Решение задач по теоретической механике. Ч.3. Динамика. Чеботарев А.С - 7 стр.

UptoLike

ВУЗ: 

ВГУ | Воронеж

Составители: 

Рубрика: 

7
§2. Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɜɟɪɞɵɯ ɬɟɥ
Ɇɨɦɟɧɬɨɦ ɢɧɟɪɰɢɢ
z
J
ɫɢɫɬɟɦɵ N ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ
z
ɧɚɡɵɜɚɟɬɫɹ ɜɟɥɢɱɢɧɚ
¦
N
k
kkz
rmJ
1
2
. Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ
ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɦɚɫɫ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ
ɨɫɢ.
k
r
– ɪɚɫɫɬɨɹɧɢɟ ɨɬ Ʉ-ɣ ɬɨɱɤɢ ɞɨ ɨɫɢ
z
. Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɧɟɨɬɪɢɰɚɬɟɥɶɧɚɹ
ɜɟɥɢɱɢɧɚ
0t
z
J
. ȿɞɢɧɢɰɵ ɢɡɦɟɪɟɧɢɹ ɦɨɦɟɧɬɨɜ ɢɧɟɪɰɢɢ
>@
2
ɦɤɝJ
z
ɜ
ɋɂ,
>
@
2
z
J
ɤɝ ɦ ɫ
ɜ ɬɟɯɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɟɞɢɧɢɰ.
Ⱦɥɹ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ (ȺɌɌ)
³³³
¸
¸
¸
¹
·
¨
¨
¨
©
§
)(
2
)(
2
)(
2
),,(
V
ȿɋɅɂ
VM
z
dvr
V
M
V
M
constdvrzyxdmrJ

UU
,
ɝɞɟ
M
– ɦɚɫɫɚ,
V
– ɨɛɴɟɦ,
dv
– ɷɥɟɦɟɧɬ ɨɛɴɟɦɚ,
V
M
U
– ɩɥɨɬɧɨɫɬɶ ɩɨɫɬɨɹɧɧɚɹ.
Ɍɟɨɪɟɦɚ ɒɬɟɣɧɟɪɚ (Ƚɸɣɝɟɧɫɚ):
2
1
MdJJ
c
zz
.
M
– ɦɚɫɫɚ ɬɟɥɚ,
c
z
– ɨɫɶ, ɩɪɨɯɨɞɹɳɚɹ ɱɟɪɟɡ ɰɟɧɬɪ ɦɚɫɫ,
1
z
– ɨɫɶ ɩɚɪɚɥɥɟɥɶɧɚɹ
c
z
,
d
– ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɨɫɹɦɢ.
Ɋɚɞɢɭɫ ɢɧɟɪɰɢɢ
R
ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ
z
– ɜɟɥɢɱɢɧɚ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɚɹ
ɪɚɜɟɧɫɬɜɭ:
2
MRJ
z
.
ɐɟɧɬɪɨɦ ɦɚɫɫ ɫɢɫɬɟɦɵ N ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ (ɰɟɧɬɪɨɦ ɬɹɠɟɫɬɢ ɞɥɹ
ɨɞɧɨɪɨɞɧɨɝɨ ɩɨɥɹ ɬɹɠɟɫɬɢ) ɧɚɡɵɜɚɟɬɫɹ ɬɨɱɤɚ ɋ, ɪɚɞɢɭɫ ɤɨɬɨɪɨɣ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ:
N
NN
C
mmm
rmrmrm
r
21
2211
.
                      §2. Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɬɜɟɪɞɵɯ ɬɟɥ

       Ɇɨɦɟɧɬɨɦ     ɢɧɟɪɰɢɢ     Jz     ɫɢɫɬɟɦɵ          N        ɦɚɬɟɪɢɚɥɶɧɵɯ              ɬɨɱɟɤ
                                                         N

ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z ɧɚɡɵɜɚɟɬɫɹ ɜɟɥɢɱɢɧɚ J z              ¦m r
                                                        k 1
                                                                 k k
                                                                    2
                                                                        . Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ
ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɦɚɫɫ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɣ
ɨɫɢ.
 rk – ɪɚɫɫɬɨɹɧɢɟ ɨɬ Ʉ-ɣ ɬɨɱɤɢ ɞɨ ɨɫɢ z . Ɇɨɦɟɧɬ ɢɧɟɪɰɢɢ ɧɟɨɬɪɢɰɚɬɟɥɶɧɚɹ
ɜɟɥɢɱɢɧɚ J z t 0 . ȿɞɢɧɢɰɵ ɢɡɦɟɪɟɧɢɹ ɦɨɦɟɧɬɨɜ ɢɧɟɪɰɢɢ >J z @ ɤɝ ˜ ɦ ɜ
                                                                   2


ɋɂ, > J z @ ɤɝ ˜ ɦ ˜ ɫ ɜ ɬɟɯɧɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɟɞɢɧɢɰ.
                      2


    Ⱦɥɹ ɚɛɫɨɥɸɬɧɨ ɬɜɟɪɞɨɝɨ ɬɟɥɚ (ȺɌɌ)
                                    §                     ·
              2                2
                                    ¨                   M¸          M           2
    J z ³ r dm ³ U ( x, y, z )r dv ¨ U const              ¸               ³r        dv ,
         (M )      (V )             ¨                  V ¸         V    (V )
                                    ©     ȿɋɅɂ            ¹
    ɝɞɟ M – ɦɚɫɫɚ,
        V – ɨɛɴɟɦ,
        dv – ɷɥɟɦɟɧɬ ɨɛɴɟɦɚ,
               M
        U         – ɩɥɨɬɧɨɫɬɶ ɩɨɫɬɨɹɧɧɚɹ.
                V

Ɍɟɨɪɟɦɚ ɒɬɟɣɧɟɪɚ (Ƚɸɣɝɟɧɫɚ):
                      J z1 J zc  Md 2 .

                                       M – ɦɚɫɫɚ ɬɟɥɚ,
                                      z c – ɨɫɶ, ɩɪɨɯɨɞɹɳɚɹ ɱɟɪɟɡ ɰɟɧɬɪ ɦɚɫɫ,
                                      z1 – ɨɫɶ ɩɚɪɚɥɥɟɥɶɧɚɹ z c ,
                                      d – ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɨɫɹɦɢ.




    Ɋɚɞɢɭɫ ɢɧɟɪɰɢɢ R ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ z – ɜɟɥɢɱɢɧɚ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɚɹ
                  2
ɪɚɜɟɧɫɬɜɭ: J z MR .
    ɐɟɧɬɪɨɦ ɦɚɫɫ ɫɢɫɬɟɦɵ N ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ (ɰɟɧɬɪɨɦ ɬɹɠɟɫɬɢ ɞɥɹ
ɨɞɧɨɪɨɞɧɨɝɨ ɩɨɥɹ ɬɹɠɟɫɬɢ) ɧɚɡɵɜɚɟɬɫɹ ɬɨɱɤɚ ɋ, ɪɚɞɢɭɫ ɤɨɬɨɪɨɣ
                              m1 r1  m2 r2  ˜ ˜ ˜  m N rN
ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ: rC                                    .
                                 m1  m2  ˜ ˜ ˜  m N

                                       7

Страницы