- Ht phuung trinh ciin bdng:
LFkr =O; LF~J' =O; Lm
(Fk)=O (5-19)
k=I k=I k=l
' - V -
Hai phuang trinh dau tuang duang v&i R' = °LFk =O con phuang trinh cu6i
tuang duang k=1
D~ng 2: Di@u ki~n cfui va du dS h~ Ive phfulg can bfulg la t6ng hinh chiSu cac lgc tren m9t 111,lc va t6ng momen cac Ive d6i v&i hai diSm A va B tuy y tri~t tieu v&i di@u ki~n AB khong vuong g6c v6i 111,lc chiSu (Hinh 11.2.11 ).
N
LF1o: =O; I.mA (~)=O; I.mB(~)=O
k=l k=I k=I (5-20)
Chung minh: di@u ki~n cfui dugc chung minh dga vao di,mg chufui: nSu m9t trong ba di@u ki~n tren khong dugc thoa man thi h~ h;rc da cho tuang duang v6i m9t lgc ho~c m9t ngiu Ive, me khong thoa man dinh lu~t 1 ( chuang 1 ). V~y h~ lgc khong thS can bfulg. DS chung minh di@u ki~n du ta nh~ xet rfulg nSu ca ba di@u ki~n tren dugc thoa man thi h~ Ive khong c6 hgp Ive, cung khong thS tuang duang v6i ngiu lgc. Tove v~y, h~ Ive khong thS c6 hgp Ive, vi nSu h~ Ive c6 hgp lgc ( R
-t O ) thi dga vao hai di@u ki~n cu6i va dva vao dinh li V arinhong:
(5-21) Tue la hgp Ive R phai di qua hai diSm A va B. Di@u nay mau thuin v6i di@u ki~n d!u:
(5-22) Vi R -t 0 ; eosa -t 0 ( do ab khong vuong g6c v&i t11,1c chiSu veeta) nen R cosa khong thS tri~t tieu.
H~ Ive cung khong tuang duang v&i m9t ngiu h;rc m O nao khac 0 vi trong truang hgp nay do veeta ehinh tri~t tieu va do cong thuc biSn thien momen chinh ( cong thuc 2-7), ta e6:
- - N - -
M0 =MA= LmA(Fk) = 0 (5-23)
DtJ,ng 3: DiSu ki~n dn va du d~ h~ Ive phiing can bfulg la t6ng momen cua cac Ive dfii v&i ba diSm a, b, c khong thing hang tri~t tieu (Hinh 11.2.12).
(5-24)
Chfrng minh: DiSu ki~n c§n la hiSn nhien vi n~u m(>t trong ba diSu ki~n tren khong thoa man thi h~ Ive tuong duong v&i ngfiu Ive khac 0, se khong thoa man dinh Iu~t 1 ( chuong I). DS chCmg minh diSu ki~n du tuong tv each chung minh cho d<;IDg 2, n~u ba diSu ki~n dugc thoa man thi h~ Ive khong c6 hgp Ive ciing khong thS tuong duong v&i m(>t ngfiu Ive khac 0. H~ Ive khong thS c6 hgp Ive vi dva vao dinh Ii varinh~ng n~u ba diSu ki~n tren thoa man thi hgp h;rc phai di qua ba diSm A, B, C khong thfulghang. Di&t nay vo Ii, v~y R =0, tire R' =0. H~ Ive ciing khong thS tuong duong v&i m(>tngfiu Ive m O nao khac 0, vi R'= 0 ta c6 ( dva vao
dinh li _ _ N _ _ bi~n thien momen chinh (2-7)):
Mo =MA= LmA(Fk)=O (5-25)
k=l