Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 32 стр.

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'
– ɢɡɦɟɧɟɧɢɟ ɞɥɢɧɵ ɩɪɭɠɢɧɵ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɟɟ ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɣ
ɞɥɢɧɨɣ;
ɞɥɹ ɫɩɢɪɚɥɶɧɨɣ ɩɪɭɠɢɧɵ:
c – ɠɟɫɬɤɨɫɬɶ, ɪɚɜɧɚɹ ɜɟɥɢɱɢɧɟ ɦɨɦɟɧɬɚ ɫɢɥɵ, ɜɵɡɵɜɚɸɳɟɝɨ
ɡɚɤɪɭɱɢɜɚɧɢɟ ɩɪɭɠɢɧɵ ɧɚ 1 ɪɚɞɢɚɧ;
'
– ɭɝɨɥ ɡɚɤɪɭɱɢɜɚɧɢɹ ɨɬ ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ.
ɉɪɢɪɚɳɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɩɪɢ ɩɟɪɟɦɟɳɟɧɢɢ ɟɟ ɢɡ
ɨɞɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɜ ɞɪɭɝɨɟ ɪɚɜɧɨ ɫɭɦɦɟ ɪɚɛɨɬ, ɩɪɨɢɡɜɟɞɟɧɧɵɯ ɧɚ ɷɬɨɦ
ɩɟɪɟɦɟɳɟɧɢɢ ɜɫɟɦɢ ɫɢɥɚɦɢ, ɩɪɢɥɨɠɟɧɧɵɦɢ ɤ ɫɢɫɬɟɦɟ, ɬ.ɟ.
T
2
– T
1
= A
12
.
ȿɫɥɢ ɫɢɫɬɟɦɚ ɧɟɢɡɦɟɧɹɟɦɚɹ, ɬɨ
)(
1212
e
ATT
, ɝɞɟ
)(
12
e
A
– ɫɭɦɦɚ ɪɚɛɨɬ ɜɧɟɲɧɢɯ ɫɢɥ.
ɉɪɨɢɡɜɨɞɧɚɹ ɨɬ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɪɚɜɧɚ ɫɭɦɦɟ
ɦɨɳɧɨɫɬɟɣ ɜɫɟɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɷɬɭ ɫɢɫɬɟɦɭ, ɬ.ɟ.
N
dt
dT
.
ȿɫɥɢ ɫɢɫɬɟɦɚ ɞɜɢɠɟɬɫɹ ɜ ɩɨɬɟɧɰɢɚɥɶɧɨɦ ɫɢɥɨɜɨɦ ɩɨɥɟ, ɬɨ ɩɨɥɧɚɹ
ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ, ɪɚɜɧɚɹ ɫɭɦɦɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ
ɷɧɟɪɝɢɣ, ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ, ɬ. ɟ. Ɍ + ɉ = const.
Ɂɚɞɚɱɚ ʋ 1. Ƚɪɭɡ Ɇ ɦɚɫɫɨɣ m
ɩɨɦɟɳɟɧ ɧɚ ɧɟɝɥɚɞɤɭɸ ɧɚɤɥɨɧɧɭɸ
ɩɥɨɫɤɨɫɬɶ, ɨɛɪɚɡɭɸɳɭɸ ɫ
ɝɨɪɢɡɨɧɬɨɦ ɭɝɨɥ
D
, ɢ ɩɪɢɤɪɟɩɥɟɧ
ɤ ɤɨɧɰɭ ɩɪɭɠɢɧɵ ɫ ɠɟɫɬɤɨɫɬɶɸ
ɫ, ɞɪɭɝɨɣ ɤɨɧɟɰ ɤɨɬɨɪɨɣ ɡɚɤɪɟɩɥɟɧ
ɧɟɩɨɞɜɢɠɧɨ. Ɉɩɪɟɞɟɥɢɬɶ
ɦɚɤɫɢɦɚɥɶɧɨɟ ɪɚɫɬɹɠɟɧɢɟ S
ɩɪɭɠɢɧɵ, ɟɫɥɢ ɜ ɧɚɱɚɥɶɧɵɣ
ɦɨɦɟɧɬ ɩɪɭɠɢɧɚ ɛɵɥɚ
ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɚ, ɚ ɝɪɭɡ ɨɬɩɭɳɟɧ
ɛɟɡ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ.
Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɬɟɥɚ ɨ
ɩɥɨɫɤɨɫɬɶ ɪɚɜɟɧ f, ɩɪɢɱɟɦ
.
D
tgf
Ɋɟɲɟɧɢɟ. ɂɦɟɟɦ: ɧɚ ɝɪɭɡ Ɇ ɞɟɣɫɬɜɭɟɬ ɜɟɫ
P
, ɭɩɪɭɝɚɹ ɫɢɥɚ
F
,
ɧɨɪɦɚɥɶɧɚɹ ɪɟɚɤɰɢɹ ɩɥɨɫɤɨɫɬɢ
N
ɢ ɫɢɥɚ ɬɪɟɧɢɹ
ɬɪ
F
, ɧɚɩɪɚɜɥɟɧɧɚɹ ɤɚɤ
F
.
ɂɡɦɟɧɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɪɚɜɧɨ ɪɚɛɨɬɟ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ
ɫɢɫɬɟɦɟ:
¦
)(
1212
FATT
. Ɍɚɤ ɤɚɤ
,0
21
vv
ɬɨ
0
21
TT
,
ɫɥɟɞɨɜɚɬɟɥɶɧɨ,
,0
12
A
ɬ. ɟ.
0)()()()(
1212121212
NAFAFAPAA
ɬɪ
.
Ɂɞɟɫɶ
D
cosfPF
ɬɪ
,
cxF
,
.mgP 
     ' – ɢɡɦɟɧɟɧɢɟ ɞɥɢɧɵ ɩɪɭɠɢɧɵ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɟɟ ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɣ
ɞɥɢɧɨɣ;
    ɞɥɹ ɫɩɢɪɚɥɶɧɨɣ ɩɪɭɠɢɧɵ:
    c – ɠɟɫɬɤɨɫɬɶ, ɪɚɜɧɚɹ ɜɟɥɢɱɢɧɟ ɦɨɦɟɧɬɚ ɫɢɥɵ, ɜɵɡɵɜɚɸɳɟɝɨ
ɡɚɤɪɭɱɢɜɚɧɢɟ ɩɪɭɠɢɧɵ ɧɚ 1 ɪɚɞɢɚɧ;
     ' – ɭɝɨɥ ɡɚɤɪɭɱɢɜɚɧɢɹ ɨɬ ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ.
    ɉɪɢɪɚɳɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɩɪɢ ɩɟɪɟɦɟɳɟɧɢɢ ɟɟ ɢɡ
ɨɞɧɨɝɨ ɩɨɥɨɠɟɧɢɹ ɜ ɞɪɭɝɨɟ ɪɚɜɧɨ ɫɭɦɦɟ ɪɚɛɨɬ, ɩɪɨɢɡɜɟɞɟɧɧɵɯ ɧɚ ɷɬɨɦ
ɩɟɪɟɦɟɳɟɧɢɢ ɜɫɟɦɢ ɫɢɥɚɦɢ, ɩɪɢɥɨɠɟɧɧɵɦɢ ɤ ɫɢɫɬɟɦɟ, ɬ.ɟ.
    T2 – T1 = A12.
    ȿɫɥɢ ɫɢɫɬɟɦɚ ɧɟɢɡɦɟɧɹɟɦɚɹ, ɬɨ
    T2  T1 A12( e ) , ɝɞɟ A12( e ) – ɫɭɦɦɚ ɪɚɛɨɬ ɜɧɟɲɧɢɯ ɫɢɥ.
    ɉɪɨɢɡɜɨɞɧɚɹ ɨɬ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɫɢɫɬɟɦɵ ɪɚɜɧɚ ɫɭɦɦɟ
                                                               dT
ɦɨɳɧɨɫɬɟɣ ɜɫɟɯ ɫɢɥ, ɞɟɣɫɬɜɭɸɳɢɯ ɧɚ ɷɬɭ ɫɢɫɬɟɦɭ, ɬ.ɟ.              N.
                                                               dt
    ȿɫɥɢ ɫɢɫɬɟɦɚ ɞɜɢɠɟɬɫɹ ɜ ɩɨɬɟɧɰɢɚɥɶɧɨɦ ɫɢɥɨɜɨɦ ɩɨɥɟ, ɬɨ ɩɨɥɧɚɹ
ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ, ɪɚɜɧɚɹ ɫɭɦɦɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɢ ɩɨɬɟɧɰɢɚɥɶɧɨɣ
ɷɧɟɪɝɢɣ, ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ, ɬ. ɟ. Ɍ + ɉ = const.

Ɂɚɞɚɱɚ ʋ 1. Ƚɪɭɡ Ɇ ɦɚɫɫɨɣ m
ɩɨɦɟɳɟɧ ɧɚ ɧɟɝɥɚɞɤɭɸ ɧɚɤɥɨɧɧɭɸ
ɩɥɨɫɤɨɫɬɶ,       ɨɛɪɚɡɭɸɳɭɸ     ɫ
ɝɨɪɢɡɨɧɬɨɦ ɭɝɨɥ D , ɢ ɩɪɢɤɪɟɩɥɟɧ
ɤ ɤɨɧɰɭ ɩɪɭɠɢɧɵ ɫ ɠɟɫɬɤɨɫɬɶɸ
ɫ, ɞɪɭɝɨɣ ɤɨɧɟɰ ɤɨɬɨɪɨɣ ɡɚɤɪɟɩɥɟɧ
ɧɟɩɨɞɜɢɠɧɨ.            Ɉɩɪɟɞɟɥɢɬɶ
ɦɚɤɫɢɦɚɥɶɧɨɟ ɪɚɫɬɹɠɟɧɢɟ S
ɩɪɭɠɢɧɵ, ɟɫɥɢ ɜ ɧɚɱɚɥɶɧɵɣ
ɦɨɦɟɧɬ          ɩɪɭɠɢɧɚ      ɛɵɥɚ
ɧɟɞɟɮɨɪɦɢɪɨɜɚɧɚ, ɚ ɝɪɭɡ ɨɬɩɭɳɟɧ
ɛɟɡ        ɧɚɱɚɥɶɧɨɣ    ɫɤɨɪɨɫɬɢ.
Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɬɟɥɚ ɨ
ɩɥɨɫɤɨɫɬɶ ɪɚɜɟɧ f, ɩɪɢɱɟɦ
 f  tgD .
    Ɋɟɲɟɧɢɟ. ɂɦɟɟɦ: ɧɚ ɝɪɭɡ Ɇ ɞɟɣɫɬɜɭɟɬ ɜɟɫ P , ɭɩɪɭɝɚɹ ɫɢɥɚ F ,
ɧɨɪɦɚɥɶɧɚɹ ɪɟɚɤɰɢɹ ɩɥɨɫɤɨɫɬɢ N ɢ ɫɢɥɚ ɬɪɟɧɢɹ Fɬɪ , ɧɚɩɪɚɜɥɟɧɧɚɹ ɤɚɤ F .
    ɂɡɦɟɧɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɪɚɜɧɨ ɪɚɛɨɬɟ ɫɢɥ, ɩɪɢɥɨɠɟɧɧɵɯ ɤ
ɫɢɫɬɟɦɟ: T2  T1     ¦A12 ( F ) . Ɍɚɤ ɤɚɤ v1 v 2 0, ɬɨ T1 T2 0 ,
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, A12  0, ɬ. ɟ. A12 A12 ( P )  A12 ( Fɬɪ )  A12 ( F )  A12 ( N )   0.
    Ɂɞɟɫɶ Fɬɪ   fP cos D , F cx , P mg.

                                       32

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