(t)cos(101 t), ɝɞɟ |
a(t) 80sin |
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Ɍ |
ɚ 12,56 ɫ, Ɍ 0,063 ɫ. |
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Ɂɚɞɚɱɚ ʋ 2. ɉɧɟɜɦɚɬɢɱɟɫɤɢɣ ɨɬɛɨɣɧɵɣ ɦɨɥɨɬɨɤ ɩɪɢɜɨɞɢɬɫɹ ɜ ɞɜɢɠɟɧɢɟ ɫɠɚɬɵɦ ɜɨɡɞɭɯɨɦ, ɩɨɫɬɭɩɚɸɳɢɦ ɜ ɤɨɪɩɭɫ ɦɨɥɨɬɤɚ ɱɟɪɟɡ ɲɥɚɧɝ Ⱥ.
Ⱦɚɜɥɟɧɢɟ ɜɨɡɞɭɯɚ, ɩɪɢɥɨɠɟɧɧɨɟ ɤ ɩɨɪɲɧɸ D ɦɨɥɨɬɤɚ, ɢɡɦɟɧɹɟɬɫɹ ɫɨɝɥɚɫɧɨ ɭɪɚɜɧɟɧɢɸ
S H0 H1 cos(pt) H3 cos(3pt),
ɝɞɟ ɪ, ɇ0, ɇ1, ɇ3 – ɩɨɫɬɨɹɧɧɵɟ ɜɟɥɢɱɢɧɵ. ȼ ɤɨɪɩɭɫ ɦɨɥɨɬɤɚ ɜɦɨɧɬɢɪɨɜɚɧɚ ɩɪɭɠɢɧɚ ȼ ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɠɟɫɬɤɨɫɬɢ ɫ. ɉɪɭɠɢɧɚ ɭɩɢɪɚɟɬɫɹ ɥɟɜɵɦ ɤɨɧɰɨɦ ɜ ɩɨɪɲɟɧɶ, ɚ ɩɪɚɜɵɦ –
Ɋɟɲɟɧɢɟ.
ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɩɨ ɝɨɪɢɡɨɧɬɚɥɢ ɧɚɩɪɚɜɨ, ɜɡɹɜ ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɜ ɩɨɥɨɠɟɧɢɢ ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɩɨɪɲɧɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ ɇ0 ɢ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ ɩɪɭɠɢɧɵ Fɫɬ . ȼ ɷɬɨɦ ɩɨɥɨɠɟɧɢɢ ɩɪɭɠɢɧɚ ɫɠɚɬɚ ɧɚ d ɫɢɥɨɣ ɇ0. ɉɪɢ ɷɬɨɦ ɜɨɡɧɢɤɚɟɬ ɫɢɥɚ ɭɩɪɭɝɨɫɬɢ ɩɪɭɠɢɧɵ Fɫɬ = ɫd. Ɂɚɩɢɲɟɦ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ ɩɨɪɲɧɹ ɜ ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ ɯ: H0 cd 0 .
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ɂɡɨɛɪɚɡɢɦ ɩɨɪɲɟɧɶ ɫɦɟɳɟɧɧɵɦ ɢɡ ɧɭɥɹ ɧɚɩɪɚɜɨ ɧɚ ɯ, ɩɪɢ ɷɬɨɦ ɩɪɨɟɤɰɢɹ ɧɚ ɨɫɶ ɯ ɜɨɡɧɢɤɲɟɣ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ F ɪɚɜɧɚ:
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Fx cdx |
c(d x) . |
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Ʉɪɨɦɟ ɬɨɝɨ, ɤ ɩɨɪɲɧɸ ɩɪɢɥɨɠɟɧɵ ɫɢɥɵ: Ɋ – ɜɟɫ, N – ɧɨɪɦɚɥɶɧɚɹ |
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ɪɟɚɤɰɢɹ ɤɨɪɩɭɫɚ, S – ɫɢɥɚ ɞɚɜɥɟɧɢɹ ɫɠɚɬɨɝɨ ɜɨɡɞɭɯɚ. |
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ɋɨɫɬɚɜɢɦ |
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ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ |
ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɩɨɪɲɧɹ |
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mx Sx Fx . |
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ɍɱɢɬɵɜɚɹ (1) ɢ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ S, ɧɚɯɨɞɢɦ |
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H1 cos( pt) H3 cos(3pt) cd cx. |
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ɍɱɢɬɵɜɚɹ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ, ɡɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɜɵɧɭɠɞɟɧɧɵɯ |
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ɤɨɥɟɛɚɧɢɣ ɩɨɪɲɧɹ ɜ ɜɢɞɟ |
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x k2 x h1 cos(pt) h3 cos(3pt) , |
(9) |
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ɝɞɟ |
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k |
cg , h |
H1g , h |
H3 g . |
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ɇɚɣɞɟɦ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (2) ɜ ɜɢɞɟ |
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x2 |
A1 sin( pt) B1 cos( pt) A2 sin(3pt) B2 cos(3pt) . |
(10) |
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Ʉɨɷɮɮɢɰɢɟɧɬɵ Ⱥ1, ȼ1, Ⱥ2, ȼ2 ɧɚɣɞɟɦ, ɜɵɱɢɫɥɢɜ ɩɟɪɜɭɸ ɢ ɜɬɨɪɭɸ ɩɪɨɢɡɜɨɞɧɵɟ ɨɬ ɯ2 ɢ ɩɨɞɫɬɚɜɢɜ ɧɚɣɞɟɧɧɵɟ ɜɵɪɚɠɟɧɢɹ ɜ (2), ɚ ɡɚɬɟɦ ɩɪɢɪɚɜɧɹɜ ɤɨɷɮɮɢɰɢɟɧɬɵ, ɫɬɨɹɳɢɟ ɜ ɩɪɚɜɨɣ ɢ ɥɟɜɨɣ ɱɚɫɬɹɯ ɭɪɚɜɧɟɧɢɹ ɩɪɢ ɫɢɧɭɫɟ ɢ ɤɨɫɢɧɭɫɟ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ
A1 0, B1 |
h1 |
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A3 |
0, B3 |
h3 |
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k 2 p2 |
k 2 9p2 |
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ɉɨɞɫɬɚɜɢɜ ɷɬɢ ɤɨɷɮɮɢɰɢɟɧɬɵ ɜ (3), ɧɚɯɨɞɢɦ ɢɫɤɨɦɨɟ ɭɪɚɜɧɟɧɢɟ ɜɵɧɭɠɞɟɧɧɵɯ ɤɨɥɟɛɚɧɢɣ ɩɨɪɲɧɹ:
x |
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h1 |
cos(pt) |
h3 |
cos(3pt). |
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k2 p2 |
k2 p2 |
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ȼɫɥɭɱɚɟ k = p ɧɚɫɬɭɩɚɸɬ ɪɟɡɨɧɚɧɫɧɵɟ ɤɨɥɟɛɚɧɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ.
ȼɫɥɭɱɚɟ k = 3p ɧɚɫɬɭɩɚɸɬ ɪɟɡɨɧɚɧɫɧɵɟ ɤɨɥɟɛɚɧɢɹ ɬɪɟɬɶɟɝɨ ɩɨɪɹɞɤɚ.
Ɍɚɤ ɤɚɤ k |
cg |
, ɬɨ ɩɨɞɛɨɪ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɭɩɪɭɝɨɫɬɢ ɩɪɭɠɢɧɵ |
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ɫɥɟɞɭɟɬ ɩɪɨɢɡɜɨɞɢɬɶ ɬɚɤ, ɱɬɨɛɵ ɨɛɟɫɩɟɱɢɬɶ ɜɵɩɨɥɧɟɧɢɟ ɧɟɪɚɜɟɧɫɬɜ k p ɢ k 3p. ɉɪɢ ɷɬɨɦ ɩɨɪɲɟɧɶ ɧɟ ɛɭɞɟɬ ɩɨɩɚɞɚɬɶ ɜ ɪɟɡɨɧɚɧɫ.
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§7. Ɍɟɨɪɟɦɚ ɨ ɞɜɢɠɟɧɢɢ ɰɟɧɬɪɚ ɦɚɫɫ |
Ɂɚɞɚɱɚ ʋ 1. |
Ȼɚɬɶ Ɇ.ɂ. |
«Ɍɟɨɪɟɬɢɱɟɫɤɚɹ |
ɦɟɯɚɧɢɤɚ ɜ |
ɩɪɢɦɟɪɚɯ ɢ ɡɚɞɚɱɚɯ», ɬɨɦ 2, |
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ɇɚɭɤɚ, 1975. |
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Ɍɨɧɤɢɣ |
ɨɞɧɨɪɨɞɧɵɣ |
ɫɬɟɪɠɟɧɶ ɈȺ ɞɥɢɧɨɣ l ɢ ɜɟɫɨɦ Ɋ |
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ɜɪɚɳɚɟɬɫɹ ɜɨɤɪɭɝ ɜɟɪɬɢɤɚɥɶɧɨɣ |
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ɨɫɢ Ɉ1Ɉ2 ɫ ɩɨɫɬɨɹɧɧɨɣ ɭɝɥɨɜɨɣ |
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ɫɤɨɪɨɫɬɶɸ Ζ. |
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Ɉɩɪɟɞɟɥɢɬɶ ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ |
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ɜɧɟɲɧɢɯ ɫɢɥ. Ɇɚɫɫɨɣ ɨɫɢ Ɉ1Ɉ2 |
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ɩɪɟɧɟɛɪɟɱɶ. |
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Ɋɟɲɟɧɢɟ. ȼ |
ɫɨɨɬɜɟɬɫɬɜɢɢ |
ɫ ɬɟɨɪɟɦɨɣ |
ɨ |
ɞɜɢɠɟɧɢɢ ɰɟɧɬɪɚ ɦɚɫɫ |
ɫɢɫɬɟɦɵ |
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n |
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ɦɚɬɟɪɢɚɥɶɧɵɯ |
ɬɨɱɟɤ Mwc |
= ¦ |
Fke |
ɞɥɹ |
ɨɩɪɟɞɟɥɟɧɢɹ ɝɥɚɜɧɨɝɨ |
ɜɟɤɬɨɪɚ |
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k 1 |
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n
ɜɧɟɲɧɢɯ ɫɢɥ ɫɢɫɬɟɦɵ R= ¦Fke ɞɨɫɬɚɬɨɱɧɨ ɧɚɣɬɢ MwC . Ɍɚɤ ɤɚɤ ɰɟɧɬɪ ɦɚɫɫ
k 1
ɫɬɟɪɠɧɹɧɚɯɨɞɢɬɫɹɜɬɨɱɤɟ ɋɧɚɪɚɫɫɬɨɹɧɢɢ l/2 ɨɬɨɫɢɜɪɚɳɟɧɢɹɢɢɦɟɟɬ, ɜɫɢɥɭ
ɩɨɫɬɨɹɧɫɬɜɚ |
ɜɟɤɬɨɪɚ |
Ζ, |
ɬɨɥɶɤɨ |
ɰɟɧɬɪɨɫɬɪɟɦɢɬɟɥɶɧɨɟ |
ɭɫɤɨɪɟɧɢɟ |
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wn OCΖ 2 |
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Ζ 2 |
, ɤɨɬɨɪɨɟ ɧɚɩɪɚɜɥɟɧɨ ɜɞɨɥɶ ɫɬɟɪɠɧɹ ɨɬ ɋ ɤ Ɉ, ɬɨ ɝɥɚɜɧɵɣ |
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ɜɟɤɬɨɪ ɜɧɟɲɧɢɯ ɫɢɥ ɫɢɫɬɟɦɵ R ɢɦɟɟɬ ɬɨ ɠɟ ɧɚɩɪɚɜɥɟɧɢɟ ɢ ɪɚɜɟɧ ɩɨ ɦɨɞɭɥɸ
Pl
C2g Ζ2 .
ȼɞɚɧɧɨɦ ɫɥɭɱɚɟ ɝɥɚɜɧɵɣ ɜɟɤɬɨɪ ɜɧɟɲɧɢɯ ɫɢɥ ɹɜɥɹɟɬɫɹ ɜɟɤɬɨɪɧɨɣ ɫɭɦɦɨɣ ɜɟɫɚ ɫɬɟɪɠɧɹ ɢ ɪɟɚɤɰɢɣ ɨɩɨɪ Ɉ1 ɢ Ɉ2.
Ɂɚɞɚɱɚ ʋ 2. Ȼɚɬɶ Ɇ.ɂ. «Ɍɟɨɪɟɬɢɱɟɫɤɚɹ ɦɟɯɚɧɢɤɚ ɜ ɩɪɢɦɟɪɚɯ ɢ ɡɚɞɚɱɚɯ», ɬɨɦ 2, ɇɚɭɤɚ,1975.
Ʉɨɥɟɫɨ ɜɟɫɨɦ Ɋ ɤɚɬɢɬɫɹ ɫɨ ɫɤɨɥɶɠɟɧɢɟɦ ɩɨ ɩɪɹɦɨɥɢɧɟɣɧɨɦɭ ɝɨɪɢɡɨɧɬɚɥɶɧɨɦɭ ɪɟɥɶɫɭ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɩɨɫɬɨɹɧɧɨɣ ɫɢɥɵ F, ɩɪɢɥɨɠɟɧɧɨɣɤɟɝɨɰɟɧɬɪɭɬɹɠɟɫɬɢ ɋ. ɇɚɣɬɢ ɫɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɦɚɫɫ ɋ ɤɨɥɟɫɚ, ɟɫɥɢɜɧɚɱɚɥɶɧɵɣɦɨɦɟɧɬɨɧɨ ɧɚɯɨɞɢɥɨɫɶ ɜ ɩɨɤɨɟ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɪɚɜɟɧ f. Ɉɫɢ x, y ɢɡɨɛɪɚɠɟɧɵɧɚɪɢɫɭɧɤɟ.
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Ɋɟɲɟɧɢɟ. Ʉ ɤɨɥɟɫɭ ɩɪɢɥɨɠɟɧɵ ɜɧɟɲɧɢɟ ɫɢɥɵ: Ɋ – ɟɝɨ ɜɟɫ, F – ɞɜɢɠɭɳɚɹ ɫɢɥɚ, R – ɧɨɪɦɚɥɶɧɚɹ ɪɟɚɤɰɢɹ ɪɟɥɶɫɚ, Fɬɪ – ɫɢɥɚ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ, ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɞɨɥɶ ɪɟɥɶɫɚ ɜ ɫɬɨɪɨɧɭ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ ɫɢɥɟ F.
ɉɪɢɦɟɧɢɦ ɬɟɨɪɟɦɭ ɨ ɞɜɢɠɟɧɢɢ ɰɟɧɬɪɚ ɦɚɫɫ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɯ ɢ ɭ:
Mxc |
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F Fɬɪ , |
Myc |
R P |
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ɉɪɢ |
ɞɜɢɠɟɧɢɢ |
ɤɨɥɟɫɚ |
yC |
r const . |
ɉɨɷɬɨɦɭ |
ɭC |
0 , ɢ ɢɡ ɜɬɨɪɨɝɨ |
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ɭɪɚɜɧɟɧɢɹ ɫɥɟɞɭɟɬ R = P. Ɍɚɤ ɤɚɤ ɩɪɢ ɤɚɱɟɧɢɢ ɤɨɥɟɫɚ ɫɨ ɫɤɨɥɶɠɟɧɢɟɦ ɫɢɥɚ |
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ɬɪɟɧɢɹ ɞɨɫɬɢɝɚɟɬ ɫɜɨɟɝɨ ɧɚɢɛɨɥɶɲɟɝɨ ɡɧɚɱɟɧɢɹ, ɬɨ Fɬɪ |
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fR . ɂɫɩɨɥɶɡɨɜɚɜ |
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ɷɬɨ ɡɧɚɱɟɧɢɟ Fɬɪ |
ɜ ɩɟɪɜɨɦ ɭɪɚɜɧɟɧɢɢ (1), ɢɦɟɟɦ xC |
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ɉɨɞɫɬɚɧɨɜɤɚ ɧɚɱɚɥɶɧɨɝɨ ɭɫɥɨɜɢɹ t = 0, ɯ |
0 (ɤɨɥɟɫɨ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ |
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ɧɚɯɨɞɢɥɨɫɶ ɜ ɩɨɤɨɟ) ɜ ɭɪɚɜɧɟɧɢɟ ɞɚɟɬ ɋ1=0. ȼɧɟɫɹ ɷɬɨ ɡɧɚɱɟɧɢɟ ɋ1 ɩɨɥɭɱɢɦ ɢɫɤɨɦɵɣ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɢ ɰɟɧɬɪɚ ɦɚɫɫ ɋ
ɤɨɥɟɫɚ: xC g F fP t . Ⱦɜɢɠɟɧɢɟ ɤɨɥɟɫɚ ɜɨɡɦɨɠɧɨ ɩɪɢ ɧɚɥɢɱɢɢ
P
ɧɟɪɚɜɟɧɫɬɜɚ F > fP.
§8. Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ
Ɂɚɞɚɱɚ ʋ 1.
ɉɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɚɬɮɨɪɦɟ Ⱥ ɞɜɢɠɭɳɟɣɫɹ ɩɨ ɢɧɟɪɰɢɢ ɫɨ
ɫɤɨɪɨɫɬɶɸ −0 , ɩɟɪɟɦɟɳɚɟɬɫɹ ɬɟɥɟɠɤɚ ȼ ɫ ɩɨɫɬɨɹɧɧɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ
ɧɟɤɨɬɨɪɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɬɟɥɟɠɤɚ ɛɵɥɚ ɡɚɬɨɪɦɨɠɟɧɚ.
Ɉɩɪɟɞɟɥɢɬɶ: ɨɛɳɭɸ ɫɤɨɪɨɫɬɶ − ɩɥɚɬɮɨɪɦɵ ɫ ɬɟɥɟɠɤɨɣ ɩɨɫɥɟ ɟɟ ɨɫɬɚɧɨɜɤɢ. M – ɦɚɫɫɚ ɩɥɚɬɮɨɪɦɵ, m – ɦɚɫɫɚ ɬɟɥɟɠɤɢ.
Ɋɟɲɟɧɢɟ. ɉɨ ɬɟɨɪɟɦɟ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ
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Ɍɚɤ ɤɚɤ ɬɨɪɦɨɠɟɧɢɟ ɬɟɥɟɠɤɢ ɩɪɨɢɫɯɨɞɢɬ ɫ «ɩɨɦɨɳɶɸ» ɜɧɭɬɪɟɧɧɟɣ ɫɢɥɵ, ɬɨ:
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↑"0" ɦɨɦɟɧɬ ɞɜɢɠɟɧɢɹ, |
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Q 0 |
Q const Q0 |
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↓"1" ɦɨɦɟɧɬ ɨɫɬɚɧɨɜɤɢ. |
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Ʉɨɥɢɱɟɫɬɜɨ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ ɫɨɯɪɚɧɹɟɬɫɹ. |
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M−0 m(−0 u0 ) M− m−, |
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Ɂɚɞɚɱɚ ʋ 2. Ⱥɪɤɭɲɚ Ⱥ.ɂ. «Ɋɭɤɨɜɨɞɫɬɜɨ ɤ ɪɟɲɟɧɢɸ ɡɚɞɚɱ ɩɨ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ», Ɇɨɫɤɜɚ, ȼɵɫɲɚɹ ɲɤɨɥɚ,1999.
Ɇɚɲɢɧɢɫɬ ɬɟɩɥɨɜɨɡɚ ɨɬɤɥɸɱɚɟɬ ɞɜɢɝɚɬɟɥɶ ɢ ɧɚɱɢɧɚɟɬ ɬɨɪɦɨɡɢɬɶ ɜ ɦɨɦɟɧɬ, ɤɨɝɞɚ ɬɟɩɥɨɜɨɡ ɢɦɟɟɬ ɫɤɨɪɨɫɬɶ 90 ɤɦ/ɱ. ɑɟɪɟɡ ɫɤɨɥɶɤɨ ɜɪɟɦɟɧɢ ɬɟɩɥɨɜɨɡ ɨɫɬɚɧɨɜɢɬɫɹ, ɟɫɥɢ ɫɢɥɚ ɬɨɪɦɨɠɟɧɢɹ ɩɨɫɬɨɹɧɧɚ ɢ ɫɨɫɬɚɜɥɹɟɬ 0,12 ɟɝɨ ɜɟɫɚ, ɚ ɞɜɢɠɟɧɢɟ ɩɪɨɢɫɯɨɞɢɬ ɩɨ ɝɨɪɢɡɨɧɬɚɥɶɧɨɦɭ ɢ ɪɨɜɧɨɦɭ ɭɱɚɫɬɤɭ ɞɨɪɨɝɢ?
Ɋɟɲɟɧɢɟ.
1.Ɍɟɩɥɨɜɨɡ ɞɜɢɠɟɬɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ, ɩɨɬɨɦɭ ɪɚɫɫɦɨɬɪɢɦ ɞɜɢɠɟɧɢɟ ɟɝɨ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɋ (ɰɟɧɬɪɚ ɦɚɫɫɵ), ɫɱɢɬɚɹ ɱɬɨ ɤ ɧɟɦɭ ɩɪɢɥɨɠɟɧɵ ɜɫɟ ɜɧɟɲɧɢɟ ɫɢɥɵ.
2.ɉɨɫɥɟ ɬɨɝɨ, ɤɚɤ ɨɬɤɥɸɱɚɟɬɫɹ ɞɜɢɝɚɬɟɥɶ ɢ ɜɤɥɸɱɚɟɬɫɹ ɬɨɪɦɨɡɧɨɟ ɭɫɬɪɨɣɫɬɜɨ, ɧɚ ɬɟɩɥɨɜɨɡ ɞɟɣɫɬɜɭɸɬ ɬɪɢ ɫɢɥɵ: ɫɢɥɚ ɬɹɠɟɫɬɢ G, ɧɨɪɦɚɥɶɧɚɹ ɪɟɚɤɰɢɹ ɪɟɥɶɫɨɜ R ɢ ɫɢɥɚ ɬɨɪɦɨɠɟɧɢɹ F. ȼ ɧɚɱɚɥɟ ɬɨɪɦɨɠɟɧɢɹ ɫɤɨɪɨɫɬɶ
V0 = 90ɤɦ/ɱ = 25ɦ/ɫ, ɜ ɤɨɧɰɟ V = 0. Ɍɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɩɭɬɶ S ɢ ɜɪɟɦɹ t, ɡɚ ɤɨɬɨɪɨɟ ɷɬɨɬ ɩɭɬɶ ɩɪɨɣɞɟɧ.
3. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɜɪɟɦɟɧɢ ɬɨɪɦɨɠɟɧɢɹ ɩɪɢɦɟɧɢɦ ɬɟɨɪɟɦɭ ɨɛ
Ω
ɢɡɦɟɧɟɧɢɢ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ: Q1 Q0 ³F (t)dt .
0
ɋɩɪɨɟɰɢɪɨɜɚɜ ɜɟɤɬɨɪɵ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɨɫɶ (ɨɫɶ ɯ), ɭɜɢɞɢɦ, ɱɬɨ ɩɪɨɟɤɰɢɢ ɫɢɥ G ɢ Rn ɪɚɜɧɵ ɧɭɥɸ, ɚ ɩɪɨɟɤɰɢɹ ɫɢɥɵ F ɩɨɥɭɱɚɟɬɫɹ ɪɚɜɧɨɣ
ɟɟ ɦɨɞɭɥɸ, ɧɨ ɫɨ ɡɧɚɤɨɦ ɦɢɧɭɫ: ɩɪɨɟɤɰɢɹ ɫɤɨɪɨɫɬɢ V0 ɬɚɤɠɟ ɪɚɜɧɚ ɟɟ ɦɨɞɭɥɸ, ɩɨɷɬɨɦɭ –Ft = –mV0.
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4. Ɋɟɲɚɟɦ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɨɬɧɨɫɢɬɟɥɶɧɨ t: t = mV0/F. Ɍɚɤ ɤɚɤ ɫɢɥɚ
ɬɨɪɦɨɠɟɧɢɹ |
F = 0, |
12G = 0,12mg, |
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Ɂɚɞɚɱɚ ʋ 3. Ⱥɪɤɭɲɚ Ⱥ.ɂ. «Ɋɭɤɨɜɨɞɫɬɜɨ ɤ ɪɟɲɟɧɢɸ ɡɚɞɚɱ ɩɨ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɟ», Ɇɨɫɤɜɚ, ȼɵɫɲɚɹ ɲɤɨɥɚ,1999.
Ʉɚɤɨɣ ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɤɨɥɟɫ ɡɚɬɨɪɦɨɠɟɧɧɨɝɨ ɚɜɬɨɦɨɛɢɥɹ ɨ ɞɨɪɨɝɭ (ɫɱɢɬɚɬɶ, ɱɬɨ ɡɚɬɨɪɦɨɠɟɧɵ ɜɫɟ ɱɟɬɵɪɟ ɤɨɥɟɫɚ), ɟɫɥɢ ɜ ɦɨɦɟɧɬ ɜɵɤɥɸɱɟɧɢɹ ɞɜɢɝɚɬɟɥɹ ɢ ɧɚɠɚɬɢɹ ɬɨɪɦɨɡɚ ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɚɜɬɨɦɨɛɢɥɹ V0 = 60 ɤɦ/ɱ ɢ ɚɜɬɨɦɨɛɢɥɶ ɨɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɱɟɪɟɡ 5 ɫ ɩɨɫɥɟ ɧɚɱɚɥɚ ɬɨɪɦɨɠɟɧɢɹ.
Ɋɟɲɟɧɢɟ.
1.ȼ ɡɚɞɚɱɟ ɢɡɜɟɫɬɧɨ ɜɪɟɦɹ ɞɜɢɠɟɧɢɹ ɡɚɬɨɪɦɨɠɟɧɧɨɝɨ ɚɜɬɨɦɨɛɢɥɹ, ɬ. ɟ. ɢɦɟɟɬɫɹ ɜ ɜɢɞɭ ɢɦɩɭɥɶɫ ɫɢɥɵ, ɩɨɷɬɨɦɭ ɞɥɹ ɪɟɲɟɧɢɹ ɩɪɢɦɟɧɢɦ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ.
2.ɇɚ ɡɚɬɨɪɦɨɠɟɧɧɵɣ ɚɜɬɨɦɨɛɢɥɶ ɞɟɣɫɬɜɭɟɬ ɞɟɜɹɬɶ ɫɢɥ; G – ɜɟɫ ɚɜɬɨɦɨɛɢɥɹ, ɱɟɬɵɪɟ ɪɟɚɤɰɢɢ ɩɨɜɟɪɯɧɨɫɬɢ ɞɨɪɨɝɢ, ɩɪɢɥɨɠɟɧɧɵɟ ɤ ɤɚɠɞɨɦɭ
ɤɨɥɟɫɭ Ri , ɢ ɱɟɬɵɪɟ ɫɢɥɵ ɬɪɟɧɢɹ Ri f , ɬɚɤɠɟ ɩɪɢɥɨɠɟɧɧɵɟ ɤ ɤɨɥɟɫɚɦ.
ɉɪɢɧɢɦɚɹ ɚɜɬɨɦɨɛɢɥɶ ɡɚ ɦɚɬɟɪɢɚɥɶɧɭɸ ɬɨɱɤɭ, ɫɱɢɬɚɟɦ, ɱɬɨ ɜɫɟ ɷɬɢ ɫɢɥɵ ɩɪɢɥɨɠɟɧɵ ɜ ɰɟɧɬɪɟ ɬɹɠɟɫɬɢ ɚɜɬɨɦɨɛɢɥɹ, ɢ ɬɨɝɞɚ, ɡɚɦɟɧɢɜ ɱɟɬɵɪɟ
ɪɟɚɤɰɢɢ ɩɨɜɟɪɯɧɨɫɬɢ ɢɯ ɫɭɦɦɨɣ Rn ɢ ɱɟɬɵɪɟ ɫɢɥɵ ɬɪɟɧɢɹ ɢɯ ɫɭɦɦɨɣ Rf ,
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ɩɨɥɭɱɢɦ ɬɨɥɶɤɨ ɬɪɢ ɫɢɥɵ G , |
Rn , Rf . |
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3. ɋɢɥɵ |
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ɱɢɫɥɟɧɧɨ ɪɚɜɧɵ ɞɪɭɝ ɞɪɭɝɭ ɢ ɜɡɚɢɦɧɨ |
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ɭɪɚɜɧɨɜɟɲɢɜɚɸɬɫɹ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɦɩɭɥɶɫ ɫɨɡɞɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ ɫɢɥɨɣ ɬɪɟɧɢɹ Rf Rn f Gf .
4. ɂɦɩɭɥɶɫ ɫɢɥɵ ɬɪɟɧɢɹ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɞɟɣɫɬɜɭɟɬ ɜ ɫɬɨɪɨɧɭ, ɩɪɨɬɢɜɨɩɨɥɨɠɧɭɸ ɞɜɢɠɟɧɢɸ, ɩɨɷɬɨɦɭ ɬɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɨɥɢɱɟɫɬɜɚ
t
ɞɜɢɠɟɧɢɹ ɞɥɹ ɞɚɧɧɨɣ ɡɚɞɚɱɢ ɢɦɟɟɬ ɜɢɞ mV1 mV0 ³Gfdt .
0
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ɇɨ ɚɜɬɨɦɨɛɢɥɶ ɱɟɪɟɡ |
t =5ɫ |
ɨɫɬɚɧɚɜɥɢɜɚɟɬɫɹ, mV0 |
Gft , ɩɨɷɬɨɦɭ |
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V0 |
fGt , Ɉɬɤɭɞɚ, |
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ɜ ɜɢɞɭ, ɱɬɨ V0 = 60 |
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§9. Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɦɨɦɟɧɬɚ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ
Ɂɚɞɚɱɚ ʋ 1. Ȼɚɬɶ Ɇ.ɂ. «Ɍɟɨɪɟɬɢɱɟɫɤɚɹ ɦɟɯɚɧɢɤɚ ɜ ɩɪɢɦɟɪɚɯ ɢ ɡɚɞɚɱɚɯ», ɬɨɦ 2, ɇɚɭɤɚ,1975.
ɉɪɢ ɜɪɚɳɟɧɢɢ ɛɚɪɚɛɚɧɚ 1 ɜɟɫɨɦ Ɋ1 ɢ ɪɚɞɢɭɫɨɦ r ɜɨɤɪɭɝ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ z ɧɚ ɟɝɨ ɛɨɤɨɜɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɧɚɦɚɬɵɜɚɟɬɫɹ ɧɢɬɶ, ɤɨɬɨɪɚɹ ɩɪɢɜɨɞɢɬ ɜ ɞɜɢɠɟɧɢɟ ɝɪɭɡ 2 ɜɟɫɨɦ Ɋ2, ɫɤɨɥɶɡɹɳɢɣ ɩɨ ɧɟɩɨɞɜɢɠɧɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɥɨɫɤɨɫɬɢ. Ɉɩɪɟɞɟɥɢɬɶ ɭɝɥɨɜɨɟ ɭɫɤɨɪɟɧɢɟ ɛɚɪɚɛɚɧɚ, ɟɫɥɢ ɤ ɧɟɦɭ ɩɪɢɥɨɠɟɧ ɜɪɚɳɚɸɳɢɣ ɦɨɦɟɧɬ mɜɪ, ɚ ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɝɪɭɡɚ ɨ ɩɥɨɫɤɨɫɬɶ ɪɚɜɟɧ f. ȼɵɫɨɬɨɣ ɝɪɭɡɚ ɩɪɟɧɟɛɪɟɱɶ.
Ɋɟɲɟɧɢɟ. ɉɪɢɦɟɧɢɦ ɬɟɨɪɟɦɭ ɨɛ ɢɡɦɟɧɟɧɢɢ ɝɥɚɜɧɨɝɨ ɦɨɦɟɧɬɚ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ ɫɢɫɬɟɦɵ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z,
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¦momZ (Fke ) . |
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ɂɡɨɛɪɚɡɢɦ ɜɧɟɲɧɢɟ ɫɢɥɵ ɢ ɦɨɦɟɧɬɵ ɫɢɫɬɟɦɵ: mɜɪ – ɜɪɚɳɚɸɳɢɣ ɦɨɦɟɧɬ; Ɋ1 – ɜɟɫ ɛɚɪɚɛɚɧɚ, Ɋ2 – ɜɟɫ ɝɪɭɡɚ, R1 ɢ R1' – ɫɨɫɬɚɜɥɹɸɳɢɟ ɪɟɚɤɰɢɢ ɨɫɢ ɛɚɪɚɛɚɧɚ, R2 – ɧɨɪɦɚɥɶɧɚɹ ɪɟɚɤɰɢɹ ɩɥɨɫɤɨɫɬɢ, Fɬɪ – ɫɢɥɚ ɬɪɟɧɢɹ ɩɪɢ
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ɫɤɨɥɶɠɟɧɢɢ ɝɪɭɡɚ ɨ ɩɥɨɫɤɨɫɬɶ. ɍɱɢɬɵɜɚɹ, |
ɱɬɨ R2 P2 , Fɬɪ = fP2, ɚ ɫɢɥɵ |
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Ɋ1, R1 ɢR1' ɩɪɢɥɨɠɟɧɵ ɜ ɬɨɱɤɟ, ɥɟɠɚɳɟɣ ɧɚ ɨɫɢ z, ɡɚɩɢɲɟɦ: |
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¦momZ (Fke ) mɜɪ |
fP2 r . |
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(1) |
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Ƚɥɚɜɧɵɣ ɦɨɦɟɧɬ ɤɨɥɢɱɟɫɬɜ ɞɜɢɠɟɧɢɹ LZ ɞɚɧɧɨɣ ɫɢɫɬɟɦɵ |
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ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɩɨɞɜɢɠɧɨɣ ɨɫɢ z |
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Pr2 |
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LZ L(Z1) L(Z2) JZΖ momZ (m2 |
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r2Ζ . |
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ȼɡɹɜ ɩɪɨɢɡɜɨɞɧɭɸ LZ ɩɨ ɜɪɟɦɟɧɢ ɫ ɭɱɺɬɨɦ ɬɨɝɨ, ɱɬɨ |
Ζ |
Μ , ɢɦɟɟɦ |
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P1 2P2 |
r 2Μ . |
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ɉɨɞɫɬɚɜɢɜ ɪɟɡɭɥɶɬɚɬɵ (1) ɢ (2) ɜ ɭɪɚɜɧɟɧɢɟ |
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¦momZ (Fke ) ɢ |
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k 1 |
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ɪɟɲɢɜ ɟɝɨ ɨɬɧɨɫɢɬɟɥɶɧɨ Μ , ɩɨɥɭɱɢɦ ɢɫɤɨɦɵɣ ɪɟɡɭɥɶɬɚɬ: |
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2g |
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Μ |
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(mɜɪ |
fP2r). |
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Ɂɚɞɚɱɚ ʋ 2. Ȼɚɬɶ Ɇ.ɂ. «Ɍɟɨɪɟɬɢɱɟɫɤɚɹ ɦɟɯɚɧɢɤɚ ɜ ɩɪɢɦɟɪɚɯ ɢ ɡɚɞɚɱɚɯ», ɬɨɦ 2, ɇɚɭɤɚ,1975.
ɇɚ ɛɨɤɨɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɪɭɝɥɨɝɨ ɤɨɧɭɫɚ ɫɢɦɦɟɬɪɢɱɧɨ ɪɚɫɩɨɥɨɠɟɧɵ ɞɜɚ ɝɪɭɡɚ, ɫɨɟɞɢɧɟɧɧɵɟ ɦɟɠɞɭ ɫɨɛɨɣ ɬɨɧɤɨɣ ɧɢɬɶɸ ɢ ɨɬɫɬɨɹɳɢɟ ɨɬ ɨɫɢ ɜɪɚɳɟɧɢɹ ɤɨɧɭɫɚ ɧɚ ɪɚɫɫɬɨɹɧɢɢ ɨɞɧɨɣ ɬɪɟɬɢ ɪɚɞɢɭɫɚ ɨɫɧɨɜɚɧɢɹ ɤɨɧɭɫɚ. Ʉɨɧɭɫ ɜɦɟɫɬɟ ɫ ɝɪɭɡɚɦɢ ɜɪɚɳɚɥɫɹ ɫ ɭɝɥɨɜɨɣ ɫɤɨɪɨɫɬɶɸ Ζ. ɉɨɫɥɟ ɜɧɟɡɚɩɧɨɝɨ ɪɚɡɪɵɜɚ ɧɢɬɢ ɝɪɭɡɵ ɧɚɱɚɥɢ ɨɩɭɫɤɚɬɶɫɹ ɩɨ ɧɚɩɪɚɜɥɹɸɳɢɦ ɛɨɤɨɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɨɧɭɫɚ. ȼɟɫ ɤɚɠɞɨɝɨ ɢɡ ɝɪɭɡɨɜ ɜ ɱɟɬɵɪɟ ɪɚɡɚ ɦɟɧɶɲɟ ɜɟɫɚ ɤɨɧɭɫɚ. Ɉɩɪɟɞɟɥɢɬɶ ɭɝɥɨɜɭɸ ɫɤɨɪɨɫɬɶ ɤɨɧɭɫɚ ɜ ɦɨɦɟɧɬ, ɤɨɝɞɚ ɝɪɭɡɵ ɞɨɫɬɢɝɧɭɬ ɨɫɧɨɜɚɧɢɹ ɤɨɧɭɫɚ. ɋɢɥɚɦɢ ɫɨɩɪɨɬɢɜɥɟɧɢɣ ɞɜɢɠɟɧɢɸ ɩɪɟɧɟɛɪɟɱɶ. Ƚɪɭɡɵ ɫɱɢɬɚɬɶ ɬɨɱɟɱɧɵɦɢ ɦɚɫɫɚɦɢ.
Ɋɟɲɟɧɢɟ.
ȼɡɹɜ ɧɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ ɜ ɧɢɠɧɟɣ ɨɩɨɪɟ Ⱥ ɨɫɢ ɤɨɧɭɫɚ, ɧɚɩɪɚɜɢɦ ɨɫɶ z ɩɨ ɨɫɢ ɜɪɚɳɟɧɢɹ ɤɨɧɭɫɚ. Ɉɛɨɡɧɚɱɢɦ: Ɋ – ɜɟɫ ɤɨɧɭɫɚ, r –ɪɚɞɢɭɫ ɨɫɧɨɜɚɧɢɹ ɤɨɧɭɫɚ.
ɂɡɨɛɪɚɡɢɦ ɜɧɟɲɧɢɟ ɫɢɥɵ ɦɚɬɟɪɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ, ɫɨɫɬɨɹɳɟɣ ɢɡ ɤɨɧɭɫɚ ɢ ɞɜɭɯ ɝɪɭɡɨɜ: Ɋ – ɜɟɫ ɤɨɧɭɫɚ, Ɋ1 ɢ Ɋ2 – ɜɟɫɚ ɝɪɭɡɨɜ, RȺx, RȺy, RȺz, RBx, Rȼy – ɫɨɫɬɚɜɥɹɸɳɢɟ ɪɟɚɤɰɢɣ ɨɩɨɪ Ⱥ ɢ ȼ.
ɉɪɢɦɟɧɢɦ ɬɟɨɪɟɦɭ ɨɛ ɢɡɦɟɧɟɧɢɢ ɦɨɦɟɧɬɚ ɤɨɥɢɱɟɫɬɜɚ
ɞɜɢɠɟɧɢɹ |
ɫɢɫɬɟɦɵ |
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ɦɚɬɟɪɢɚɥɶɧɵɯ |
ɬɨɱɟɤ |
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ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z. |
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Ɍɚɤ ɤɚɤ ɜɫɟ ɜɧɟɲɧɢɟ ɫɢɥɵ ɥɢɛɨ ɩɚɪɚɥɥɟɥɶɧɵ, ɥɢɛɨ ɩɟɪɟɫɟɤɚɸɬ ɨɫɶ ɜɪɚɳɟɧɢɹ z, ɬɨ ɫɭɦɦɚ ɦɨɦɟɧɬɨɜ ɜɫɟɯ ɜɧɟɲɧɢɯ ɫɢɥ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z ɪɚɜɧɚ ɧɭɥɸ:
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0 . Ɍɨ ɟɫɬɶ |
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ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, L1Z L2Z . ɂɬɚɤ, |
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ɢɦɟɟɬ ɦɟɫɬɨ ɫɥɭɱɚɣ ɫɨɯɪɚɧɟɧɢɹ ɩɪɨɟɤɰɢɢ ɦɨɦɟɧɬɚ ɤɨɥɢɱɟɫɬɜ ɞɜɢɠɟɧɢɹ
ɫɢɫɬɟɦɵ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ. ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɮɨɪɦɭɥɨɣ |
LZ |
IZΖ , |
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ɡɚɩɢɲɟɦ: |
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L1Z |
IZ1Ζ1 , L2Z |
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IZ1 , IZ2 |
– ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɜ |
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ɩɟɪɜɨɦ ɢ |
ɜɨ |
ɜɬɨɪɨɦ |
ɩɨɥɨɠɟɧɢɢ |
ɝɪɭɡɨɜ. |
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Ɍɚɤ ɤɚɤ L1Z L2Z ɢ |
L1Z |
IZ1Ζ1 , |
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Ζ I 2Ζ |
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Ɉɫɬɚɟɬɫɹ ɜɵɱɢɫɥɢɬɶ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɫɢɫɬɟɦɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z. Ɍɚɤ ɤɚɤ ɦɚɬɟɪɢɚɥɶɧɚɹ ɫɢɫɬɟɦɚ ɫɨɫɬɨɢɬ ɢɡ ɤɨɧɭɫɚ ɢ ɞɜɭɯ ɝɪɭɡɨɜ, ɬɨ ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɫɢɫɬɟɦɵ ɪɚɜɟɧ ɫɭɦɦɟ ɦɨɦɟɧɬɨɜ ɢɧɟɪɰɢɢ ɤɨɧɭɫɚ ɢ ɝɪɭɡɨɜ.
Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɤɨɧɭɫɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z ɪɚɜɟɧ |
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ɋɱɢɬɚɹ ɝɪɭɡɵ ɬɨɱɟɱɧɵɦɢ ɦɚɫɫɚɦɢ, ɢɦɟɟɦ ɞɥɹ ɧɚɱɚɥɶɧɨɝɨ ɢ ɤɨɧɟɱɧɨɝɨ ɩɨɥɨɠɟɧɢɣ ɝɪɭɡɨɜ
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4g ©3 |
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ɂɫɩɨɥɶɡɭɹ ɧɚɣɞɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɩɨɥɭɱɢɦ Ζ2 |
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Ζ1 . |
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§10. Ɍɟɨɪɟɦɚ ɨɛ ɢɡɦɟɧɟɧɢɢ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ
Ʉɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɦɚɬɟɪɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɩɪɟɞɟɥɹɸɬ ɮɨɪɦɭɥɨɣ
n |
m v2 |
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Ⱦɥɹ ɫɢɫɬɟɦ, ɫɨɜɟɪɲɚɸɳɢɯ ɫɥɨɠɧɨɟ ɞɜɢɠɟɧɢɟ, ɤɢɧɟɬɢɱɟɫɤɭɸ ɷɧɟɪɝɢɸ ɨɩɪɟɞɟɥɹɸɬ ɩɨ ɮɨɪɦɭɥɟ Ʉɺɧɢɝɚ
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ɝɞɟ T c– ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɜ ɟɟ ɨɬɧɨɫɢɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɢ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɩɨɫɬɭɩɚɬɟɥɶɧɨ ɞɜɢɠɭɳɟɣɫɹ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɫ ɧɚɱɚɥɨɦ ɜ ɰɟɧɬɪɟ ɢɧɟɪɰɢɢ. (Ɂɞɟɫɶ ɢ ɞɚɥɟɟ ɩɨɞ ɰɟɧɬɪɨɦ ɢɧɟɪɰɢɢ ɩɨɧɢɦɚɟɦ ɰɟɧɬɪ ɦɚɫɫ).
Ⱦɥɹ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɞɜɢɠɭɳɟɝɨɫɹ ɩɨɫɬɭɩɚɬɟɥɶɧɨ,
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Mv2 |
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T |
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C |
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ɝɞɟ M – ɦɚɫɫɚ ɬɟɥɚ; |
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vC |
– ɫɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ (ɢɥɢ ɥɸɛɨɣ ɞɪɭɝɨɣ ɬɨɱɤɢ). |
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Ⱦɥɹ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɜɪɚɳɚɸɳɟɝɨɫɹ ɜɨɤɪɭɝ ɨɫɢ, |
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T |
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JZΖ2 |
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ɝɞɟ JZ |
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– ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɜɪɚɳɟɧɢɹ, |
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Ζ – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ. |
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Ⱦɥɹ ɨɛɳɟɝɨ ɫɥɭɱɚɹ ɞɜɢɠɟɧɢɹ ɬɜɟɪɞɨɝɨ ɬɟɥɚ (ɜ ɬɨɦ ɱɢɫɥɟ ɢ ɞɥɹ ɩɥɨɫɤɨɝɨ |
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ɞɜɢɠɟɧɢɹ) |
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T |
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JCΖ 2 |
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ɝɞɟ JC – ɦɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɟɥɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɝɧɨɜɟɧɧɨɣ ɨɫɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɰɟɧɬɪ ɢɧɟɪɰɢɢ (ɞɥɹ ɩɥɨɫɤɨɝɨ ɞɜɢɠɟɧɢɹ ɷɬɚ ɨɫɶ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚ ɩɥɨɫɤɨɫɬɢ ɞɜɢɠɟɧɢɹ);
Ζ – ɦɝɧɨɜɟɧɧɚɹ ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ.
ɗɥɟɦɟɧɬɚɪɧɭɸ ɪɚɛɨɬɭ ɫɢɥɵ, ɩɪɢɥɨɠɟɧɧɨɣ ɤ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɟ, ɧɚ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɦ ɩɟɪɟɦɟɳɟɧɢɢ ɨɩɪɟɞɟɥɹɸɬ ɮɨɪɦɭɥɨɣ
ΓA F dr Fdscos(F,Ω ) Fx dx Fy dy Fz dz .
Ɋɚɛɨɬɚ ɫɢɥɵ ɧɚ ɤɨɧɟɱɧɨɦ ɩɟɪɟɦɟɳɟɧɢɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ
A ³ |
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³FΩ ds |
³(Fx dx Fy dy Fzdz) . |
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Ɋɚɛɨɬɚ ɩɨɫɬɨɹɧɧɨɣ ɫɢɥɵ ɧɚ ɩɪɹɦɨɥɢɧɟɣɧɨɦ ɭɱɚɫɬɤɟ ɩɭɬɢ s
AFscos(F,s) .
Ɋɚɛɨɬɚ ɫɢɥ ɬɹɠɟɫɬɢ ɥɸɛɨɣ ɫɢɫɬɟɦɵ
A12 (P) P(zC1 zC2 ) ,
ɝɞɟ P – ɜɟɫ ɜɫɟɣ ɫɢɫɬɟɦɵ;
zC1 ɢ zC2 – ɚɩɩɥɢɤɚɬɵ ɰɟɧɬɪɚ ɢɧɟɪɰɢɢ ɜ ɧɚɱɚɥɶɧɨɦ ɢ ɤɨɧɟɱɧɨɦ ɩɨɥɨɠɟɧɢɹɯ ɫɢɫɬɟɦɵ.
Ɋɚɛɨɬɚ ɭɩɪɭɝɨɣ ɫɢɥɵ Fx cx ɩɪɢ ɩɪɹɦɨɥɢɧɟɣɧɨɦ ɩɟɪɟɦɟɳɟɧɢɢ ɬɨɱɤɢ
A12 |
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ɝɞɟ x1 x2 ɤɨɨɪɞɢɧɚɬɵ ɧɚɱɚɥɶɧɨɝɨ ɢ ɤɨɧɟɱɧɨɝɨ ɩɨɥɨɠɟɧɢɣ
ɗɥɟɦɟɧɬɚɪɧɭɸ ɪɚɛɨɬɭ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ ɬɜɟɪɞɨɦɭ ɬɟɥɭ, ɩɟɪɟɦɟɳɚɸɳɟɦɭɫɹ ɩɪɨɢɡɜɨɥɶɧɨ, ɨɩɪɟɞɟɥɹɸɬ ɩɨ ɮɨɪɦɭɥɟ
ΓA >R vO MOΖ dt ,
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