Механика.Методика решения задач
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Ƚɥɚɜɚ 2. Ⱦɢɧɚɦɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ |
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ȼɵɪɚɠɟɧɢɟ (2.54) ɩɨɡɜɨɥɹɟɬ ɨɩɪɟɞɟɥɢɬɶ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t0 , |
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ɜ ɤɨɬɨɪɵɣ ɛɪɭɫɨɤ ɧɚɱɢɧɚɟɬ ɫɤɨɥɶɡɢɬɶ ɩɨ ɞɨɫɤɟ: |
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t0 |
Pmg M m |
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(2.55) |
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ɂɬɚɤ: |
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°ɩɪɢ t d t0 : |
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M m |
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(2.56) |
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Dt Pmg |
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°°ɩɪɢ t ! t0 : |
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ɂɫɩɨɥɶɡɭɹ ɩɨɥɭɱɟɧɧɵɟ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɭɫɤɨɪɟɧɢɣ ɬɟɥ ɫɢɫɬɟ- |
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ɦɵ, ɨɩɪɟɞɟɥɢɦ ɬɟɩɟɪɶ ɡɚɤɨɧɵ ɢɡɦɟɧɟɧɢɹ ɫɤɨɪɨɫɬɟɣ ɷɬɢɯ ɬɟɥ. |
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ɉɪɢ |
t d t0 ɫɤɨɪɨɫɬɢ ɛɪɭɫɤɚ ɢ ɞɨɫɤɢ ɦɟɧɹɸɬɫɹ ɨɞɢɧɚɤɨɜɵɦ |
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ɨɛɪɚɡɨɦ ɢ ɤ ɦɨɦɟɧɬɭ ɜɪɟɦɟɧɢ t ɛɭɞɭɬ ɪɚɜɧɵ:
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d t |
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(2.57) |
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M m |
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ɉɪɢ t ! t0 ɫɤɨɪɨɫɬɶ ɛɪɭɫɤɚ ɛɭɞɟɬ ɪɚɜɧɚ |
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Dt02 |
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d t , |
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2 M m |
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t0 |
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Dt02 |
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D t 2 t02 |
Pg t t0 , |
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(2.58) |
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2 M m |
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2m |
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ɚ ɫɤɨɪɨɫɬɶ ɞɨɫɤɢ – |
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Dt02 |
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d t |
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2 M m |
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2 M m |
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t0 |
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X,V |
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Ɋɢɫ. 2.8 |
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62 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ɇɚ ɪɢɫ. 2.8 ɢɡɨɛɪɚɠɟɧɵ ɩɨɥɭɱɟɧɧɵɟ ɡɚɜɢɫɢɦɨɫɬɢ ɫɤɨɪɨɫɬɟɣ ɛɪɭɫɤɚ ɢ ɞɨɫɤɢ ɨɬ ɜɪɟɦɟɧɢ.
Ɂɚɞɚɱɚ 2.5
(ɉɪɹɦɚɹ ɡɚɞɚɱɚ ɞɢɧɚɦɢɤɢ)
ɇɚɣɬɢ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ, ɞɜɢɠɭɳɟɣɫɹ ɜ ɨɞɧɨɪɨɞɧɨɦ ɢ ɩɨɫɬɨɹɧɧɨɦ ɫɢɥɨɜɨɦ ɩɨɥɟ ɫ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ X0, ɧɚɩɪɚɜɥɟɧɧɨɣ ɩɨɞ ɩɪɨɢɡɜɨɥɶɧɵɦ ɭɝɥɨɦ D ɤ ɫɢɥɟ F.
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Ɋɟɲɟɧɢɟ |
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I. ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɬɚɤ, |
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ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 2.9, ɩɪɢ ɷɬɨɦ ɧɚɱɚɥɨ |
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ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɫɨɜɩɚɞɚɟɬ ɫ ɩɨɥɨɠɟ- |
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ɧɢɟɦ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɧɚɱɚɥɶɧɵɣ |
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ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ. |
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II. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ |
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ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ |
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Ɋɢɫ. 2.9 |
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ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ: |
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dXx |
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(2.60) |
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d t |
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dXy |
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(2.61) |
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III. ɉɪɨɢɧɬɟɝɪɢɪɭɟɦ ɭɪɚɜɧɟɧɢɹ (2.60) ɢ (2.61), ɢɫɩɨɥɶɡɭɹ ɧɚ- |
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ɱɚɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɫɤɨɪɨɫɬɢ Xx (0) X0 sinD ɢ Xy (0) X0 cosD : |
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Xx (t) |
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X0 sin D , |
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(2.62) |
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Xy (t) |
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X0 cosD |
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(2.63) |
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ɂɧɬɟɝɪɢɪɭɹ ɭɪɚɜɧɟɧɢɹ (2.62) ɢ (2.63) |
ɫ ɭɱɟɬɨɦ ɧɚɱɚɥɶɧɵɯ |
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ɡɧɚɱɟɧɢɣ ɤɨɨɪɞɢɧɚɬ x0 = 0 ɢ y0 = 0, ɩɨɥɭɱɚɟɦ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɜ ɤɨɨɪɞɢɧɚɬɧɨɣ ɮɨɪɦɟ:
x(t) |
X0 sin D t , |
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X0 cosD t |
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ɂɫɤɥɸɱɢɜ ɜɪɟɦɹ ɢɡ ɭɪɚɜɧɟɧɢɣ (2.64) ɢ (2.65), ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɬɪɚɟɤɬɨɪɢɢ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ – ɭɪɚɜɧɟɧɢɟ ɩɚɪɚɛɨɥɵ:
Ƚɥɚɜɚ 2. Ⱦɢɧɚɦɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ |
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x2 ctgD x |
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜ ɨɞɧɨɪɨɞɧɨɦ ɫɢɥɨɜɨɦ ɩɨɥɟ ɦɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɞɜɢɠɟɬɫɹ ɩɨ ɩɚɪɚɛɨɥɟ.
Ɂɚɞɚɱɚ 2.6
Ɍɟɥɨ ɧɟɛɨɥɶɲɢɯ ɪɚɡɦɟɪɨɜ ɞɜɢɠɟɬɫɹ ɩɨ ɩɨɜɟɪɯɧɨɫɬɢ ɧɟɩɨɞɜɢɠɧɨɝɨ ɤɥɢɧɚ ɫ ɭɝɥɨɦ ɩɪɢ ɨɫɧɨɜɚɧɢɢ D . ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɪɚɜɧɹɥɚɫɶ ȣ0 ɢ ɫɨɫɬɚɜɥɹɥɚ ɭɝɨɥ M0 ɫ ɪɟɛɪɨɦ ɤɥɢɧɚ (ɫɦ. ɪɢɫ. 2.10).
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Ɋɢɫ. 2.10
Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɬɟɥɚ ɨ ɩɨɜɟɪɯɧɨɫɬɶ ɤɥɢɧɚ P tgD . ɇɚɣɬɢ ɭɫɬɚɧɨɜɢɜɲɭɸɫɹ ɫɤɨɪɨɫɬɶ ɫɤɨɥɶɠɟɧɢɹ ɬɟɥɚ.
Ɋɟɲɟɧɢɟ
I. ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɬɚɤ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 2.11 (ɜɢɞ ɫɛɨɤɭ) ɢ ɪɢɫ. 2.12 (ɜɢɞ ɫɜɟɪɯɭ ɧɚ ɩɨɜɟɪɯɧɨɫɬɶ ɤɥɢɧɚ). Ɉɫɶ X ɧɚɩɪɚɜɢɦ ɜɞɨɥɶ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɩɚɪɚɥɥɟɥɶɧɨ ɪɟɛɪɭ ɤɥɢɧɚ (ɪɢɫ. 2.12). ɉɪɢ ɷɬɨɦ ɨɫɶ Y ɧɚɩɪɚɜɢɦ ɩɨ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɪɟɛɪɭ ɤɥɢɧɚ, ɚ ɨɫɶ Z ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɧɚɤɥɨɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɥɢɧɚ (ɪɢɫ. 2.11).
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mgsinD |
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Ɋɢɫ. 2.11 |
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Ɋɢɫ. 2.12 |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ɇɚ ɪɢɫ. 2.11 ɢ 2.12 ɢɡɨɛɪɚɠɟɧɵ ɬɚɤɠɟ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɬɟɥɨ ɜ ɩɪɨɰɟɫɫɟ ɞɜɢɠɟɧɢɹ: ɫɢɥɚ ɬɹɠɟɫɬɢ mg, ɫɢɥɚ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɨɩɨɪɵ N ɢ ɫɢɥɚ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ Fɬɪ.
ɋɤɨɪɨɫɬɶ ɬɟɥɚ X(t) ɫɨɫɬɚɜɥɹɟɬ ɫ ɨɫɶɸ X ɭɝɨɥ ij(t) (ɫɦ. ɪɢɫ. 2.12), ɤɨɬɨɪɵɣ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɜɪɟɦɟɧɢ.
II. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɜɵɛɪɚɧɧɵɟ ɨɫɢ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ:
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dXx |
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Fɬɪ cosM |
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(2.67) |
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mg sin D Fɬɪ sin M , |
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N mg cosD . |
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ɂɫɩɨɥɶɡɭɟɦ ɡɚɤɨɧ Ⱥɦɨɧɬɨɧɚ – Ʉɭɥɨɧɚ (ɫɦ. ɩ. 2.1.2.ȼ) ɞɥɹ ɫɢɥɵ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɢ ɭɱɬɟɦ ɡɚɞɚɧɧɭɸ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɫɜɹɡɶ
ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɪɟɧɢɹ |
P ɫ ɭɝɥɨɦ |
D ɩɪɢ ɨɫɧɨɜɚɧɢɢ ɧɚɤɥɨɧɧɨɣ |
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ɩɥɨɫɤɨɫɬɢ: |
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Fɬɪ |
PN tgD N . |
(2.70) |
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Ɂɚɩɢɲɟɦ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɮɭɧɤɰɢɢ ɭɝɥɚ ij, ɜɵɪɚɡɢɜ ɢɯ |
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ɱɟɪɟɡ ɩɪɨɟɤɰɢɢ ɫɤɨɪɨɫɬɢ ɬɟɥɚ: |
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cosM |
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Xx |
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III. ɉɨɥɭɱɟɧɚ ɩɨɥɧɚɹ ɫɢɫɬɟɦɚ |
ɭɪɚɜɧɟɧɢɣ (2.67) – (2.71) ɞɥɹ |
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ɨɩɪɟɞɟɥɟɧɢɹ ɩɪɨɟɤɰɢɣ ɫɤɨɪɨɫɬɢ ɬɟɥɚ ɧɚ ɨɫɢ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ, ɪɟɲɢɬɶ ɤɨɬɨɪɭɸ ɜ ɨɛɳɟɦ ɜɢɞɟ ɞɨɫɬɚɬɨɱɧɨ ɫɥɨɠɧɨ ɢɡ-ɡɚ ɧɚɥɢɱɢɹ ɜ ɧɟɣ ɞɜɭɯ ɫɜɹɡɚɧɧɵɯ ɧɟɥɢɧɟɣɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ. Ɉɞɧɚɤɨ ɧɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɧɚɯɨɞɢɬɶ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɫɤɨɪɨɫɬɢ ɬɟɥɚ. ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɭɫɬɚɧɨɜɢɜɲɭɸɫɹ ɫɤɨɪɨɫɬɶ ɬɟɥɚ, ɬ.ɟ. ɡɧɚɱɟɧɢɟ ɫɤɨɪɨɫɬɢ ɜ ɬɨ ɜɪɟɦɹ, ɤɨɝɞɚ ɫɭɦɦɚ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɬɟɥɨ, ɫɬɚɧɟɬ ɪɚɜɧɨɣ ɧɭɥɸ.
Ɋɚɫɫɦɨɬɪɢɦ ɢɡɦɟɧɟɧɢɟ ɯɚɪɚɤɬɟɪɚ ɞɜɢɠɟɧɢɹ ɬɟɥɚ ɫɨ ɜɪɟɦɟɧɟɦ. ȼ ɩɥɨɫɤɨɫɬɢ ɞɜɢɠɟɧɢɹ ɧɚ ɬɟɥɨ ɞɟɣɫɬɜɭɸɬ ɞɜɟ ɫɢɥɵ: ɫɢɥɚ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɢ ɩɪɨɟɤɰɢɹ ɫɢɥɵ ɬɹɠɟɫɬɢ. ɂɡ (2.69) ɢ (2.70) ɩɨɥɭ-
ɱɢɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɫɢɥɵ ɬɪɟɧɢɹ: |
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Fɬɪ mg sin D . |
(2.77) |
Ʉɚɤ ɜɢɞɢɦ, ɦɨɞɭɥɶ ɫɢɥɵ ɬɪɟɧɢɹ ɪɚɜɟɧ ɜɟɥɢɱɢɧɟ ɩɪɨɟɤɰɢɢ ɫɢɥɵ ɬɹɠɟɫɬɢ ɧɚ ɧɚɤɥɨɧɧɭɸ ɩɥɨɫɤɨɫɬɶ. Ⱦɟɣɫɬɜɭɸɳɢɟ ɧɚ ɬɟɥɨ ɫɢɥɵ ɛɭ-
Ƚɥɚɜɚ 2. Ⱦɢɧɚɦɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ |
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ɞɭɬ ɩɨɜɨɪɚɱɢɜɚɬɶ ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ ɬɟɥɚ ɞɨ ɬɟɯ ɩɨɪ, ɩɨɤɚ ɨɧ ɧɟ ɫɨɜɩɚɞɟɬ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɫ ɨɫɶɸ Y. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɫɤɨɪɟɧɢɟ ɨɛɪɚɬɢɬɫɹ ɜ ɧɨɥɶ, ɤɨɝɞɚ ɫɢɥɚ ɬɪɟɧɢɹ ɛɭɞɟɬ ɧɚɩɪɚɜɥɟɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɫɨɫɬɚɜɥɹɸɳɟɣ ɫɢɥɵ ɬɹɠɟɫɬɢ ɜ ɩɥɨɫɤɨɫɬɢ ɞɜɢɠɟɧɢɹ ɬɟɥɚ. Ⱦɚɥɶɧɟɣɲɟɟ ɞɜɢɠɟɧɢɟ ɛɭɞɟɬ ɩɪɨɢɫɯɨɞɢɬɶ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ Xɭɫɬ, ɧɚɩɪɚɜɥɟɧɧɨɣ ɜɞɨɥɶ ɨɫɢ Y.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɨɫɬɚɬɨɱɧɨ ɧɚɣɬɢ ɭɪɚɜɧɟɧɢɟ, ɫɜɹɡɵɜɚɸɳɟɟ ɩɪɨɟɤɰɢɸ ɫɤɨɪɨɫɬɢ ɬɟɥɚ ɧɚ ɨɫɶ Y ɫ ɦɨɞɭɥɟɦ ɟɝɨ ɫɤɨɪɨɫɬɢ. Ⱦɥɹ ɷɬɨɝɨ ɩɪɟɨɛɪɚɡɭɟɦ ɩɨɥɭɱɟɧɧɭɸ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (2.67) – (2.71) ɤ ɜɢɞɭ:
dXx
dt dXy
dt
XXx g sin D ,
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(2.72)
(2.73)
ɉɪɨɢɡɜɨɞɧɭɸ ɨɬ ɦɨɞɭɥɹ ɫɤɨɪɨɫɬɢ ɩɨ ɜɪɟɦɟɧɢ ɩɪɟɞɫɬɚɜɢɦ ɜ ɜɢɞɟ:
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ɉɨɞɫɬɚɧɨɜɤɚ (2.72) ɢ (2.73) ɜ (2.74) ɩɪɢɜɨɞɢɬ ɤ ɭɪɚɜɧɟɧɢɸ: |
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ɭɫɥɨɜɢɣ (X(0) X0 , |
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ɂɧɬɟɝɪɢɪɭɹ |
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(2.75) ɫ |
ɭɱɟɬɨɦ |
ɧɚɱɚɥɶɧɵɯ |
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M(0) M0 ), ɩɨɥɭɱɚɟɦ: |
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X0 (1 sin M0 ) . |
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(2.76) |
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ɉɨɞɫɬɚɜɥɹɹ X = Xy = Xɭɫɬ ɜ (2.76), ɧɚɯɨɞɢɦ ɢɫɤɨɦɵɣ ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɞɜɢɠɟɧɢɹ:
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ɉɪɨɚɧɚɥɢɡɢɪɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɭɫɬɚɧɨɜɢɜɲɟɣɫɹ ɫɤɨɪɨɫɬɢ ɜ ɞɜɭɯ ɱɚɫɬɧɵɯ ɫɥɭɱɚɹɯ.
ȿɫɥɢ ij0 = ʌ/2 (ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɧɚɩɪɚɜɥɟɧɚ ɜɧɢɡ ɩɨ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ), ɬɨ Xɭɫɬ = X0. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɜɢɠɟɧɢɟ ɬɟɥɚ ɫɪɚɡɭ ɩɪɨɢɫɯɨɞɢɬ ɫ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɶɸ, ɩɨɫɤɨɥɶɤɭ ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɧɟɝɨ ɫɢɥɵ ɫɤɨɦɩɟɧɫɢɪɨɜɚɧɵ.
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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ɉɪɢ ij0 = –ʌ/2 |
ɫɤɨɪɨɫɬɶ |
ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɞɜɢɠɟɧɢɹ ɪɚɜɧɚ |
Xɭɫɬ = 0. ɇɚɱɚɥɶɧɚɹ |
ɫɤɨɪɨɫɬɶ, |
ɧɚɩɪɚɜɥɟɧɧɚɹ ɜɜɟɪɯ ɩɨ ɧɚɤɥɨɧɧɨɣ |
ɩɥɨɫɤɨɫɬɢ, ɩɪɢɜɨɞɢɬ ɤ ɪɚɜɧɨɡɚɦɟɞɥɟɧɧɨɦɭ ɞɜɢɠɟɧɢɸ. ɉɪɢ ɷɬɨɦ ɢ ɩɪɨɟɤɰɢɹ ɫɢɥɵ ɬɹɠɟɫɬɢ, ɢ ɫɢɥɚ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɧɚɩɪɚɜɥɟɧɵ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɫɤɨɪɨɫɬɢ. ɑɟɪɟɡ ɧɟɤɨɬɨɪɨɟ ɜɪɟɦɹ ɫɤɨɪɨɫɬɶ ɬɟɥɚ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɨɥɶ. ɋɢɥɚ ɬɪɟɧɢɹ ɫɬɚɧɨɜɢɬɫɹ ɫɢɥɨɣ ɬɪɟɧɢɹ ɩɨɤɨɹ ɢ ɦɟɧɹɟɬ ɧɚɩɪɚɜɥɟɧɢɟ ɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɟ. Ⱦɜɢɠɟɧɢɹ ɜɧɢɡ ɩɨ ɧɚɤɥɨɧɧɨɣ ɩɥɨɫɤɨɫɬɢ ɧɟ ɩɪɨɢɫɯɨɞɢɬ, ɬ.ɤ. ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɫɢɥɵ ɬɪɟɧɢɹ ɩɨɤɨɹ ɜ ɭɫɥɨɜɢɹɯ ɞɚɧɧɨɣ ɡɚɞɚɱɢ ɫɨɜɩɚɞɚɟɬ ɩɨ ɦɨɞɭɥɸ ɫɨ ɡɧɚɱɟɧɢɟɦ ɩɪɨɟɤɰɢɢ ɫɢɥɵ ɬɹɠɟɫɬɢ ɧɚ ɧɚɤɥɨɧɧɭɸ ɩɥɨɫɤɨɫɬɶ.
Ɂɚɞɚɱɚ 2.7
ɋɬɚɥɶɧɨɣ ɲɚɪɢɤ ɪɚɞɢɭɫɚ r ɧɚɱɢɧɚɟɬ ɞɜɢɝɚɬɶɫɹ ɜ ɫɨɫɭɞɟ, ɡɚɩɨɥɧɟɧɧɨɦ ɝɥɢɰɟɪɢɧɨɦ, ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ ɬɹɠɟɫɬɢ. ɇɚɣɬɢ ɡɚɜɢɫɢɦɨɫɬɶ ɫɤɨɪɨɫɬɢ ɲɚɪɢɤɚ ɨɬ ɜɪɟɦɟɧɢ X(t), ɚ ɬɚɤɠɟ ɨɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɶ ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɞɜɢɠɟɧɢɹ ɲɚɪɢɤɚ Xɭɫɬ. Ʉɨɷɮɮɢɰɢɟɧɬ ɜɹɡɤɨɝɨ ɬɪɟɧɢɹ ɜ ɝɥɢɰɟɪɢɧɟ ɪɚɜɟɧ K, ɩɥɨɬɧɨɫɬɶ ɝɥɢɰɟɪɢɧɚ – U1, ɩɥɨɬɧɨɫɬɶ ɫɬɚɥɢ – U2. ɋɱɢɬɚɬɶ, ɱɬɨ ɫɢɥɚ ɜɹɡɤɨɝɨ ɬɪɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ ɋɬɨɤɫɚ: Fɜ = 6SrXK .
Ɋɟɲɟɧɢɟ
I. ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ, ɫɜɹɡɚɧɧɭɸ ɫ ɫɨɫɭɞɨɦ, ɬɚɤ, ɤɚɤ
ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 2.13. ɇɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ ɫɨ- |
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ɜɦɟɫɬɢɦ ɫ ɩɨɥɨɠɟɧɢɟɦ ɲɚɪɢɤɚ ɜ ɦɨɦɟɧɬ ɧɚ- |
FȺ |
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ɱɚɥɚ ɟɝɨ ɞɜɢɠɟɧɢɹ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɫɥɨɜɢ- |
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Fɜ |
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ɟɦ ɡɚɞɚɱɢ ɧɚɱɚɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɲɚɪɢɤɚ ɪɚɜɧɚ |
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ɧɭɥɸ: X 0 |
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II. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɲɚɪɢ- |
mg |
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ɤɚ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɶ X ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ: |
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ma |
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ɝɞɟ Fɜ – ɫɢɥɚ ɜɹɡɤɨɝɨ ɬɪɟɧɢɹ, ɚ FA |
– ɫɢɥɚ |
Ɋɢɫ. 2.13 |
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Ⱥɪɯɢɦɟɞɚ. |
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ɂɫɩɨɥɶɡɭɟɦ ɡɚɤɨɧ Ⱥɪɯɢɦɟɞɚ ɢ ɮɨɪɦɭɥɭ ɋɬɨɤɫɚ, ɨɩɢɫɵɜɚɸ- |
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ɳɢɟ ɫɜɨɣɫɬɜɚ ɷɬɢɯ ɫɢɥ: |
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FA |
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6SrXK . |
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Ƚɥɚɜɚ 2. Ⱦɢɧɚɦɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ |
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Ɂɞɟɫɶ V |
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– ɨɛɴɟɦ ɲɚɪɢɤɚ. |
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ȼɵɪɚɡɢɦ ɬɚɤɠɟ ɦɚɫɫɭ ɲɚɪɢɤɚ ɱɟɪɟɡ ɟɝɨ ɩɥɨɬɧɨɫɬɶ: |
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m U2V . |
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III. ɉɨɞɫɬɚɜɥɹɹ (2.80) – (2.82) ɜ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ (2.79), |
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ɩɨɥɭɱɚɟɦ: |
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U V |
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Ⱦɥɹ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (2.83) ɩɪɢɜɟɞɟɦ ɟɝɨ ɤ ɜɢɞɭ |
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6SrK |
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ɢ ɫɞɟɥɚɟɦ ɡɚɦɟɧɭ ɩɟɪɟɦɟɧɧɵɯ: |
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A BX |
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Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ (2.85) ɩɨ ɜɪɟɦɟɧɢ, ɩɨɥɭɱɚɟɦ: |
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ɋ ɭɱɟɬɨɦ (2.86) ɜɵɪɚɠɟɧɢɟ (2.84) ɩɪɢɧɢɦɚɟɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ: |
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Ɋɟɲɢɦ ɩɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧ- |
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ɧɵɯ ɫ ɭɱɟɬɨɦ ɧɚɱɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ ɫɤɨɪɨɫɬɢ X 0 |
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ɂɫɩɨɥɶɡɭɹ ɮɨɪɦɭɥɭ (2.85), ɜɟɪɧɟɦɫɹ ɤ ɫɬɚɪɨɣ ɩɟɪɟɦɟɧɧɨɣ X: |
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ɉɨɞɫɬɚɜɢɜ ɡɧɚɱɟɧɢɹ ɤɨɧɫɬɚɧɬ A ɢ B ɢɡ (2.84), ɚ ɬɚɤɠɟ ɡɧɚɱɟɧɢɟ V, ɩɨɥɭɱɢɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɫɤɨɪɨɫɬɢ ɲɚɪɢɤɚ:
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ɉɪɢ t !! 2r 2 U2 ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɲɚɪɢɤɚ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟ
9K
ɢɡɦɟɧɹɟɬɫɹ ɢ ɪɚɜɧɚ
X |
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68 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
Ɂɚɞɚɱɚ 2.8
Ȼɪɭɫɨɤ ɫɤɨɥɶɡɢɬ ɩɨ ɝɥɚɞɤɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɫɨ ɫɤɨɪɨɫɬɶɸ ȣ0 ɢ ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ ɩɨɩɚɞɚɟɬ ɜ ɨɛɥɚɫɬɶ, ɨɝɪɚɧɢɱɟɧɧɭɸ
ɡɚɛɨɪɨɦ ɜ ɮɨɪɦɟ ɩɨɥɭɨɤɪɭɠɧɨɫɬɢ (ɪɢɫ. 2.14). Ɉɩɪɟɞɟɥɢɬɶ ɜɪɟɦɹ, ɱɟɪɟɡ ɤɨɬɨɪɨɟ ɛɪɭɫɨɤ ɩɨɤɢɧɟɬ ɷɬɭ ɨɛɥɚɫɬɶ. Ɋɚɞɢɭɫ ɤɪɢɜɢɡɧɵ ɡɚɛɨɪɚ R, ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɛɪɭɫɤɚ ɨ ɩɨɜɟɪɯɧɨɫɬɶ ɡɚɛɨɪɚ P.
Ɋɚɡɦɟɪɵ ɛɪɭɫɤɚ ɦɧɨɝɨ ɦɟɧɶɲɟ R. |
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I. ȼɵɛɟɪɟɦ ɩɪɨɢɡɜɨɥɶɧɭɸ ɢɧɟɪ- |
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ɰɢɚɥɶɧɭɸ ɫɢɫɬɟɦɭ ɨɬɫɱɟɬɚ, ɠɟɫɬɤɨ |
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ɫɜɹɡɚɧɧɭɸ ɫ ɡɚɛɨɪɨɦ. ɂɡɨɛɪɚɡɢɦ ɧɚ |
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ɬɚɧɝɟɧɰɢɚɥɶɧɭɸ ɨɫɶ, ɡɚɞɚɧ- |
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ɧɭɸ ɨɪɬɨɦ IJ , ɧɚɩɪɚɜɥɟɧɧɭɸ ɜɞɨɥɶ |
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ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɛɪɭɫɤɚ, ɢ ɧɨɪ- |
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ɦɚɥɶɧɭɸ ɨɫɶ, ɡɚɞɚɧɧɭɸ ɨɪɬɨɦ n , ɧɚ- |
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Ɋɢɫ. 2.14 |
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ɩɪɚɜɥɟɧɧɭɸ ɤ ɰɟɧɬɪɭ ɤɪɢɜɢɡɧɵ ɬɪɚ- |
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ɟɤɬɨɪɢɢ |
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨ ɫɤɨɪɨɫɬɢ |
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(ɫɦ. ɬɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ ɜ Ƚɥɚɜɟ 1).
II. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɛɪɭɫɤɚ ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɧɟɪɰɢɚɥɶɧɨɣ ɫɢɫɬɟɦɵ ɨɬɫɱɟɬɚ, ɠɟɫɬɤɨ ɫɜɹɡɚɧɧɨɣ ɫ ɡɚɛɨɪɨɦ, ɜ ɩɪɨɟɤɰɢ-
ɹɯ ɧɚ ɬɚɧɝɟɧɰɢɚɥɶɧɭɸ ɢ ɧɨɪɦɚɥɶɧɭɸ ɨɫɢ: |
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ma |
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man |
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ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɡɚɤɨɧɨɦ Ⱥɦɨɧɬɨɧɚ – Ʉɭɥɨɧɚ ɞɥɹ ɫɢɥɵ ɬɪɟɧɢɹ |
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ɫɤɨɥɶɠɟɧɢɹ: |
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III ɂɡ (2.95) – (2.97) ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɦɨ- |
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ɞɭɥɹ ɫɤɨɪɨɫɬɢ ɛɪɭɫɤɚ: |
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Ɋɟɲɚɹ ɭɪɚɜɧɟɧɢɟ (2.98) ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ, ɩɨ- |
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ɥɭɱɢɦ: |
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Ƚɥɚɜɚ 2. Ⱦɢɧɚɦɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ |
69 |
1 P t C .
X R
Ʉɨɧɫɬɚɧɬɭ ɋ
(X(0) X0 ): C |
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ɜ (2.100) ɨɩɪɟɞɟɥɢɦ ɢɡ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɨɞɭɥɶ ɫɤɨɪɨɫɬɢ ɛɪɭɫɤɚ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t, ɤɨɝɞɚ ɛɪɭɫɨɤ ɟɳɟ ɞɜɢɠɟɬɫɹ ɜɞɨɥɶ ɡɚɛɨɪɚ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
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Ɂɚɦɟɬɢɦ, ɱɬɨ ɞɥɹ ɥɸɛɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ t ɢ ɩɪɢ ɥɸɛɨɣ, ɧɟ ɪɚɜɧɨɣ ɧɭɥɸ, ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ X0, ɫɤɨɪɨɫɬɶ ɛɪɭɫɤɚ X > 0. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɛɪɭɫɨɤ ɧɟ ɨɫɬɚɧɨɜɢɬɫɹ, ɚ ɨɛɹɡɚɬɟɥɶɧɨ ɩɪɨɣɞɟɬ ɜɫɸ ɨɛɥɚɫɬɶ, ɨɝɪɚɧɢɱɟɧɧɭɸ ɡɚɛɨɪɨɦ, ɩɨɫɤɨɥɶɤɭ ɫ ɭɦɟɧɶɲɟɧɢɟɦ ɫɤɨɪɨɫɬɢ ɞɜɢɠɟɧɢɹ ɛɪɭɫɤɚ ɭɦɟɧɶɲɚɟɬɫɹ ɢ ɫɢɥɚ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ ɦɟɠɞɭ ɛɪɭɫɤɨɦ ɢ ɡɚɛɨɪɨɦ.
ɉɭɬɶ, ɩɪɨɣɞɟɧɧɵɣ ɬɟɥɨɦ ɡɚ ɜɪɟɦɹ d t |
ɫ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ t, |
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ɩɪɢ ɞɜɢɠɟɧɢɢ ɜɞɨɥɶ ɡɚɛɨɪɚ, ɪɚɜɟɧ: |
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ɉɭɬɶ, ɩɪɨɣɞɟɧɧɵɣ ɬɟɥɨɦ ɡɚ ɜɪɟɦɹ t ɞɜɢɠɟɧɢɹ ɜɞɨɥɶ ɡɚɛɨɪɚ ɩɨɥɭɱɢɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ (2.102) ɩɨ ɜɪɟɦɟɧɢ:
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Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɜɪɟɦɟɧɢ, ɱɟɪɟɡ ɤɨɬɨɪɨɟ ɛɪɭɫɨɤ ɩɨɤɢɧɟɬ ɨɛ- |
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ɥɚɫɬɶ, ɨɝɪɚɧɢɱɟɧɧɭɸ ɡɚɛɨɪɨɦ ɩɪɟɨɛɪɚɡɭɟɦ (2.103) ɤ ɜɢɞɭ: |
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ɉɨɫɤɨɥɶɤɭ |
ɞɥɢɧɚ ɡɚɛɨɪɚ s SR ɢɫɤɨɦɨɟ ɜɪɟɦɹ |
ɞɜɢɠɟɧɢɹ |
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ɛɪɭɫɤɚ ɜɞɨɥɶ ɡɚɛɨɪɚ t0 |
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70 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
ɉɪɢ ɦɚɥɵɯ ɡɧɚɱɟɧɢɹɯ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɪɟɧɢɹ ( SP 1 ) ɜɪɟɦɹ
ɞɜɢɠɟɧɢɹ ɛɪɭɫɤɚ t0 ɛɭɞɟɬ ɪɚɜɧɨ |
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(1 |
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Ɂɚɞɚɱɚ 2.9
ɇɚ ɫɬɨɥɟ ɥɟɠɢɬ ɞɨɫɤɚ ɦɚɫɫɨɣ Ɇ = 1 ɤɝ, ɚ ɧɚ ɞɨɫɤɟ – ɝɪɭɡ ɦɚɫɫɨɣ m = 2 ɤɝ. Ʉɚɤɭɸ ɫɢɥɭ F ɧɭɠɧɨ ɩɪɢɥɨɠɢɬɶ ɤ ɞɨɫɤɟ, ɱɬɨɛɵ ɨɧɚ ɜɵɫɤɨɥɶɡɧɭɥɚ ɢɡ-ɩɨɞ ɝɪɭɡɚ? Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɦɟɠɞɭ ɝɪɭɡɨɦ ɢ ɞɨɫɤɨɣ ɪɚɜɟɧ P1 = 0,25, ɚ ɦɟɠɞɭ ɞɨɫɤɨɣ ɢ ɫɬɨɥɨɦ – P2 = 0,5.
Ɋɟɲɟɧɢɟ
I.ȼɵɛɟɪɟɦ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɬɚɤ, ɤɚɤ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 2.15
ɢɢɡɨɛɪɚɡɢɦ ɫɢɥɵ, ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɬɟɥɚ ɫɢɫɬɟɦɵ.
Y
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Ɋɢɫ. 2.15 |
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ɇɚ ɝɪɭɡ ɜ ɩɪɨɰɟɫɫɟ ɞɜɢɠɟɧɢɹ ɞɟɣɫɬɜɭɸɬ ɫɢɥɚ ɬɹɠɟɫɬɢ mg, ɫɢɥɚ ɬɪɟɧɢɹ Fɬɪ1 ɢ ɫɢɥɚ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɞɨɫɤɢ N. Ʉ ɞɨɫɤɟ ɩɪɢ-
ɥɨɠɟɧɵ ɝɨɪɢɡɨɧɬɚɥɶɧɚɹ ɫɢɥɚ F, ɫɢɥɚ ɬɹɠɟɫɬɢ Mg, ɫɢɥɚ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ ɫɬɨɥɚ R, ɫɢɥɚ ɧɨɪɦɚɥɶɧɨɝɨ ɞɚɜɥɟɧɢɹ ɝɪɭɡɚ N ɢ ɫɢɥɵ ɬɪɟɧɢɹ ɫɨ ɫɬɨɪɨɧɵ ɝɪɭɡɚ ɢ ɫɬɨɥɚ Fɬɪ1 ɢ Fɬɪ2 . ɋɢɥɚɦɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɨɡ-
ɞɭɯɚ ɩɪɟɧɟɛɪɟɝɚɟɦ.
ɉɪɨɚɧɚɥɢɡɢɪɭɟɦ ɯɚɪɚɤɬɟɪ ɞɜɢɠɟɧɢɹ ɬɟɥ ɫɢɫɬɟɦɵ. ȿɫɥɢ ɩɪɢɥɨɠɟɧɧɚɹ ɤ ɞɨɫɤɟ ɫɢɥɚ F ɦɚɥɚ, ɬɨ ɝɪɭɡ ɢ ɞɨɫɤɚ ɞɜɢɠɭɬɫɹ ɫ ɨɞɢɧɚɤɨɜɵɦ ɭɫɤɨɪɟɧɢɟɦ (ɢɥɢ ɩɨɤɨɹɬɫɹ), ɚ ɫɢɥɚ ɬɪɟɧɢɹ Fɬɪ1 ɦɟɠɞɭ ɝɪɭɡɨɦ ɢ
ɞɨɫɤɨɣ ɹɜɥɹɟɬɫɹ ɫɢɥɨɣ ɬɪɟɧɢɹ ɩɨɤɨɹ. ɋ ɭɜɟɥɢɱɟɧɢɟɦ ɜɧɟɲɧɟɣ ɫɢɥɵ