D:\j_Iam_TEMP\_jh\_jh_ex2.wxmx

Умова

Визначити силу гідростатичного тиску на бокову грань "MN" параболоїдального каналу висотою Hmn = 1 метр та шириною B = 3 метри. Рівень води у каналі H = 2 метри.
--> kill(all);

\[\]\[\tag{%o0} \ensuremath{\mathrm{done}}\]

--> ro:1000;g:9.81;H:2;B:3;Hmn:1;

\[\]\[\tag{%o1} 1000\]

\[\]\[\tag{%o2} 9.81\]

\[\]\[\tag{%o3} 2\]

\[\]\[\tag{%o4} 3\]

\[\]\[\tag{%o5} 1\]

--> fh(x):=x··2;fHab(x):=Hmn;fH(x):=H;Fx(h):=sqrt(h);diff(Fx(h),h);

\[\]\[\tag{%o6} \mathop{fh}(x)\mathop{:=}{{x}^{2}}\]

\[\]\[\tag{%o7} \mathop{fHab}(x)\mathop{:=}\ensuremath{\mathrm{Hmn}}\]

\[\]\[\tag{%o8} \mathop{fH}(x)\mathop{:=}H\]

\[\]\[\tag{%o9} \mathop{Fx}(h)\mathop{:=}\sqrt{h}\]

\[\]\[\tag{%o10} \frac{1}{2 \sqrt{h}}\]

--> dh(h):=1/(2·sqrt(h));

\[\]\[\tag{%o11} \mathop{dh}(h)\mathop{:=}\frac{1}{2 \sqrt{h}}\]

--> plot2d([fh(x),fHab(x),fH(x)],[x,0,2]);

\[\]\[\tag{%o12} false\]

Figure 1:D:\j_Iam_TEMP\_jh\_jh_ex2.png
Diagram
--> plot2d(Fx(h),[h,0,H]);

\[\]\[\tag{%o13} false\]

Figure 2:D:\j_Iam_TEMP\_jh\_jh_ex2a.png
Diagram
--> fp(h):=ro·g·(H−h);

\[\]\[\tag{%o14} \mathop{fp}(h)\mathop{:=}\ensuremath{\mathrm{ro}} g\, \left( H\mathop{-}h\right) \]

P_x - горизонтальна проекція сили тиску

--> P_x:integrate(fp(h)·B,h,0,Hmn);

\[\]\[\tag{%o15} 44145.0\]

--> mP_x:integrate(fp(h)·(H−h)·B·h,h,0,Hmn);

\[\]\[\tag{%o16} 26977.5\]

--> mP_x_down:integrate(fp(h)·B·h,h,0,Hmn);

\[\]\[\tag{%o17} 19620.0\]

--> h_D:mP_x/P_x;

\[\]\[\tag{%o18} 0.6111111111111112\]

--> h_D_:mP_x_down/P_x;

\[\]\[\tag{%o19} 0.4444444444444444\]

--> h_D+h_D_;

\[\]\[\tag{%o20} 1.0555555555555556\]

Figure 3:D:\j_Iam_TEMP\_jh\_jh_ex2_P_x.jpg
Diagram

P_z - вертикальна проекція сили тиску

--> P_z:integrate(fp(h)·B·dh(h),h,0,Hmn);

\[\]\[\tag{%o21} 49050.0\]

--> mP_z:integrate(fp(h)·B·dh(h)·(Fx(h)),h,0,Hmn);

\[\]\[\tag{%o22} 22072.5\]

--> x_C:mP_z/P_z;

\[\]\[\tag{%o23} 0.45\]

Figure 4:D:\j_Iam_TEMP\_jh\_jh_ex2_P_z.jpg
Diagram

P - Сила гідростатичного тиску

--> P:sqrt(P_x··2+P_z··2);

\[\]\[\tag{%o24} 65990.02595089655\]

--> phi_rad:atan(P_z/P_x),numer;

\[\]\[\tag{%o25} 0.83798122500839\]

--> phi_grad:atan(P_z/P_x)·(180/%pi),numer;

\[\]\[\tag{%o26} 48.01278750418334\]

--> d:tan(P_z/P_x);

\[\]\[\tag{%o27} 2.0199703317182265\]

Plot2d
--> fh_D(x):=h_D_;

\[\]\[\tag{%o28} \mathop{fh\_ D}(x)\mathop{:=}\ensuremath{\mathrm{h\_ D\_ }}\]

--> plot2d([fh(x),fHab(x),fH(x),fh_D(x),[discrete,[0.44,0.44],[0,1]],
        [discrete,[0.44],[0.45]]],[x,0,1.5],
    [legend, "parabola","Top_box", "Water", "P_x", "P_z", "point"],
   [style, [lines, 5,5], lines, [lines, 3,1], lines, lines, [points, 3,2]],
   [point_type, circle]);

\[\]\[\tag{%o29} false\]

Figure 5:D:\j_Iam_TEMP\_jh\_jh_ex2c.png
Diagram
eq:y=x_C;plot2d([fh(x),fHab(x),fH(x),fh_D(x),eq],[x,0,2],[y,0,4]);

Answer

--> P_x;P_z;P;h_D_;x_C;phi_grad;

\[\]\[\tag{%o30} 44145.0\]

\[\]\[\tag{%o31} 49050.0\]

\[\]\[\tag{%o32} 65990.02595089655\]

\[\]\[\tag{%o33} 0.4444444444444444\]

\[\]\[\tag{%o34} 0.45\]

\[\]\[\tag{%o35} 48.01278750418334\]

Figure 6:D:\j_Iam_TEMP\_jh\_jh_ex2.jpg
Diagram
Scale screen 300 px
--> P_x/300;P_z/300;P/300;

\[\]\[\tag{%o36} 147.15\]

\[\]\[\tag{%o37} 163.5\]

\[\]\[\tag{%o38} 219.9667531696552\]

Created with wxMaxima.

The source of this Maxima session can be downloaded here.