Механика.Методика решения задач
.pdfȽɥɚɜɚ 2. Ⱦɢɧɚɦɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ |
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F ɫɢɥɚ ɬɪɟɧɢɹ Fɬɪ1 ɜɨɡɪɚɫɬɚɟɬ ɢ ɩɪɢ ɧɟɤɨɬɨɪɨɦ ɡɧɚɱɟɧɢɢ ɜɧɟɲɧɟɣ |
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ɫɢɥɵ F0 ɞɨɫɬɢɝɚɟɬ ɫɜɨɟɝɨ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ, |
ɪɚɜɧɨɝɨ ɫɢɥɟ |
ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ. ɗɬɨ ɡɧɚɱɟɧɢɟ ɫɢɥɵ F0 ɢ ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ
ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ.
II. Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɹ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɢ ɞɨɫɤɢ ɜ ɩɪɨɟɤɰɢɹɯ
ɧɚ ɨɫɢ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ: |
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ma |
Fɬɪ1 , |
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(2.107) |
0 |
N mg , |
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(2.108) |
MA |
F Fɬɪ1 Fɬɪ2 , |
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(2.109) |
0 |
R N Mg . |
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(2.110) |
Ɂɞɟɫɶ a ɢ A – ɩɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɣ ɝɪɭɡɚ ɢ ɞɨɫɤɢ ɧɚ ɨɫɶ X. |
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ɂɫɩɨɥɶɡɭɟɦ ɡɚɤɨɧ Ⱥɦɨɧɬɨɧɚ – Ʉɭɥɨɧɚ, |
ɨɩɢɫɵɜɚɸɳɢɣ ɫɜɨɣ- |
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ɫɬɜɨ ɫɢɥɵ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ: |
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Fɬɪ2 P2 R . |
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(2.111) |
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Ɂɧɚɱɟɧɢɟ ɫɢɥɵ F0 , |
ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɝɪɭɡ, ɩɪɢ ɤɨɬɨɪɨɦ ɧɚɱ- |
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ɧɟɬɫɹ ɟɝɨ ɫɤɨɥɶɠɟɧɢɟ ɩɨ ɞɨɫɤɟ, ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɢɡ ɭɫɥɨɜɢɣ: |
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a |
A , |
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(2.112) |
Fɬɪ1 P1N . |
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(2.113) |
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III. Ɋɟɲɢɦ ɩɨɥɭɱɟɧɧɭɸ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ (2.107) – (2.113) |
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ɨɬɧɨɫɢɬɟɥɶɧɨ F0 : |
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F0 |
M m a Fɬɪ2 |
M m g P1 P2 . |
(2.114) |
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɞɨɫɤɚ ɜɵɫɤɨɥɶɡɧɭɥɚ ɢɡ-ɩɨɞ ɝɪɭɡɚ, ɧɟɨɛɯɨɞɢɦɨ ɩɪɢɥɨɠɢɬɶ ɤ ɞɨɫɤɟ ɫɢɥɭ F, ɭɞɨɜɥɟɬɜɨɪɹɸɳɭɸ
ɭɫɥɨɜɢɸ: |
M m g P1 P2 . |
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F t F0 |
(2.115) |
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ɉɨɞɫɬɚɜɥɹɹ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɡɚɞɚɧɧɵɟ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ, |
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ɩɨɥɭɱɚɟɦ: |
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F t F0 |
22,5 ɇ . |
(2.116) |
Ɂɚɞɚɱɚ 2.10
(Ɉɛɪɚɬɧɚɹ ɡɚɞɚɱɚ ɞɢɧɚɦɢɤɢ)
ɇɚɣɬɢ ɦɨɞɭɥɶ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɫɢɥɵ F, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɱɚɫɬɢɰɭ ɦɚɫɫɨɣ m ɩɪɢ ɟɟ ɞɜɢɠɟɧɢɢ ɜ ɩɥɨɫɤɨɫɬɢ XY ɩɨ ɷɥɥɢɩɬɢɱɟɫɤɨɣ ɬɪɚɟɤɬɨɪɢɢ. Ɂɚɤɨɧ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɤɨɨɪɞɢɧɚɬɧɨɣ ɮɨɪɦɟ ɢɦɟɟɬ
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ɜɢɞ x(t) |
Asin Zt , y(t) B cos Zt , ɝɞɟ A, B, Z – ɩɨɫɬɨɹɧɧɵɟ ɜɟ- |
ɥɢɱɢɧɵ. |
Ɋɟɲɟɧɢɟ |
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I.ɂɫɩɨɥɶɡɭɟɦ ɞɟɤɚɪɬɨɜɭ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ, ɤɚɤ ɩɪɟɞɥɨɠɟɧɨ
ɜɭɫɥɨɜɢɢ ɡɚɞɚɱɢ.
II.Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰɵ ɜ ɩɪɨɟɤɰɢɹɯ ɧɚ ɨɫɢ ɡɚɞɚɧɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ:
ma |
x |
m |
d2 |
x |
F |
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(2.117) |
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d t 2 |
x |
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ma |
y |
m |
d2 |
y |
F . |
(2.118) |
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d t 2 |
y |
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ɂɫɩɨɥɶɡɭɟɦ ɡɚɞɚɧɧɵɣ ɜ ɭɫɥɨɜɢɢ ɡɚɞɚɱɢ ɡɚɤɨɧ ɞɜɢɠɟɧɢɹ: |
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x(t) |
Asin |
Zt , |
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(2.119) |
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y(t) |
B cos Zt . |
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(2.120) |
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III. Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ (2.119) ɢ (2.120) ɞɜɚɠɞɵ ɩɨ ɜɪɟɦɟɧɢ, ɩɨɥɭɱɢɦ:
d2 x |
AZ2 sin Zt , |
(2.121) |
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d t 2 |
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d2 y |
BZ2 cos Zt . |
(2.122) |
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d t 2 |
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ɉɨɞɫɬɚɜɥɹɹ (2.121) ɢ (2.122) ɜ (2.117) ɢ (2.118), ɩɨɥɭɱɚɟɦ ɡɚ-
ɤɨɧ ɢɡɦɟɧɟɧɢɹ ɩɪɨɟɤɰɢɣ ɫɢɥɵ F, ɞɟɣɫɬɜɭɸɳɟɣ ɧɚ ɱɚɫɬɢɰɭ ɩɪɢ ɟɟ ɞɜɢɠɟɧɢɢ ɜ ɩɥɨɫɤɨɫɬɢ XY ɩɨ ɷɥɥɢɩɬɢɱɟɫɤɨɣ ɬɪɚɟɤɬɨɪɢɢ:
F (t) |
mAZ2 sin Zt , |
(2.123) |
x |
mBZ2 cos Zt . |
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F (t) |
(2.124) |
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y |
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ɂɫɩɨɥɶɡɭɹ (2.123) ɢ (2.124), ɩɨɥɭɱɚɟɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɦɨɞɭɥɹ ɫɢɥɵ F:
F F 2 |
F 2 |
mZ2 A2 sin 2 Zt B2 cos2 Zt . |
(2.125) |
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y |
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Ɂɚɩɢɲɟɦ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɫɢɥɵ ɜ ɜɟɤɬɨɪɧɨɦ ɜɢɞɟ:
F (t) F (t)i F (t) j |
mAZ2 sin Zt i mBZ2 cos Zt j |
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y |
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mZ2r(t) , |
(2.126) |
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Ƚɥɚɜɚ 2. Ⱦɢɧɚɦɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ |
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ɝɞɟ r – ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɱɚɫɬɢɰɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫɢɥɚ F , ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɱɚɫɬɢɰɭ, ɧɚɩɪɚɜɥɟɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨ ɪɚɞɢɭɫ-ɜɟɤɬɨɪɭ ɱɚɫɬɢɰɵ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ.
Ɂɚɞɚɱɚ 2.11
Ɉɞɧɨɪɨɞɧɵɣ ɭɩɪɭɝɢɣ ɫɬɟɪɠɟɧɶ ɞɜɢɠɟɬɫɹ ɩɨ ɝɥɚɞɤɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɩɨɫɬɨɹɧɧɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɫɢɥɵ F0 , ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɩɨ ɟɝɨ ɬɨɪɰɭ. Ⱦɥɢɧɚ
ɫɬɟɪɠɧɹ ɢ ɩɥɨɳɚɞɶ ɟɝɨ ɬɨɪɰɚ ɜ ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɦ ɫɨɫɬɨɹɧɢɢ ɪɚɜɧɵ l0 ɢ S0 , ɦɨɞɭɥɶ ɘɧɝɚ ɦɚɬɟɪɢɚɥɚ ɫɬɟɪɠɧɹ – E, ɤɨɷɮɮɢɰɢɟɧɬ
ɉɭɚɫɫɨɧɚ – ȝ. Ɉɩɪɟɞɟɥɢɬɶ ɡɚɜɢɫɢɦɨɫɬɢ ɧɚɩɪɹɠɟɧɢɹ ɭɩɪɭɝɢɯ ɫɢɥ V (x) ɢ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɞɟɮɨɪɦɚɰɢɢ H (x) ɨɬ ɤɨɨɪɞɢɧɚɬɵ x ɜɞɨɥɶ
ɫɬɟɪɠɧɹ, ɚ ɬɚɤɠɟ ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɭɞɥɢɧɟɧɢɟ ɫɬɟɪɠɧɹ.
Ɋɟɲɟɧɢɟ
I. ȼɵɛɟɪɟɦ ɞɟɤɚɪɬɨɜɭ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɫ ɨɫɶɸ X (ɫɦ. ɪɢɫ. 2.16), ɧɚɩɪɚɜɥɟɧɧɨɣ ɜɞɨɥɶ ɫɬɟɪɠɧɹ. ɉɪɢ ɭɫɤɨɪɟɧɧɨɦ ɞɜɢɠɟɧɢɢ ɫɬɟɪɠɧɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɫɢɥɵ F0 ɜ ɧɟɦ ɜɨɡɧɢɤɚɸɬ
ɜɧɭɬɪɟɧɧɢɟ ɭɩɪɭɝɢɟ ɫɢɥɵ ɢ ɩɪɨɞɨɥɶɧɵɟ ɞɟɮɨɪɦɚɰɢɢ, ɪɚɡɥɢɱɧɵɟ ɜ ɪɚɡɧɵɯ ɫɟɱɟɧɢɹɯ, ɚ ɬɚɤɠɟ ɢɡɦɟɧɟɧɢɹ ɩɨɩɟɪɟɱɧɵɯ ɪɚɡɦɟɪɨɜ. Ⱦɟɮɨɪɦɚɰɢɢ, ɜɨɡɧɢɤɚɸɳɢɟ ɜ ɫɬɟɪɠɧɟ, ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɦɚɥɵɦɢ, ɚ ɧɚɩɪɹɠɟɧɢɹ ɜɨɡɧɢɤɚɸɳɢɯ ɩɪɢ ɷɬɨɦ ɭɩɪɭɝɢɯ ɫɢɥ V (x) – ɩɨɞɱɢɧɹɸɳɢɦɢɫɹ
ɡɚɤɨɧɭ Ƚɭɤɚ (2.11).
[(x) |
[(x+dx) |
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V(x) |
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V(x+dx) |
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x+dx |
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Ɋɢɫ. 2.16
II. Ɋɚɫɫɦɨɬɪɢɦ ɫɥɨɣ dx ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɝɨ ɫɬɟɪɠɧɹ ɫ ɤɨɨɪɞɢɧɚɬɨɣ x ɜɞɨɥɶ ɧɟɝɨ (ɫɦ. ɪɢɫ. 2.16). Ɇɚɫɫɚ dm ɜɵɞɟɥɟɧɧɨɝɨ ɷɥɟɦɟɧɬɚ ɫɬɟɪɠɧɹ ɧɟ ɢɡɦɟɧɹɟɬɫɹ ɩɪɢ ɞɟɮɨɪɦɚɰɢɢ ɢ ɨɫɬɚɟɬɫɹ ɪɚɜɧɨɣ
dm |
m |
dx , |
(2.127) |
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l |
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ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
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ɝɞɟ m – ɦɚɫɫɚ ɜɫɟɝɨ ɫɬɟɪɠɧɹ. |
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ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɷɥɟɦɟɧɬɚ ɫɬɟɪɠɧɹ ɜ |
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ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ X ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɧɚɩɪɹɠɟɧɢɣ ɭɩɪɭɝɢɯ ɫɢɥ ɡɚɩɢ- |
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ɲɟɦ ɜ ɜɢɞɟ: |
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dma S (x dx [(x dx))V (x dx [ (x dx)) |
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S (x [(x))V (x [(x)) | S (x) |
wV |
dx . |
(2.128) |
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wx |
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Ɂɞɟɫɶ a – ɩɪɨɟɤɰɢɹ ɭɫɤɨɪɟɧɢɹ ɫɬɟɪɠɧɹ ɧɚ ɨɫɶ X, ȟ(x) ɢ ȟ(x+dx) – ɫɦɟɳɟɧɢɹ ɥɟɜɨɣ ɢ ɩɪɚɜɨɣ ɝɪɚɧɢɰ ɜɵɞɟɥɟɧɧɨɝɨ ɮɪɚɝɦɟɧɬɚ ɩɪɢ ɞɟ-
ɮɨɪɦɚɰɢɢ (ɫɦ. ɪɢɫ.2.16); S (x [ (x)) ɢ S (x dx [ (x dx)) – ɩɥɨ-
ɳɚɞɢ ɩɨɩɟɪɟɱɧɵɯ ɫɟɱɟɧɢɣ ɫɬɟɪɠɧɹ ɧɚ ɝɪɚɧɢɰɚɯ ɜɵɞɟɥɟɧɧɨɣ ɨɛɥɚɫɬɢ. ɉɨɫɤɨɥɶɤɭ ɞɟɮɨɪɦɚɰɢɢ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɦɚɥɵɦɢ, ɬɨ ɜ ɜɵɪɚɠɟɧɢɢ (2.128) ɨɬɛɪɨɲɟɧɵ ɱɥɟɧɵ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɦɚɥɨɫɬɢ ɧɟ ɬɨɥɶɤɨ ɩɨ dx,
ɧɨ ɢ ɩɨ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɩɪɨɞɨɥɶɧɨɣ ɞɟɮɨɪɦɚɰɢɢ H(x) |
w[ |
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Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ (2.128) ɞɨɩɨɥɧɢɦ ɝɪɚɧɢɱɧɵɦ |
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ɭɫɥɨɜɢɟɦ ɞɥɹ ɧɚɩɪɹɠɟɧɢɹ ɭɩɪɭɝɢɯ ɫɢɥ: |
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V (x 0) 0 . |
(2.129) |
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ɍɫɤɨɪɟɧɢɟ a, ɨɞɢɧɚɤɨɜɨɟ ɜ ɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɪɟɠɢɦɟ ɞɥɹ ɜɫɟɯ ɬɨɱɟɤ ɫɬɟɪɠɧɹ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫɨ ɜɬɨɪɵɦ ɡɚɤɨɧɨɦ ɇɶɸɬɨɧɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ:
a |
F0 |
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(2.130) |
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ɉɪɢ ɞɜɢɠɟɧɢɢ ɫɬɟɪɠɧɹ ɫ ɭɫɤɨɪɟɧɢɟɦ ɜɨɡɧɢɤɚɸɳɢɟ ɧɟɨɞɧɨɪɨɞɧɵɟ ɩɪɨɞɨɥɶɧɵɟ ɞɟɮɨɪɦɚɰɢɢ ɩɪɢɜɨɞɹɬ ɤ ɪɚɡɥɢɱɧɵɦ ɜ ɪɚɡɧɵɯ ɫɟɱɟɧɢɹɯ ɩɨɩɟɪɟɱɧɵɦ ɞɟɮɨɪɦɚɰɢɹɦ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɡɦɟɧɟɧɢɸ ɩɥɨɳɚɞɢ ɩɨɩɟɪɟɱɧɵɯ ɫɟɱɟɧɢɣ ɫɬɟɪɠɧɹ S(x):
S (x) S0 (1 PH(x))2 | S0 (1 2PH(x)) . |
(2.131) |
ɇɚɩɪɹɠɟɧɢɹ ɭɩɪɭɝɢɯ ɫɢɥ ɫɜɹɡɚɧɵ ɫ ɩɪɨɞɨɥɶɧɵɦɢ ɞɟɮɨɪɦɚ- |
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ɰɢɹɦɢ ɡɚɤɨɧɨɦ Ƚɭɤɚ: |
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V (x) EH (x) . |
(2.132) |
III. ɉɪɟɨɛɪɚɡɭɹ ɡɚɩɢɫɚɧɧɭɸ ɫɢɫɬɟɦɭ |
ɭɪɚɜɧɟɧɢɣ (2.127), |
(2.128) ɢ (2.130) – (2.132), ɩɨɥɭɱɚɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɧɚɩɪɹɠɟɧɢɣ ɭɩɪɭɝɢɯ ɫɢɥ:
F0 |
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2P |
V (x) · wV (x) |
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Ƚɥɚɜɚ 2. Ⱦɢɧɚɦɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ |
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ɂɧɬɟɝɪɢɪɭɹ (2.133) ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ ɫ ɭɱɟɬɨɦ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ (2.129), ɩɨɥɭɱɚɟɦ:
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V (x)2 . |
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Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (2.134) ɨɬɧɨɫɢɬɟɥɶɧɨ V (x) ɢɦɟɟɬ ɜɢɞ: |
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Ɂɚɩɢɲɟɦ ɢɫɤɨɦɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɧɚɩɪɹɠɟɧɢɣ ɭɩɪɭɝɢɯ ɫɢɥ ɜɞɨɥɶ ɫɬɟɪɠɧɹ ɫ ɭɱɟɬɨɦ ɦɚɥɨɫɬɢ ɜɬɨɪɨɝɨ ɫɥɚɝɚɟɦɨɝɨ ɜ ɩɨɞɤɨɪɟɧɧɨɦ ɜɵɪɚɠɟɧɢɢ (2.135):
V (x) |
F0 x |
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S0l0 . |
(2.136) |
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɞɟɮɨɪɦɚɰɢɣ ɜɞɨɥɶ ɫɬɟɪɠɧɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɤɨɧɨɦ Ƚɭɤɚ ɢɦɟɟɬ ɜɢɞ:
H (x) |
F0 x |
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S0l0 E . |
(2.137) |
ɉɨɥɧɨɟ ɩɪɨɞɨɥɶɧɨɟ ɭɞɥɢɧɟɧɢɟ ɫɬɟɪɠɧɹ ɧɚɯɨɞɢɦ, ɢɧɬɟɝɪɢɪɭɹ
(2.137):
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(2.138) |
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ɂɫɤɨɦɨɟ ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɭɞɥɢɧɟɧɢɟ ɫɬɟɪɠɧɹ ɪɚɜɧɨ: |
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(2.139) |
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2.4. Ɂɚɞɚɱɢ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɝɨ ɪɟɲɟɧɢɹ
Ɂɚɞɚɱɚ 1
ɇɚ ɧɟɩɨɞɜɢɠɧɨɦ ɤɥɢɧɟ ɫ ɭɝɥɨɦ ɩɪɢ ɨɫɧɨɜɚɧɢɢ D ɞɢɬɫɹ ɬɟɥɨ ɦɚɫɫɨɣ m1 , ɤ ɤɨɬɨɪɨɦɭ ɩɪɢ-
ɤɪɟɩɥɟɧɚ ɥɟɝɤɚɹ ɧɟɪɚɫɬɹɠɢɦɚɹ ɧɢɬɶ, |
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ɩɟɪɟɤɢɧɭɬɚɹ ɱɟɪɟɡ ɧɟɜɟɫɨɦɵɣ ɛɥɨɤ, ɠɟ- |
m1 |
ɫɬɤɨ ɫɜɹɡɚɧɧɵɣ ɫ ɤɥɢɧɨɦ. Ʉ ɞɪɭɝɨɦɭ |
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ɤɨɧɰɭ ɧɢɬɢ ɩɪɢɤɪɟɩɥɟɧɨ ɬɟɥɨ ɦɚɫɫɨɣ |
D |
m2 , ɧɟ ɤɚɫɚɸɳɟɟɫɹ ɤɥɢɧɚ (ɫɦ. ɪɢɫ.). Ɉɬ- |
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= 30q ɧɚɯɨ-
m2
76 ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ
ɧɨɲɟɧɢɟ ɦɚɫɫ ɬɟɥ K m2 
m1 2 / 3 . Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɦɟɠɞɭ ɩɟɪɜɵɦ ɬɟɥɨɦ ɢ ɩɥɨɫɤɨɫɬɶɸ ɪɚɜɟɧ P = 0.1. ɇɚɣɬɢ ɜɟɥɢɱɢɧɭ ɢ ɧɚ-
ɩɪɚɜɥɟɧɢɟ ɭɫɤɨɪɟɧɢɹ ɜɬɨɪɨɝɨ ɬɟɥɚ. |
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Ɉɬɜɟɬ: ɭɫɤɨɪɟɧɢɟ ɜɬɨɪɨɝɨ ɬɟɥɚ ɧɚɩɪɚɜɥɟɧɨ ɜɧɢɡ ɢ ɪɚɜɧɨ |
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a2 g K sin D P cosD K 1 |
0.05g . |
Ɂɚɞɚɱɚ 2
ɇɚ ɧɚɤɥɨɧɧɭɸ ɩɨɜɟɪɯɧɨɫɬɶ, ɫɨɫɬɚɜɥɹɸɳɭɸ ɭɝɨɥ D ɫ ɝɨɪɢɡɨɧɬɨɦ, ɩɨɥɨɠɢɥɢ ɞɜɚ ɛɪɭɫɤɚ 1 ɢ 2 (ɫɦ. ɪɢɫ.). Ɇɚɫɫɵ ɛɪɭɫɤɨɜ ɪɚɜɧɵ m1
ɢ m2, ɤɨɷɮɮɢɰɢɟɧɬɵ ɬɪɟɧɢɹ ɦɟɠɞɭ ɩɨɜɟɪɯ- |
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ɧɨɫɬɶɸ ɢ ɷɬɢɦɢ ɛɪɭɫɤɚɦɢ – P1 ɢ P2, ɩɪɢɱɟɦ |
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P1 > P2. ɇɚɣɬɢ ɫɢɥɭ ɞɚɜɥɟɧɢɹ ɨɞɧɨɝɨ ɛɪɭɫɤɚ |
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ɧɚ ɞɪɭɝɨɣ, ɜɨɡɧɢɤɚɸɳɭɸ ɜ ɩɪɨɰɟɫɫɟ ɢɯ |
D |
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ɫɤɨɥɶɠɟɧɢɹ, ɢ ɭɝɥɵ D, ɩɪɢ ɤɨɬɨɪɵɯ ɛɭɞɟɬ |
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ɫɤɨɥɶɠɟɧɢɟ ɛɪɭɫɤɨɜ. |
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Ɉɬɜɟɬ: F |
P1 P2 m1m2 g cosD |
; D ! arctg |
P1m1 P2m2 |
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Ɂɚɞɚɱɚ 3
Ɇɚɬɟɪɢɚɥɶɧɚɹ ɬɨɱɤɚ ɦɚɫɫɨɣ m ɞɜɢɠɟɬɫɹ ɩɨ ɝɥɚɞɤɨɣ ɜɧɭɬɪɟɧɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɜɟɪɬɢɤɚɥɶɧɨɝɨ ɰɢɥɢɧɞɪɚ ɪɚɞɢɭɫɨɦ R. ɇɚɣɬɢ ɦɨɞɭɥɶ ɫɢɥɵ ɞɚɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɧɚ ɫɬɟɧɤɭ ɰɢɥɢɧɞɪɚ ɜ ɬɨɬ ɦɨɦɟɧɬ, ɤɨɝɞɚ ɟɟ ɫɤɨɪɨɫɬɶ ɫɨɫɬɚɜɥɹɟɬ ɭɝɨɥ D ɫ ɝɨɪɢɡɨɧɬɨɦ ɢ ɩɨ ɦɨɞɭɥɸ ɪɚɜɧɚ X0 .
Ɉɬɜɟɬ: F mX02 
R cos2 D .
Ɂɚɞɚɱɚ 4
ɑɚɫɬɢɰɚ ɞɜɢɠɟɬɫɹ ɜɞɨɥɶ ɨɫɢ X ɩɨ ɡɚɤɨɧɭ x Dt 2 Et3 , ɝɞɟ D
ɢ E – ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ. ȼ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t 0 |
ɫɢɥɚ, |
ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɱɚɫɬɢɰɭ, ɪɚɜɧɚ F0 . ɇɚɣɬɢ ɦɨɞɭɥɢ ɫɢɥɵ ɜ ɬɨɱɤɟ |
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ɩɨɜɨɪɨɬɚ ɢ ɜ ɦɨɦɟɧɬ, ɤɨɝɞɚ ɱɚɫɬɢɰɚ ɨɩɹɬɶ ɨɤɚɠɟɬɫɹ ɜ ɬɨɱɤɟ x |
0 . |
Ɉɬɜɟɬ: F0 , 2F0 . |
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Ƚɥɚɜɚ 2. Ⱦɢɧɚɦɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ |
77 |
Ɂɚɞɚɱɚ 5
ɇɚ ɝɥɚɞɤɨɣ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɥɟɠɢɬ ɤɥɢɧ ɦɚɫɫɨɣ M ɫ ɭɝɥɨɦ ɩɪɢ ɨɫɧɨɜɚɧɢɢ D. Ɍɟɥɨ ɦɚɫɫɨɣ m ɫɤɨɥɶɡɢɬ ɩɨ ɧɚɤɥɨɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɥɢɧɚ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɦɟɠɞɭ ɤɥɢɧɨɦ ɢ ɬɟɥɨɦ ɪɚɜɟɧ P . ɇɚɣɬɢ ɝɨɪɢɡɨɧɬɚɥɶɧɵɟ ɩɪɨɟɤɰɢɢ ɭɫɤɨɪɟɧɢɣ ɬɟɥɚ ɢ ɤɥɢɧɚ,
ɚ ɬɚɤɠɟ ɫɢɥɵ N ɢ R, ɫ ɤɨɬɨɪɵɦɢ ɬɟɥɨ ɞɚɜɢɬ ɧɚ ɤɥɢɧ ɢ ɤɥɢɧ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɭɸ ɩɨɜɟɪɯɧɨɫɬɶ.
Ɉɬɜɟɬ: |
g sin D P cosD |
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mg sin D P cosD |
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am |
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sin D tgD P |
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sin D tgD P |
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cosD |
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m cosD |
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m cosD P sin D |
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mg |
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M ¸g . |
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sin D tgD P |
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sin D tgD P |
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Ɂɚɞɚɱɚ 6
ɇɢɬɶ ɩɟɪɟɤɢɧɭɬɚ ɱɟɪɟɡ ɥɟɝɤɢɣ ɜɪɚɳɚɸɳɢɣɫɹ ɛɟɡ ɬɪɟɧɢɹ ɛɥɨɤ. ɇɚ ɨɞɧɨɦ ɤɨɧɰɟ ɧɢɬɢ ɩɪɢɤɪɟɩɥɟɧ ɝɪɭɡ ɦɚɫɫɨɣ M, ɚ ɩɨ ɞɪɭɝɨɣ ɫɜɢɫɚɸɳɟɣ ɱɚɫɬɢ ɧɢɬɢ ɫɤɨɥɶɡɢɬ ɦɭɮɬɨɱɤɚ ɦɚɫɫɨɣ m ɫ ɩɨɫɬɨɹɧɧɵɦ ɭɫ-
ɤɨɪɟɧɢɟɦ ac |
ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɢɬɢ. ɇɚɣɬɢ ɫɢɥɭ ɬɪɟɧɢɹ, ɫ ɤɨɬɨɪɨɣ |
ɧɢɬɶ ɞɟɣɫɬɜɭɟɬ ɧɚ ɦɭɮɬɨɱɤɭ. |
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Ɉɬɜɟɬ: Fɬɪ |
2g ac mM m M . |
Ɂɚɞɚɱɚ 7
ɉɭɥɹ, ɩɪɨɛɢɜɚɹ ɞɨɫɤɭ ɬɨɥɳɢɧɨɣ h, ɢɡɦɟɧɹɟɬ ɫɜɨɸ ɫɤɨɪɨɫɬɶ ɨɬ X0 ɞɨ X . ɇɚɣɬɢ ɜɪɟɦɹ ɞɜɢɠɟɧɢɹ ɩɭɥɢ ɜ ɞɨɫɤɟ, ɫɱɢɬɚɹ ɫɢɥɭ ɫɨɩɪɨ-
ɬɢɜɥɟɧɢɹ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɣ ɤɜɚɞɪɚɬɭ ɫɤɨɪɨɫɬɢ. |
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Ɉɬɜɟɬ: t |
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Ɂɚɞɚɱɚ 8
ɑɟɪɟɡ ɛɥɨɤ, ɨɫɶ ɤɨɬɨɪɨɝɨ ɝɨɪɢɡɨɧɬɚɥɶɧɚ, ɩɟɪɟɤɢɧɭɬɚ ɧɟɪɚɫɬɹɠɢɦɚɹ ɜɟɪɟɜɤɚ ɞɥɢɧɨɣ l. Ɂɚ ɤɨɧɰɵ ɜɟɪɟɜɤɢ ɞɟɪɠɚɬɫɹ ɞɜɟ ɨɛɟɡɶɹɧɵ ɨɞɢɧɚɤɨɜɨɣ ɦɚɫɫɨɣ, ɧɚɯɨɞɹɳɢɟɫɹ ɧɚ ɨɞɢɧɚɤɨɜɨɦ ɪɚɫɫɬɨɹɧɢɢ l ɨɬ
78 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ɛɥɨɤɚ. Ɉɛɟɡɶɹɧɵ ɧɚɱɢɧɚɸɬ ɨɞɧɨɜɪɟɦɟɧɧɨ ɩɨɞɧɢɦɚɬɶɫɹ ɜɜɟɪɯ, ɩɪɢɱɟɦ ɨɞɧɚ ɢɡ ɧɢɯ ɩɨɞɧɢɦɚɟɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɜɟɪɟɜɤɢ ɫɨ ɫɤɨɪɨɫɬɶɸ X , ɚ ɞɪɭɝɚɹ ɫɨ ɫɤɨɪɨɫɬɶɸ 2X. ɑɟɪɟɡ ɤɚɤɢɟ ɢɧɬɟɪɜɚɥɵ ɜɪɟɦɟɧɢ ɤɚɠɞɚɹ ɢɡ ɨɛɟɡɶɹɧ ɞɨɫɬɢɝɧɟɬ ɛɥɨɤɚ? Ɇɚɫɫɚɦɢ ɛɥɨɤɚ ɢ ɜɟɪɟɜɤɢ ɩɪɟɧɟɛɪɟɱɶ.
2l
Ɉɬɜɟɬ: t1 t2 3X .
Ɂɚɞɚɱɚ 9
ɋɢɫɬɟɦɚ ɬɪɟɯ ɬɟɥ, ɫɜɹɡɚɧɧɵɯ ɦɟɠɞɭ ɫɨɛɨɣ ɫ ɩɨɦɨɳɶɸ ɞɜɭɯ ɧɢɬɟɣ ɢ ɬɪɟɯ ɛɥɨɤɨɜ, ɢɡɨɛɪɚɠɟɧɚ ɧɚ ɪɢɫɭɧɤɟ. Ⱦɜɚ ɬɟɥɚ ɩɨɞɜɟɲɟɧɵ ɧɚ ɧɢɬɹɯ, ɚ ɬɪɟɬɶɟ ɧɚɯɨɞɢɬɫɹ ɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ. Ɉɫɢ ɤɪɚɣɧɢɯ ɛɥɨɤɨɜ, ɜ ɨɬɥɢɱɢɟ ɨɬ ɨɫɢ ɫɪɟɞɧɟɝɨ ɛɥɨɤɚ, ɡɚɤɪɟɩɥɟɧɵ (ɫɦ. ɪɢɫ.).
m1
m3 
m2
ɋɱɢɬɚɹ ɡɚɞɚɧɧɵɦɢ ɦɚɫɫɵ m1 ɢ m2, ɨɩɪɟɞɟɥɢɬɶ ɦɚɫɫɭ m3, ɩɪɢ ɤɨɬɨɪɨɣ ɨɫɶ ɫɪɟɞɧɟɝɨ ɛɥɨɤɚ ɛɭɞɟɬ ɨɫɬɚɜɚɬɶɫɹ ɧɟɩɨɞɜɢɠɧɨɣ. Ɍɪɟɧɢɟɦ ɢ ɦɚɫɫɚɦɢ ɛɥɨɤɨɜ ɢ ɧɢɬɟɣ ɩɪɟɧɟɛɪɟɱɶ.
Ɉɬɜɟɬ: m |
2m1m2 |
. |
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3 |
m2 |
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m1 |
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Ɂɚɞɚɱɚ 10
Ʉɚɤɨɜ ɞɨɥɠɟɧ ɛɵɬɶ ɦɢɧɢɦɚɥɶɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɫɤɨɥɶɠɟɧɢɹ P ɦɟɠɞɭ ɲɢɧɚɦɢ ɚɜɬɨɦɨɛɢɥɹ ɢ ɚɫɮɚɥɶɬɨɦ, ɱɬɨɛɵ ɚɜ-
ɬɨɦɨɛɢɥɶ ɦɨɝ ɩɪɨɣɬɢ ɡɚɤɪɭɝɥɟɧɢɟ ɫ ɪɚɞɢɭɫɨɦ R = 200 ɦ ɧɚ ɫɤɨɪɨɫɬɢ X = 100 ɤɦ/ɱ?
Ɉɬɜɟɬ: P |
X2 |
| 0.4 . |
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Rg |
Ƚɥɚɜɚ 2. Ⱦɢɧɚɦɢɤɚ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɢ ɩɪɨɫɬɟɣɲɢɯ ɫɢɫɬɟɦ |
79 |
Ɂɚɞɚɱɚ 11.
Ɉɞɧɨɪɨɞɧɵɣ ɭɩɪɭɝɢɣ ɫɬɟɪɠɟɧɶ ɦɚɫɫɨɣ m ɩɨɞɜɟɫɢɥɢ ɡɚ ɨɞɢɧ ɤɨɧɟɰ ɤ ɩɨɬɨɥɤɭ. Ⱦɥɢɧɚ ɢ ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɫɬɟɪɠɧɹ ɜ ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɦ ɫɨɫɬɨɹɧɢɢ – l0 ɢ S0 , ɦɨɞɭɥɶ ɘɧɝɚ ɦɚɬɟɪɢɚɥɚ
ɫɬɟɪɠɧɹ ɪɚɜɟɧ E, ɚ ɤɨɷɮɮɢɰɢɟɧɬ ɉɭɚɫɫɨɧɚ – ȝ. Ɉɩɪɟɞɟɥɢɬɶ ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɭɞɥɢɧɟɧɢɟ ɫɬɟɪɠɧɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ ɬɹɠɟɫɬɢ, ɚ ɬɚɤɠɟ ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɢɡɦɟɧɟɧɢɟ ɟɝɨ ɨɛɴɟɦɚ.
Ɉɬɜɟɬ: |
ǻl |
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mg |
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; |
ǻV |
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(1 2P)mg |
. |
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l |
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2ES |
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V |
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0 |
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0 |
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2ES |
0 |
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0 |
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80 |
ɆȿɏȺɇɂɄȺ. ɆȿɌɈȾɂɄȺ Ɋȿɒȿɇɂə ɁȺȾȺɑ |
ȽɅȺȼȺ 3 ɁȺɄɈɇɕ ɂɁɆȿɇȿɇɂə ɂɆɉɍɅɖɋȺ ɂ ɆȿɏȺɇɂɑȿɋɄɈɃ
ɗɇȿɊȽɂɂ ɋɂɋɌȿɆɕ ɆȺɌȿɊɂȺɅɖɇɕɏ ɌɈɑȿɄ
3.1. Ɍɟɨɪɟɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ
ɐɟɧɬɪ ɦɚɫɫ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ (ɫɢɫɬɟɦɵ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ) – ɬɨɱɤɚ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɪɚɞɢɭɫ-ɜɟɤɬɨɪ ɤɨɬɨɪɨɣ rɰɦ ɪɚɜɟɧ
rɰɦ |
¦mi ri |
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i |
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, |
(3.1) |
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ɝɞɟ m ¦mi |
m |
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– ɦɚɫɫɚ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ, ri |
ɢ mi – ɪɚɞɢɭɫ- |
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i |
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ɜɟɤɬɨɪ ɢ ɦɚɫɫɚ i-ɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɫɢɫɬɟɦɵ.
ɋɤɨɪɨɫɬɶ ɰɟɧɬɪɚ ɦɚɫɫ ȣɰɦ – ɮɢɡɢɱɟɫɤɚɹ ɜɟɥɢɱɢɧɚ, ɪɚɜɧɚɹ
|
ȣɰɦ |
¦miȣi |
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i |
. |
(3.2) |
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ɝɞɟ ȣi |
m |
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– ɫɤɨɪɨɫɬɶ i-ɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɫɢɫɬɟɦɵ. |
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ɍɫɤɨɪɟɧɢɟ ɰɟɧɬɪɚ ɦɚɫɫ aɰɦ |
– ɮɢɡɢɱɟɫɤɚɹ ɜɟɥɢɱɢɧɚ, ɪɚɜɧɚɹ |
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aɰɦ |
¦miai |
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i |
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. |
(3.3) |
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ɝɞɟ ai |
m |
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– ɭɫɤɨɪɟɧɢɟ i-ɨɣ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɫɢɫɬɟɦɵ. |
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3.1.1. ɂɦɩɭɥɶɫ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ
ɂɦɩɭɥɶɫ (ɤɨɥɢɱɟɫɬɜɨ ɞɜɢɠɟɧɢɹ) ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ p
– ɮɢɡɢɱɟɫɤɚɹ ɜɟɥɢɱɢɧɚ, ɪɚɜɧɚɹ ɩɪɨɢɡɜɟɞɟɧɢɸ ɦɚɫɫɵ ɦɚɬɟɪɢɚɥɶɧɨɣ ɬɨɱɤɢ ɧɚ ɟɟ ɫɤɨɪɨɫɬɶ:
p mȣ . |
(3.4) |
ɂɦɩɭɥɶɫ ɦɟɯɚɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ P – ɮɢɡɢɱɟɫɤɚɹ ɜɟɥɢɱɢɧɚ,
ɪɚɜɧɚɹ ɫɭɦɦɟ ɢɦɩɭɥɶɫɨɜ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ, ɢɡ ɤɨɬɨɪɵɯ ɫɨɫɬɨɢɬ ɫɢɫɬɟɦɚ:
P ¦ pi |
¦miȣi mȣɰɦ { pɰɦ . |
(3.5) |
i |
i |
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