Reja: Statik moment. Og`irlik markazi

> restart;with(Student[Calculus1]):
> p:=phi->a*(1+cos(phi));
> s:=int(p(phi)^2,phi=0..2*Pi)/2;
> Mx:=int(p(phi)^3*sin(phi),phi=0..2*Pi)/3;
> My:=int(p(phi)^3*cos(phi),phi=0..2*Pi)/3;
> Xc:=My/s; Yc:=Mx/s;
2. Inersiya momenti. Agar A₁, A₂,.., A_n nuqtalarga joylashgan va massalari mos ravishda m₁,m₂,…, m_n bo`lgan n ta moddiy nuqtalarning sistemasi berilgan bo`lib, nuqtalar A_i(x_i;y_i) koordinatalarga ega bo`lsa, bu moddiy nuqtalar sistemasining Ox, Oy va koordinatalar boshiga nisbatan inersiya momentlari
I_x , I_y va I₀ lar (mos ravishda)

bo`ladi.
Oldingi banddagiga o`xshash ishlarni bajarib, (29) ko`rinishda berilgan birjinsli tekis egri chiziq uchun
(34)
flomulalarni olish mumkin.
43-misol. Birjinsli radiusi r ga teng bo`lgan yarim aylananing o`z diametriga nisbag`tan inersiya momenti topilsin.
Yechish. Yarim aylanani 23- rasmdagidek joylashtirib olsak, uning diametri Ox o`qiga joylashadi va Oy o`qqa nisbatan simmetrik bo`ladi. Uning tenglamasi
.
bundan ;
(34) ning birinchi formulasiga asosan:
.
> restart;with(plots):
> implicitplot(x^2+y^2=1^2, x=-1..1, y=0..1,color=red, thickness=2);

> f:=x->sqrt(r^2-x^2);
> Ix:=Int(f(x)^2*sqrt(1+diff(f(x),x)^2),x=-r..r) assuming r>0;

> Ix:=2*int(f(x)^2*sqrt(1+diff(f(x),x)^2),x=0..r);

> value(%);

> r:=1:Ix:value(%);