_jh_ex2b_3a

D:\j_Iam_TEMP\_jh\_jh_ex2b_3a_v2.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.
Аналітичні та чисельні розрахунки методом K123.

Figure 1:

Figure 2:D:\j_Iam_TEMP\_jh\_jh_ex2b_3a_v8_01.svg
Diagram

-->	kill(all);

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

-->	load(draw)$

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

\[\]\[\tag{%o2} 1000\]

\[\]\[\tag{%o3} 9.81\]

\[\]\[\tag{%o4} 2\]

\[\]\[\tag{%o5} 3\]

\[\]\[\tag{%o6} 1\]

\[\]\[\tag{%o7} 1000\]

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

\[\]\[\tag{%o8} \mathop{fh}(x)\mathop{:=}{{x}^{2}}\]

\[\]\[\tag{%o9} \mathop{fHab}(x)\mathop{:=}\ensuremath{\mathrm{Hmn}}\]

\[\]\[\tag{%o10} \mathop{fH}(x)\mathop{:=}H\]

\[\]\[\tag{%o11} \mathop{Fx}(h)\mathop{:=}\sqrt{h}\]

-->	dh(h):=1/(2·sqrt(h));

\[\]\[\tag{%o12} \mathop{dh}(h)\mathop{:=}\frac{1}{2 \sqrt{h}}\]

-->	del1:Hmn/10,numer;del2:Hmn/5,numer;del3:Hmn/3,numer;

\[\]\[\tag{%o13} 0.1\]

\[\]\[\tag{%o14} 0.2\]

\[\]\[\tag{%o15} 0.3333333333333333\]

-->	del4:Hmn/2.5,numer;del5:Hmn/2,numer;

\[\]\[\tag{%o16} 0.4\]

\[\]\[\tag{%o17} 0.5\]

-->

line_w:1;

\[\]\[\tag{%o18} 1\]

-->	x_view_st:−2; x_view:6.5; rec_start:2;

\[\]\[\tag{%o19} \mathop{-}2\]

\[\]\[\tag{%o20} 6.5\]

\[\]\[\tag{%o21} 2\]

-->

draw2d(xrange = [x_view_st,x_view],
yrange = [0,2.5],
   font      = "Arial",
            font_size = 16,
title="Parabola",
   grid = true,
   proportional_axes=xy,
line_width=4,color=black,
explicit(fh(x),x,0,Fx(H)+del1),
   line_width=4,color=grey,
explicit(fh(x),x,Hmn,Fx(H)),
   line_type = solid,
    line_width=3,color=blue,
water:polygon([[0,H],[Fx(H),H]]),
   line_width=3,color=blue,line_type = dots,
water_hm:polygon([[0,Hmn],[Fx(Hmn),Hmn]]),
   line_type = solid,
   color=black,line_width=1,
   head_both = true,
   head_length = 0.2,
   head_angle = 10,
vector([rec_start,H+del2],[B,0]),
   label(["B",rec_start+B/2,H+del3]),
   points_joined = true,
   points([[rec_start,2],[rec_start,2.3]]),
   points([[rec_start+B,2],[rec_start+B,2.3]]),
   points([[rec_start+B,2],[rec_start+B+0.3,2]]),
   points([[rec_start+B,0],[rec_start+B,0.3]]),
   vector([rec_start+B+0.4,0],[0,2]),
   points_joined = false,
   label_orientation = 'vertical,
label(["H",rec_start+B+0.2,Hmn]),
   points_joined = true,
points([[Fx(Hmn),Hmn],[Fx(Hmn)+1,Hmn]]),
    points([[Fx(Hmn),0],[Fx(Hmn)+1,0]]),
   vector([Fx(Hmn)+0.7,0],[0,Hmn]),
      points_joined = false,
label(["Hmn",1+0.5,Hmn/2]),
   color = black,
   label_orientation = 'horizontal,
   head_both = false,
   line_type = solid,
   head_length = 0.2,
head_angle = 5,
   color = black,
   /* Axis h,x */
   line_width=2,
vector([0,0],[0,H+del4]),
   label(["h",0−del2,H+del2]),
   vector([0,0],[x_view−del4,0]),
   label(["x",x_view−2·del4,del1]),
   line_width=1,
   /* vector p_a */
   vector([0.5,(H+del3)],[0,−0.3]),
   label(["p_a",0.5+0.3,2+0.2]),
      color = black, fill_color = "#eeeeee",
rectangle([rec_start,0],[rec_start+B,2]),
color = black,
   fill_color = "#cccccc",
line_width=4,
rectangle([rec_start,0],[rec_start+B,1]),
            /* GPL */
font      = "Arial",
            font_size = 16,
   color = "#0e406e",
label_orientation = 'vertical,
label(["www.k123.org.ua ",x_view−0.5,1.3]),
   label(["Kopanytsia Y (c) 2025",x_view−0.2,1.3]),
   color=black,point_type = filled_circle,
   point_size = 2,
   points_joined = false,
points([[0,0]]),
   label_orientation = 'horizontal,
   label(["m",0−del2,0+del1]),
   points([[Hmn,Fx(Hmn)]]),
   label(["n",Hmn+del2,Fx(Hmn)])
)$

D:\j_Iam_TEMP\_jh\_jh_ex2b_3a_v8_01.svg

Figure 3:

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

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

Figure 4:

-->	plot2d(Fx(h),[h,0,H], [color,black],[style, [lines, 5,5]]);

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

Figure 5:

-->	fp(h):=ro·g·(H−h);

\[\]\[\tag{%o25} \mathop{fp}(h)\mathop{:=}\ensuremath{\mathrm{ro}} g\, \left( H\mathop{-}h\right) \]

Standart

-->	h_C_st:H−Hmn/2,numer;

\[\]\[\tag{%o26} 1.5\]

-->	p_c_st:ro·g·h_C_st;

\[\]\[\tag{%o27} 14715.0\]

-->	w_st:B·Hmn;

\[\]\[\tag{%o28} 3\]

-->	P_x_st:p_c_st·w_st;

\[\]\[\tag{%o29} 44145.0\]

-->	I:(B·Hmn··3)/12,numer;

\[\]\[\tag{%o30} 0.25\]

-->	h_D_st:h_C_st+(I)/(h_C_st·w_st);

\[\]\[\tag{%o31} 1.5555555555555556\]

-->	h_D_down_st:H−h_D_st;

\[\]\[\tag{%o32} 0.4444444444444444\]

P_x - горизонтальна проекція сили тиску

-->	scale:100000;

\[\]\[\tag{%o33} 100000\]

-->	P_x:integrate(fp(h)·B,h,0,Hmn);

\[\]\[\tag{%o34} 44145.0\]

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

\[\]\[\tag{%o35} 68670.0\]

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

\[\]\[\tag{%o36} 19620.0\]

-->	h_D:mP_x/P_x;

\[\]\[\tag{%o37} 1.5555555555555556\]

-->	h_D_:mP_x_down/P_x;

\[\]\[\tag{%o38} 0.4444444444444444\]

Test the calculation results (Тест результатів розрахунків)

-->

h_D+h_D_;

\[\]\[\tag{%o39} 2.0\]

-->	fv(y):=(2−0.25·y)/2;

\[\]\[\tag{%o40} \mathop{fv}(y)\mathop{:=}\frac{2\mathop{-}0.25 y}{2}\]

equestion

-->	P_plot_(ro_,g_,B_,Hmn_,H_):=(−((B_·(Hmn_^2−2·H_·Hmn_)·g_·ro_)/2));

\[\]\[\tag{%o41} \mathop{P\_ plot\_ }\left( \ensuremath{\mathrm{ro\_ }}\mathop{,}\ensuremath{\mathrm{g\_ }}\mathop{,}\ensuremath{\mathrm{B\_ }}\mathop{,}\ensuremath{\mathrm{Hmn\_ }}\mathop{,}\ensuremath{\mathrm{H\_ }}\right) \mathop{:=}\mathop{-}\left( \frac{\ensuremath{\mathrm{B\_ }} \left( {{\ensuremath{\mathrm{Hmn\_ }}}^{2}}\mathop{-}2 \ensuremath{\mathrm{H\_ }} \ensuremath{\mathrm{Hmn\_ }}\right) \ensuremath{\mathrm{g\_ }} \ensuremath{\mathrm{ro\_ }}}{2}\right) \]

-->	P_plot_(1000,9.81,3,1,2);

\[\]\[\tag{%o42} 44145.0\]

-->	mP_plot_(ro_,g_,B_,Hmn_,H_):=(−(((2·Hmn_^3−3·H_·Hmn_^2)·g_·ro_)/2));

\[\]\[\tag{%o43} \mathop{mP\_ plot\_ }\left( \ensuremath{\mathrm{ro\_ }}\mathop{,}\ensuremath{\mathrm{g\_ }}\mathop{,}\ensuremath{\mathrm{B\_ }}\mathop{,}\ensuremath{\mathrm{Hmn\_ }}\mathop{,}\ensuremath{\mathrm{H\_ }}\right) \mathop{:=}\mathop{-}\left( \frac{\left( 2 {{\ensuremath{\mathrm{Hmn\_ }}}^{3}}\mathop{-}3 \ensuremath{\mathrm{H\_ }} {{\ensuremath{\mathrm{Hmn\_ }}}^{2}}\right) \ensuremath{\mathrm{g\_ }} \ensuremath{\mathrm{ro\_ }}}{2}\right) \]

-->	_h_D_plot_(ro_,g_,B_,Hmn_,H_):=((2·Hmn_^2−3·H_·Hmn_)/(B_·Hmn_−2·B_·H_));

\[\]\[\tag{%o44} \mathop{\_ h\_ D\_ plot\_ }\left( \ensuremath{\mathrm{ro\_ }}\mathop{,}\ensuremath{\mathrm{g\_ }}\mathop{,}\ensuremath{\mathrm{B\_ }}\mathop{,}\ensuremath{\mathrm{Hmn\_ }}\mathop{,}\ensuremath{\mathrm{H\_ }}\right) \mathop{:=}\frac{2 {{\ensuremath{\mathrm{Hmn\_ }}}^{2}}\mathop{-}3 \ensuremath{\mathrm{H\_ }} \ensuremath{\mathrm{Hmn\_ }}}{\ensuremath{\mathrm{B\_ }} \ensuremath{\mathrm{Hmn\_ }}\mathop{-}2 \ensuremath{\mathrm{B\_ }} \ensuremath{\mathrm{H\_ }}}\]

-->	_h_D_plot_(1000,9.81,3,1,2),numer;

\[\]\[\tag{%o45} 0.4444444444444444\]

Numer method

-->	Ps:0;mPs:0;ni:10000;dhi:Hmn/ni;

\[\]\[\tag{%o46} 0\]

\[\]\[\tag{%o47} 0\]

\[\]\[\tag{%o48} 10000\]

\[\]\[\tag{%o49} \frac{1}{10000}\]

-->	for i:1 thru ni step 1 do (Psi:ro·g·(H−dhi·i)·B·dhi,Ps:Ps+Psi)$

-->	for i:1 thru ni step 1 do (mPsi:ro·g·(H−dhi·i)·(dhi·i)·B·dhi,mPs:mPs+mPsi)$

-->	Ps;mPs;h_D_downi:mPs/Ps;

\[\]\[\tag{%o52} 44143.528499999935\]

\[\]\[\tag{%o53} 19621.471450949975\]

\[\]\[\tag{%o54} 0.4444925930864363\]

---- The end NM

-->	P_xv:P_x/scale;

\[\]\[\tag{%o55} 0.44145\]

-->

draw2d(xrange = [−2,x_view],
yrange = [0,3],
   font      = "Arial",
            font_size = 16,
title="Parabola",
   grid = true,
   proportional_axes=xy,
line_width=4,color=black,
explicit(fh(x),x,0,Fx(H)+del1),
   line_width=4,color=grey,
explicit(fh(x),x,Hmn,Fx(H)),
   line_type = solid,
    line_width=3,color=blue,
water:polygon([[0,H],[Fx(H),H]]),
   line_width=3,color=blue,line_type = dots,
water_hm:polygon([[0,Hmn],[Fx(Hmn),Hmn]]),
   line_type = solid,
   color=black,line_width=1,
   head_both = true,
   head_length = 0.2,
   head_angle = 10,
vector([rec_start,H+del2],[B,0]),
   label(["B",rec_start+B/2,H+del3]),
   points_joined = true,
   points([[rec_start,2],[rec_start,2.3]]),
   points([[rec_start+B,2],[rec_start+B,2.3]]),
   points([[rec_start+B,2],[rec_start+B+0.3,2]]),
   points([[rec_start+B,0],[rec_start+B,0.3]]),
vector([rec_start+B+0.3,0],[0,2]),
   points_joined = false,
   label_orientation = 'vertical,
label(["H",rec_start+B+0.2,Hmn]),
   points_joined = true,
points([[Fx(Hmn),Hmn],[Fx(Hmn)+1,Hmn]]),
    points([[Fx(Hmn),0],[Fx(Hmn)+1,0]]),
   vector([Fx(Hmn)+0.7,0],[0,Hmn]),
      points_joined = false,
label(["Hmn",1+0.5,Hmn/2]),
   color = black,
   label_orientation = 'horizontal,
   head_both = false,
   line_type = solid,
   head_length = 0.2,
head_angle = 5,
   color = black,
vector([0.5,(H+del3)],[0,−del3]),
   label(["p_a",0.5+del3,H+del2]),
   /* Axis h,x */
   line_width=2,
vector([0,0],[0,H+del4]),
   label(["h",0−del2,H+del2]),
   vector([0,0],[x_view−del4,0]),
   label(["x",x_view−2·del4,del1]),
   line_width=1,
      color = black, fill_color = "#eeeeee",
rectangle([rec_start,0],[rec_start+B,2]),
color = black,
   fill_color = "#cccccc",
line_width=4,
rectangle([rec_start,0],[rec_start+B,1]),
            /* GPL */
font      = "Arial",
            font_size = 16,
   color = "#0e406e",
label_orientation = 'vertical,
label(["www.k123.org.ua ",x_view−0.5,1.3]),
   label(["Kopanytsia Y (c) 2025",x_view−0.2,1.3]),
   color=black,point_type = filled_circle,
   point_size = 2,
   points_joined = false,
points([[0,0]]),
   label_orientation = 'horizontal,
   label(["m",0+del2,0+del2]),
   points([[Hmn,Fx(Hmn)]]),
   label(["n",Hmn+del2,Fx(Hmn)]),
   /*   flat */
   points([[rec_start+B/2,0]]),
   label(["m",rec_start+B/2+del2,0+del2]),
   points([[rec_start+B/2,Fx(Hmn)]]),
   label(["n",rec_start+B/2+del2,Fx(Hmn)+del2]),
   color = red,
   points([[rec_start+B/2,h_D_]]),
   label(["D",rec_start+B/2+del2,h_D_+del2]),
   label(["D",0+del1,h_D_+del2]),
   color = black,
    /* Epura */
line_width=1,color=blue,line_type = short_long_dashes,
fill_color = "#ffffff",
poly:polygon([[0−0.1,0],[0−0.1,2],[−1−0.1,0],[0−0.1,0]]),
line_width=1,color=blue,line_type = solid,
fill_color = lightblue,
poly:polygon([[0−0.1,0],[0−0.1,1],[−0.5−0.1,1],[−1−0.1,0],[0−0.1,0]]),
   line_type = solid,
head_length = 0.3,
head_angle = 5,
color = blue,
line_width=3,
vector([−fv(0)−0.05,0],[fv(0)−0.05,0]),
vector([−fv(1)−0.05,0.25],[fv(1)−0.05,0]),
vector([−fv(2)−0.05,0.5],[fv(2)−0.05,0]),
vector([−fv(3)−0.05,0.75],[fv(3)−0.05,0]),
vector([−fv(4)−0.05,1],[fv(4)−0.05,0]),
   color = red,
label(["Px",0−del4,h_D_+del2]),
label(["0.44",0−0.8,h_D_]),
   line_type = solid,
head_length = 0.2,
head_angle = 5,
line_width=2,
vector([0−P_xv−del1,h_D_],[P_xv,0]),
   points([[0−del1,h_D_]]),
points_joined = true,
   line_width=0.5,
   point_size    = 0.1,
points([[−1,h_D_],[rec_start+B,h_D_]]),
   points([[rec_start+B/2,0],[rec_start+B/2,H]]),
   label_orientation = 'vertical,
   head_both = true,
   line_type = solid,
   head_length = 0.2,
head_angle = 5,
   vector([1,h_D_],[0,−h_D_]),
   label(["h_D",1−del2,h_D_/2]),
/* label_orientation = 'horizontal,
label(["h_D = 0.56 m, P_x=44145.0 N",2,Hmn+del3]), */
   fill_color = white,
   rectangle([0,3−del4],[4,3]),
   label_orientation = 'vertical,
   head_both = true,
   line_type = solid,
   head_length = 0.2,
head_angle = 5,
label_orientation = 'horizontal,
label(["h_D = 0.44 m, P_x=44145.0",2,3−del2])
)$

-->

h_D;P_x;

\[\]\[\tag{%o74} 1.5555555555555556\]

\[\]\[\tag{%o75} 44145.0\]

Figure 6:

Figure 7:D:\j_Iam_TEMP\_jh\_jh_ex2b_3a_v8_02.svg
Diagram

Figure 8:D:\j_Iam_TEMP\_jh\_jh_ex2_P_x.jpg
Diagram

P_x - чисельний алгоритм методу К123

-->	P_x_sum:0;dh:Hmn/n;h:0;

\[\]\[\tag{%o57} 0\]

\[\]\[\tag{%o58} \frac{1}{1000}\]

\[\]\[\tag{%o59} 0\]

-->	Pi_x(h):=ro·g·(H−h)·B;

\[\]\[\tag{%o60} \mathop{Pi\_ x}(h)\mathop{:=}\ensuremath{\mathrm{ro}} g\, \left( H\mathop{-}h\right) B\]

-->	for i:1 while h < Hmn do (Pi:Pi_x(h)·dh,P_x_sum:P_x_sum+Pi,h:h+dh);P_x_:P_x_sum;

\[\]\[\tag{%o61} \ensuremath{\mathrm{done}}\]

\[\]\[\tag{%o62} 44159.71499999999\]

-->	Px_test:integrate(Pi_x(hi),hi,0,Hmn);

\[\]\[\tag{%o63} 44145.0\]

-->	mP_x_sum:0;dh:Hmn/n;h:0;

\[\]\[\tag{%o64} 0\]

\[\]\[\tag{%o65} \frac{1}{1000}\]

\[\]\[\tag{%o66} 0\]

-->	for i:1 while h < Hmn do (Pi:Pi_x(h)·dh·h,mP_x_sum:mP_x_sum+Pi,h:h+dh);mP_x_:mP_x_sum;

\[\]\[\tag{%o67} \ensuremath{\mathrm{done}}\]

\[\]\[\tag{%o68} 19605.280095000024\]

-->

kill(h);

\[\]\[\tag{%o69} \ensuremath{\mathrm{done}}\]

-->	mPx_test:integrate(Pi_x(h)·(h),h,0,Hmn);

\[\]\[\tag{%o70} 19620.0\]

-->	h_D:mP_x_sum/P_x_sum;h_D_test:mPx_test/Px_test;

\[\]\[\tag{%o71} 0.44396301232922425\]

\[\]\[\tag{%o72} 0.4444444444444444\]

-->	Rel_ERROR:(100/h_D_test)·(h_D_test−h_D),numer;

\[\]\[\tag{%o73} 0.10832222592453838\]

P_z - вертикальна проекція сили тиску

-->

kill(h);

\[\]\[\tag{%o74} \ensuremath{\mathrm{done}}\]

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

\[\]\[\tag{%o75} 1000\]

\[\]\[\tag{%o76} 9.81\]

\[\]\[\tag{%o77} 2\]

\[\]\[\tag{%o78} 3\]

\[\]\[\tag{%o79} 1\]

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

\[\]\[\tag{%o80} 49050.0\]

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

\[\]\[\tag{%o81} 22072.5\]

-->	x_C:mP_z/P_z;

\[\]\[\tag{%o82} 0.45\]

-->	Pz(ro_,g_,B_,Hmn_,H_):=−((B_·sqrt(Hmn_)·(2·Hmn_−6·H_)·g_·ro_)/6);

\[\]\[\tag{%o83} \mathop{Pz}\left( \ensuremath{\mathrm{ro\_ }}\mathop{,}\ensuremath{\mathrm{g\_ }}\mathop{,}\ensuremath{\mathrm{B\_ }}\mathop{,}\ensuremath{\mathrm{Hmn\_ }}\mathop{,}\ensuremath{\mathrm{H\_ }}\right) \mathop{:=}\mathop{-}\left( \frac{\ensuremath{\mathrm{B\_ }} \sqrt{\ensuremath{\mathrm{Hmn\_ }}} \left( 2 \ensuremath{\mathrm{Hmn\_ }}\mathop{-}6 \ensuremath{\mathrm{H\_ }}\right) \ensuremath{\mathrm{g\_ }} \ensuremath{\mathrm{ro\_ }}}{6}\right) \]

-->	Pz(1000,9.81,3,1,2);

\[\]\[\tag{%o84} 49050.0\]

-->	mPz(ro_,g_,B_,Hmn_,H_):=−((B_·(Hmn_^2−2·H_·Hmn_)·g_·ro_)/4);

\[\]\[\tag{%o85} \mathop{mPz}\left( \ensuremath{\mathrm{ro\_ }}\mathop{,}\ensuremath{\mathrm{g\_ }}\mathop{,}\ensuremath{\mathrm{B\_ }}\mathop{,}\ensuremath{\mathrm{Hmn\_ }}\mathop{,}\ensuremath{\mathrm{H\_ }}\right) \mathop{:=}\mathop{-}\left( \frac{\ensuremath{\mathrm{B\_ }} \left( {{\ensuremath{\mathrm{Hmn\_ }}}^{2}}\mathop{-}2 \ensuremath{\mathrm{H\_ }} \ensuremath{\mathrm{Hmn\_ }}\right) \ensuremath{\mathrm{g\_ }} \ensuremath{\mathrm{ro\_ }}}{4}\right) \]

-->	fx_C(ro_,g_,B_,Hmn_,H_):=(−((B_·(Hmn_^2−2·H_·Hmn_)·g_·ro_)/4))/(−((B_·sqrt(Hmn_)·(2·Hmn_−6·H_)·g_·ro_)/6));

\[\]\[\tag{%o86} \mathop{fx\_ C}\left( \ensuremath{\mathrm{ro\_ }}\mathop{,}\ensuremath{\mathrm{g\_ }}\mathop{,}\ensuremath{\mathrm{B\_ }}\mathop{,}\ensuremath{\mathrm{Hmn\_ }}\mathop{,}\ensuremath{\mathrm{H\_ }}\right) \mathop{:=}\frac{\mathop{-}\left( \frac{\ensuremath{\mathrm{B\_ }} \left( {{\ensuremath{\mathrm{Hmn\_ }}}^{2}}\mathop{-}2 \ensuremath{\mathrm{H\_ }} \ensuremath{\mathrm{Hmn\_ }}\right) \ensuremath{\mathrm{g\_ }} \ensuremath{\mathrm{ro\_ }}}{4}\right) }{\mathop{-}\left( \frac{\ensuremath{\mathrm{B\_ }} \sqrt{\ensuremath{\mathrm{Hmn\_ }}} \left( 2 \ensuremath{\mathrm{Hmn\_ }}\mathop{-}6 \ensuremath{\mathrm{H\_ }}\right) \ensuremath{\mathrm{g\_ }} \ensuremath{\mathrm{ro\_ }}}{6}\right) }\]

-->	x_C(ro_,g_,B_,Hmn_,H_):=((3·Hmn_^2−6·H_·Hmn_)/(4·sqrt(Hmn_)·Hmn_−12·H_·sqrt(Hmn_)));

\[\]\[\tag{%o87} \mathop{x\_ C}\left( \ensuremath{\mathrm{ro\_ }}\mathop{,}\ensuremath{\mathrm{g\_ }}\mathop{,}\ensuremath{\mathrm{B\_ }}\mathop{,}\ensuremath{\mathrm{Hmn\_ }}\mathop{,}\ensuremath{\mathrm{H\_ }}\right) \mathop{:=}\frac{3 {{\ensuremath{\mathrm{Hmn\_ }}}^{2}}\mathop{-}6 \ensuremath{\mathrm{H\_ }} \ensuremath{\mathrm{Hmn\_ }}}{4 \sqrt{\ensuremath{\mathrm{Hmn\_ }}} \ensuremath{\mathrm{Hmn\_ }}\mathop{-}12 \ensuremath{\mathrm{H\_ }} \sqrt{\ensuremath{\mathrm{Hmn\_ }}}}\]

-->	x_C(1000,9.81,3,1,2),numer;

\[\]\[\tag{%o88} 0.45\]

Figure 9:D:\j_Iam_TEMP\_jh\_jh_ex2_P_z.jpg
Diagram

-->	P_zv:P_z/scale;

\[\]\[\tag{%o89} 0.4905\]

-->

ytop=3;

\[\]\[\tag{%o90} \ensuremath{\mathrm{ytop}}\mathop{=}3\]

-->

draw2d(xrange = [−1.1,x_view],
yrange = [0,3],
   font      = "Arial",
            font_size = 16,
title="Parabola",
   grid = true,
   proportional_axes=xy,
    /*   vertical W */
   line_width=0.5,
   color=blue,
   fill_color = lightblue,
Wall:polygon([[0,0],[0,2],[1,2],[1,1],[0.75,0.75··2],[0.5,0.5··2],[0.25,0.25··2],[0,0]]),
W:polygon([[0,0],[0,2]]),
W:polygon([[0.25,0.25··2],[0.25,2]]),
W:polygon([[0.5,0.5··2],[0.5,2]]),
W:polygon([[0.75,0.75··2],[0.75,2]]),
W:polygon([[1,1··2],[1,2]]),
   /*   -- the end W */
line_width=4,color=black,
explicit(fh(x),x,0,Fx(H)+del1),
   line_width=4,color=grey,
explicit(fh(x),x,Hmn,Fx(H)),
   line_type = solid,
    line_width=3,color=blue,
water:polygon([[0,H],[Fx(H),H]]),
   line_width=3,color=blue,line_type = dots,
water_hm:polygon([[0,Hmn],[Fx(Hmn),Hmn]]),
   line_type = solid,
   color=black,line_width=1,
   head_both = true,
   head_length = 0.2,
   head_angle = 10,
vector([rec_start,H+del2],[B,0]),
label(["B",rec_start+B/2,H+del3]),
   points_joined = true,
   points([[rec_start,2],[rec_start,2.3]]),
   points([[rec_start+B,2],[rec_start+B,2.3]]),
   points([[rec_start+B,2],[rec_start+B+0.3,2]]),
   points([[rec_start+B,0],[rec_start+B,0.3]]),
   vector([rec_start+B+0.3,0],[0,2]),
   points_joined = false,
   label_orientation = 'vertical,
label(["H",rec_start+B+0.2,Hmn]),
   points_joined = true,
points([[Fx(Hmn),Hmn],[Fx(Hmn)+1,Hmn]]),
    points([[Fx(Hmn),0],[Fx(Hmn)+1,0]]),
   vector([Fx(Hmn)+0.7,0],[0,Hmn]),
      points_joined = false,
label(["Hmn",1+0.5,Hmn/2]),
   color = black,
   label_orientation = 'horizontal,
   head_both = false,
   line_type = solid,
   head_length = 0.2,
head_angle = 5,
   color = black,
vector([0.8,(H+del3)],[0,−del3]),
   label(["p_a",0.8+del3,H+del2]),
   /* Axis h,x */
   line_width=2,
vector([0,0],[0,H+del4]),
   label(["h",0−del2,H+del2]),
   vector([0,0],[x_view−del4,0]),
   label(["x",x_view−2·del4,del1]),
      line_width=1,
      color = black, fill_color = "#eeeeee",
rectangle([rec_start,0],[rec_start+B,2]),
color = black,
   fill_color = "#cccccc",
line_width=4,
rectangle([rec_start,0],[rec_start+B,1]),
   color = red,
   points_joined = true,
   line_width=1,
   points([[rec_start+B/2,0],[rec_start+B/2,H]]),
    /*   flat */
   color=black,
   point_type = filled_circle,
   point_size = 2,
   points_joined = false,
   points([[rec_start+B/2,0]]),
   label(["m",rec_start+B/2+del2,0+del2]),
   points([[rec_start+B/2,Fx(Hmn)]]),
   label(["n",rec_start+B/2+del2,Fx(Hmn)+del2]),

            /* GPL */
font      = "Arial",
            font_size = 16,
   color = "#0e406e",
label_orientation = 'vertical,
label(["www.k123.org.ua ",x_view−0.5,1.3]),
   label(["Kopanytsia Y (c) 2025",x_view−0.2,1.3]),
   color=black,
   point_type = filled_circle,
   point_size = 2,
   points_joined = false,
points([[0,0]]),
   label_orientation = 'horizontal,
   label(["m",0−del2,0+del1]),
   points([[Hmn,Fx(Hmn)]]),
   label(["n",Hmn+del2,Fx(Hmn)]),
    /* Epura
  line_width=1,color=blue,line_type = short_long_dashes,
  fill_color = "#ffffff",
poly:polygon([[0−0.1,0],[0−0.1,2],[−1−0.1,0],[0−0.1,0]]),
  line_width=1,color=blue,line_type = solid,
  fill_color = lightblue,
poly:polygon([[0−0.1,0],[0−0.1,1],[−0.5−0.1,1],[−1−0.1,0],[0−0.1,0]]),
   line_type = solid,
  head_length = 0.3,
head_angle = 5,
  color = blue,
  line_width=3,
  vector([−fv(0)−0.05,0],[fv(0)−0.05,0]),
  vector([−fv(1)−0.05,0.25],[fv(1)−0.05,0]),
  vector([−fv(2)−0.05,0.5],[fv(2)−0.05,0]),
  vector([−fv(3)−0.05,0.75],[fv(3)−0.05,0]),
  vector([−fv(4)−0.05,1],[fv(4)−0.05,0]),
   color = red,
  label(["Px",0,h_D+del2]),
  label(["0.56",0-0.8,h_D]),
   line_type = solid,
  head_length = 0.2,
head_angle = 5,
  line_width=2,
vector([0−P_xv-del1,h_D],[P_xv,0]),
   points([[0-del1,h_D]]),
points_joined = true,
   line_width=0.5,
   point_size    = 0.1,
  points([[-1,h_D],[4,h_D]]),
   points([[2.5,0],[2.5,H]]),
   label_orientation = 'vertical,
   head_both = true,
   line_type = solid,
   head_length = 0.2,
head_angle = 5,
   vector([1,h_D],[0,-h_D]),
   label(["h_D",1-del2,h_D/2]),   */
/* label_orientation = 'vertical,
color = "#654321",
label(["Vertical orientation",x_C-0.2,h_D+0.6]), */
label_orientation = 'vertical,
    color = red,
label(["Pz",x_C−0.1,h_D+0.4]),
line_type = solid,
head_length = 0.2,
head_angle = 5,
line_width=1.5,
   vector([x_C,(h_D+P_zv)],[0,−(P_zv)]),
label_orientation = 'horizontal,
    color = red,
line_type = solid,
line_width=1,
   head_both = true,
   head_length = 0.2,
   head_angle = 10,
vector([0,H+del2],[x_C,0]),
   label(["x_C",x_C/2,H+del3+del1]),
   points_joined = true,
   point_size = 0.1,
   line_type = solid,
   points([[0,H],[0,H+del3]]),
   line_type = dashes,
   points([[x_C,0],[x_C,H+del3]]),
   fill_color = white,
   rectangle([0,3−del4],[4,3]),
   label_orientation = 'vertical,
   head_both = true,
   line_type = solid,
   head_length = 0.2,
head_angle = 5,
label_orientation = 'horizontal,
label(["x_C = 0.45 m, P_z=49050.0 N",2,3−del2]),
   label(["0.45",x_C,del1])
)$

-->

x_C;P_z;

\[\]\[\tag{%o119} 0.45\]

\[\]\[\tag{%o120} 49050.0\]

Figure 10:
Diagram

Figure 11:D:\j_Iam_TEMP\_jh\_jh_ex2b_3a_v8_03.svg
Diagram

P_z - чисельний алгоритм методу К123

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

\[\]\[\tag{%o92} 1000\]

\[\]\[\tag{%o93} 9.81\]

\[\]\[\tag{%o94} 2\]

\[\]\[\tag{%o95} 3\]

\[\]\[\tag{%o96} 1\]

-->	P_z_sum:0;dh:Hmn/n;h:0;

\[\]\[\tag{%o97} 0\]

\[\]\[\tag{%o98} \frac{1}{1000}\]

\[\]\[\tag{%o99} 0\]

coordinate - x {for h=i*dh x=sqrt(h)}

-->	fB(i,dh):=sqrt(i·dh); fB(1,dh),numer;

\[\]\[\tag{%o100} \mathop{fB}\left( i\mathop{,}\ensuremath{\mathrm{dh}}\right) \mathop{:=}\sqrt{i\, \ensuremath{\mathrm{dh}}}\]

\[\]\[\tag{%o101} 0.03162277660168379\]

-->	Pi_x(h):=ro·g·(H−h)·B; Pi_x(0.2);

\[\]\[\tag{%o102} \mathop{Pi\_ x}(h)\mathop{:=}\ensuremath{\mathrm{ro}} g\, \left( H\mathop{-}h\right) B\]

\[\]\[\tag{%o103} 52974.0\]

-->	fdb(i,dh):=(fB((i+1),dh)−fB((i),dh))/dh;fdb(1000,dh),numer;

\[\]\[\tag{%o104} \mathop{fdb}\left( i\mathop{,}\ensuremath{\mathrm{dh}}\right) \mathop{:=}\frac{\mathop{fB}\left( i\mathop{+}1\mathop{,}\ensuremath{\mathrm{dh}}\right) \mathop{-}\mathop{fB}\left( i\mathop{,}\ensuremath{\mathrm{dh}}\right) }{\ensuremath{\mathrm{dh}}}\]

\[\]\[\tag{%o105} 0.4998750624609638\]

-->	for i: 0 while h < Hmn do (db:fdb(i,dh),Pi:Pi_x(h)·db·dh,P_z_sum:P_z_sum+Pi,h:h+dh)$

-->	P_z_:P_z_sum,numer;h;

\[\]\[\tag{%o107} 49064.52275520602\]

\[\]\[\tag{%o108} 1\]

FOR MOMENT P_z coordinate - x {for h=i*dh x=sqrt(h)}

-->	mP_z_sum:0;dh:Hmn/n;h:0;

\[\]\[\tag{%o109} 0\]

\[\]\[\tag{%o110} \frac{1}{1000}\]

\[\]\[\tag{%o111} 0\]

-->	for i: 0 while h < Hmn do (db:fdb(i,dh),Pi:Pi_x(h)·db·dh·fB(i,dh),mP_z_sum:mP_z_sum+Pi,h:h+dh)$

-->	mP_z_:mP_z_sum,numer;h;

\[\]\[\tag{%o113} 22003.335303355238\]

\[\]\[\tag{%o114} 1\]

-->	x_C_:mP_z_/P_z_,numer; x_C;

\[\]\[\tag{%o115} 0.44845713496765977\]

\[\]\[\tag{%o116} 0.45\]

-->	Rel_ERROR:(100/x_C)·(x_C−x_C_),numer;

\[\]\[\tag{%o117} 0.3428588960756102\]

P - Сила гідростатичного тиску

-->

P_x;P_z;

\[\]\[\tag{%o118} 44145.0\]

\[\]\[\tag{%o119} 49050.0\]

-->	P:sqrt(P_x··2+P_z··2);

\[\]\[\tag{%o120} 65990.02595089655\]

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

\[\]\[\tag{%o121} 0.83798122500839\]

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

\[\]\[\tag{%o122} 48.01278750418334\]

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

\[\]\[\tag{%o123} 2.0199703317182265\]

-->	k:P_z/P_x;

\[\]\[\tag{%o124} 1.1111111111111112\]

Проекції вектора сили й координати (Projections of the force vector and coordinates)

-->	scale:100000;

\[\]\[\tag{%o125} 100000\]

-->	P_xv:P_x/scale;P_zv:P_z/scale;Pv:P/scale;

\[\]\[\tag{%o126} 0.44145\]

\[\]\[\tag{%o127} 0.4905\]

\[\]\[\tag{%o128} 0.6599002595089656\]

-->

kill(D);

\[\]\[\tag{%o129} \ensuremath{\mathrm{done}}\]

-->	h_D;x_C;phi_rad;k;

\[\]\[\tag{%o130} 0.44396301232922425\]

\[\]\[\tag{%o131} 0.45\]

\[\]\[\tag{%o132} 0.83798122500839\]

\[\]\[\tag{%o133} 1.1111111111111112\]

Equastions K123

-->	P_plot_x(ro_,g_,B_,Hmn_,H_):=(−((B_·(Hmn_^2−2·H_·Hmn_)·g_·ro_)/2));

\[\]\[\tag{%o134} \mathop{P\_ plot\_ x}\left( \ensuremath{\mathrm{ro\_ }}\mathop{,}\ensuremath{\mathrm{g\_ }}\mathop{,}\ensuremath{\mathrm{B\_ }}\mathop{,}\ensuremath{\mathrm{Hmn\_ }}\mathop{,}\ensuremath{\mathrm{H\_ }}\right) \mathop{:=}\mathop{-}\left( \frac{\ensuremath{\mathrm{B\_ }} \left( {{\ensuremath{\mathrm{Hmn\_ }}}^{2}}\mathop{-}2 \ensuremath{\mathrm{H\_ }} \ensuremath{\mathrm{Hmn\_ }}\right) \ensuremath{\mathrm{g\_ }} \ensuremath{\mathrm{ro\_ }}}{2}\right) \]

-->	Pz(ro_,g_,B_,Hmn_,H_):=−((B_·sqrt(Hmn_)·(2·Hmn_−6·H_)·g_·ro_)/6);

\[\]\[\tag{%o135} \mathop{Pz}\left( \ensuremath{\mathrm{ro\_ }}\mathop{,}\ensuremath{\mathrm{g\_ }}\mathop{,}\ensuremath{\mathrm{B\_ }}\mathop{,}\ensuremath{\mathrm{Hmn\_ }}\mathop{,}\ensuremath{\mathrm{H\_ }}\right) \mathop{:=}\mathop{-}\left( \frac{\ensuremath{\mathrm{B\_ }} \sqrt{\ensuremath{\mathrm{Hmn\_ }}} \left( 2 \ensuremath{\mathrm{Hmn\_ }}\mathop{-}6 \ensuremath{\mathrm{H\_ }}\right) \ensuremath{\mathrm{g\_ }} \ensuremath{\mathrm{ro\_ }}}{6}\right) \]

-->	sqrt(((−((B_·(Hmn_^2−2·H_·Hmn_)·g_·ro_)/2)))··2+(−((B_·sqrt(Hmn_)·(2·Hmn_−6·H_)·g_·ro_)/6))··2);

\[\]\[\tag{%o136} \sqrt{\frac{{{\ensuremath{\mathrm{B\_ }}}^{2}} {{\left( {{\ensuremath{\mathrm{Hmn\_ }}}^{2}}\mathop{-}2 \ensuremath{\mathrm{H\_ }} \ensuremath{\mathrm{Hmn\_ }}\right) }^{2}} {{\ensuremath{\mathrm{g\_ }}}^{2}} {{\ensuremath{\mathrm{ro\_ }}}^{2}}}{4}\mathop{+}\frac{{{\ensuremath{\mathrm{B\_ }}}^{2}} \ensuremath{\mathrm{Hmn\_ }} {{\left( 2 \ensuremath{\mathrm{Hmn\_ }}\mathop{-}6 \ensuremath{\mathrm{H\_ }}\right) }^{2}} {{\ensuremath{\mathrm{g\_ }}}^{2}} {{\ensuremath{\mathrm{ro\_ }}}^{2}}}{36}}\]

-->	P(ro_,g_,B_,Hmn_,H_):=sqrt((B_^2·(Hmn_^2−2·H_·Hmn_)^2·g_^2·ro_^2)/4+(B_^2·Hmn_·(2·Hmn_−6·H_)^2·g_^2·ro_^2)/36);

\[\]\[\tag{%o137} \mathop{P}\left( \ensuremath{\mathrm{ro\_ }}\mathop{,}\ensuremath{\mathrm{g\_ }}\mathop{,}\ensuremath{\mathrm{B\_ }}\mathop{,}\ensuremath{\mathrm{Hmn\_ }}\mathop{,}\ensuremath{\mathrm{H\_ }}\right) \mathop{:=}\sqrt{\frac{{{\ensuremath{\mathrm{B\_ }}}^{2}} {{\left( {{\ensuremath{\mathrm{Hmn\_ }}}^{2}}\mathop{-}2 \ensuremath{\mathrm{H\_ }} \ensuremath{\mathrm{Hmn\_ }}\right) }^{2}} {{\ensuremath{\mathrm{g\_ }}}^{2}} {{\ensuremath{\mathrm{ro\_ }}}^{2}}}{4}\mathop{+}\frac{{{\ensuremath{\mathrm{B\_ }}}^{2}} \ensuremath{\mathrm{Hmn\_ }} {{\left( 2 \ensuremath{\mathrm{Hmn\_ }}\mathop{-}6 \ensuremath{\mathrm{H\_ }}\right) }^{2}} {{\ensuremath{\mathrm{g\_ }}}^{2}} {{\ensuremath{\mathrm{ro\_ }}}^{2}}}{36}}\]

-->	P(1000,9.81,3,1,2);

\[\]\[\tag{%o138} 65990.02595089655\]

-->	P_test(ro_,g_,B_,Hmn_,H_):=(B_/2)·ro_·g_·sqrt(((Hmn_^2−2·H_·Hmn_)^2)+(Hmn_·(2·Hmn_−6·H_)^2)/9);

\[\]\[\tag{%o139} \mathop{P\_ test}\left( \ensuremath{\mathrm{ro\_ }}\mathop{,}\ensuremath{\mathrm{g\_ }}\mathop{,}\ensuremath{\mathrm{B\_ }}\mathop{,}\ensuremath{\mathrm{Hmn\_ }}\mathop{,}\ensuremath{\mathrm{H\_ }}\right) \mathop{:=}\frac{\ensuremath{\mathrm{B\_ }}}{2} \ensuremath{\mathrm{ro\_ }} \ensuremath{\mathrm{g\_ }} \sqrt{{{\left( {{\ensuremath{\mathrm{Hmn\_ }}}^{2}}\mathop{-}2 \ensuremath{\mathrm{H\_ }} \ensuremath{\mathrm{Hmn\_ }}\right) }^{2}}\mathop{+}\frac{\ensuremath{\mathrm{Hmn\_ }} {{\left( 2 \ensuremath{\mathrm{Hmn\_ }}\mathop{-}6 \ensuremath{\mathrm{H\_ }}\right) }^{2}}}{9}}\]

-->	P_test(1000,9.81,3,1,2),numer;

\[\]\[\tag{%o140} 65990.02595089655\]

-->	k:(−((B·sqrt(Hmn)·(2·Hmn−6·H)·g·ro)/6))/((−((B·(Hmn^2−2·H·Hmn)·g·ro)/2)));

\[\]\[\tag{%o172} 1.1111111111111112\]

-->

\[\]\[\tag{%o173} 1.1111111111111112\]

-->	fk(Hmn_,H_):=(sqrt(Hmn_)·(2·Hmn_−6·H_))/(3·(Hmn_^2−2·H_·Hmn_));

\[\]\[\tag{%o174} \mathop{fk}\left( \ensuremath{\mathrm{Hmn\_ }}\mathop{,}\ensuremath{\mathrm{H\_ }}\right) \mathop{:=}\frac{\sqrt{\ensuremath{\mathrm{Hmn\_ }}} \left( 2 \ensuremath{\mathrm{Hmn\_ }}\mathop{-}6 \ensuremath{\mathrm{H\_ }}\right) }{3 \left( {{\ensuremath{\mathrm{Hmn\_ }}}^{2}}\mathop{-}2 \ensuremath{\mathrm{H\_ }} \ensuremath{\mathrm{Hmn\_ }}\right) }\]

-->	fk(1,2),numer;

\[\]\[\tag{%o175} 1.1111111111111112\]

Equastions of vector P

-->	eq:h_D=−k·x_C+D;

\[\]\[\tag{%o176} 0.44396301232922425\mathop{=}0.44396301232922364\]

-->	Di:h_D+k·x_C;

\[\]\[\tag{%o177} 0.9439630123292242\]

-->	D:2832833/3001000,numer;

\[\]\[\tag{%o179} 0.9439630123292236\]

-->	y(x):=−k·x+D;

\[\]\[\tag{%o180} \mathop{y}(x)\mathop{:=}\mathop{-}k x\mathop{+}D\]

-->

y(0.45);

\[\]\[\tag{%o181} 0.44396301232922364\]

-->	y1(x):=x··2;

\[\]\[\tag{%o182} \mathop{y1}(x)\mathop{:=}{{x}^{2}}\]

-->	eq1:−k·x+D=x··2;

\[\]\[\tag{%o183} 0.9439630123292236\mathop{-}1.1111111111111112 x\mathop{=}{{x}^{2}}\]

-->	solutions:solve([eq1],[x]);

\[\]\[rat: replaced 0.9439630123292236 by 2832833/3001000 = 0.9439630123292236 \]\[rat: replaced -1.1111111111111112 by -10/9 = -1.1111111111111112\]

\[\]\[\tag{%o184} \left[ x\mathop{=}\mathop{-}\left( \frac{\sqrt{9137579034730}\mathop{+}1500500}{2700900}\right) \mathop{,}x\mathop{=}\frac{\sqrt{9137579034730}\mathop{-}1500500}{2700900}\right] \]

-->	x:(sqrt(9137579034730)−1500500)/2700900,numer;

\[\]\[\tag{%o185} 0.5636428127589562\]

Full equastion coordinates

-->	h_D_;P_z;P_x;x_C;

\[\]\[\tag{%o186} 0.4444444444444444\]

\[\]\[\tag{%o187} 49050.0\]

\[\]\[\tag{%o188} 44145.0\]

\[\]\[\tag{%o189} 0.45\]

-->	eq_all:x··2=h_D_down+(P_z/P_x)·(x_C−x);

\[\]\[\tag{%o190} 0.3176932203748278\mathop{=}\ensuremath{\mathrm{h\_ D\_ down}}\mathop{-}0.1262697919543958\]

Numer method

-->	f1(x):=−k·x+Di;f2(x):=x··2;h1:0;h2:0;ni:1000;x1:0;x2:sqrt(Hmn);xi:x2/ni;j:0;

\[\]\[\tag{%o191} \mathop{f1}(x)\mathop{:=}\mathop{-}k x\mathop{+}\ensuremath{\mathrm{Di}}\]

\[\]\[\tag{%o192} \mathop{f2}(x)\mathop{:=}{{x}^{2}}\]

\[\]\[\tag{%o193} 0\]

\[\]\[\tag{%o194} 0\]

\[\]\[\tag{%o195} 1000\]

\[\]\[\tag{%o196} 0\]

\[\]\[\tag{%o197} 1\]

\[\]\[\tag{%o198} \frac{1}{1000}\]

\[\]\[\tag{%o199} 0\]

-->	Di;f1(xi·1000),numer;f2(xi·1000),numer;

\[\]\[\tag{%o200} 0.9439630123292242\]

\[\]\[\tag{%o201} \mathop{-}0.16714809878188697\]

\[\]\[\tag{%o202} 1.0\]

-->	for j: 1 while f1(xi·j)−f2(xi·j)>0 do (j: j+1,x2:xi·j);

\[\]\[\tag{%o203} \ensuremath{\mathrm{done}}\]

-->

x2,numer;

\[\]\[\tag{%o204} 0.564\]

--------- the end NM

-->

plot2d([y1(x),y(x),[discrete,[x_coord],[h_coord]],
       [discrete,[x_coord,xx],[h_coord,hh]],
       [discrete,[x_coord,xx],[h_coord,h_coord]],
       [discrete,[x_coord,x_coord],[h_coord,hh]],
   [discrete,[0,2],[2,2]]],[x,0,2.1],[y,0,2.1],
   [style, [lines, 5,5], [lines, 3,1], [points, 5,2],[lines, 3,2],[lines, 3,2],[lines, 3,2],
       [lines, 3,1]]);

\[\]\[plotting: range must be of the form [variable, min, max]; found: \]\[ [0.5636428127589562,0,2.1] \]\[\] \texttt{%error -- an error. To debug this try: debugmode(true);}\[\]

Figure 12:
Diagram

Figure 13:D:\j_Iam_TEMP\_jh\_jh_ex2b_3a_v2_F6.svg
Diagram

Draw

-->	/* y = 1-1-4 */;

-->	fv(y):=(2−0.25·y)/2;

\[\]\[\tag{%o206} \mathop{fv}(y)\mathop{:=}\frac{2\mathop{-}0.25 y}{2}\]

draw 4 x 3.5

-->	x_view_st:−2; x_view:6.5; rec_start:2;

\[\]\[\tag{%o207} \mathop{-}2\]

\[\]\[\tag{%o208} 6.5\]

\[\]\[\tag{%o209} 2\]

-->

draw2d(xrange = [−1.1,x_view],
yrange = [0,3.5],
   font      = "Arial",
            font_size = 16,
title="Parabola",
   grid = true,
   proportional_axes=xy,
    /*   vertical W */
   fill_color = lightblue,
   Wall:polygon([[0,0],[0,2],[1,2],[1,1],[0.75,0.75··2],[0.5,0.5··2],[0.25,0.25··2],[0,0]]),
   W:polygon([[0,0],[0,2]]),
   W:polygon([[0.25,0.25··2],[0.25,2]]),
   W:polygon([[0.5,0.5··2],[0.5,2]]),
   W:polygon([[0.75,0.75··2],[0.75,2]]),
   W:polygon([[1,1··2],[1,2]]),
line_width=4,color=black,
explicit(y1(x),x,0,1.44),
   line_width=4,color=grey,
explicit(y1(x),x,1,sqrt(2)),
   color=red,line_width=1,
   explicit(y(x),x,0,x_coord),
   line_width=3,color=blue,
poly:polygon([[0,2],[sqrt(2),2]]),
   color=black,line_width=1,
   head_both = true,
   head_length = 0.2,
   head_angle = 10,
vector([2,2.2],[1,0]),
   label(["B",2.5,2.3]),
   points_joined = true,
   points([[2,2],[2,2.3]]),
   points([[3,2],[3,2.3]]),
   points([[3,2],[3.3,2]]),
   points([[3,0],[3,0.3]]),
   vector([3.3,0],[0,2]),
   points_joined = false,
   label_orientation = 'vertical,
label(["H",3.2,1]),
   label_orientation = 'horizontal,
   head_both = false,
   line_type = solid,
   head_length = 0.2,
head_angle = 5,
   color = black,
vector([Fx(H)·2/3,(H+del3)],[0,−0.3]),
   label(["p_a",Fx(H)·2/3+del3,H+del2]),
line_width=3,color=blue,line_type = dots,
poly:polygon([[0,1],[1,1]]),
color=black,point_type = filled_circle,point_size = 1.5,
points([[0,0],[1,1]]),
color=red,point_type = filled_circle,
points([[x_coord,h_coord],[x_C,h_D]]),
/*    color = red,
label(["Proection center pressure",x_C+0.7,h_coord]),
   color = navy,
label(["Horizontal proection vector (default)",x_C+1.3,h_D+0.1]), */
       color = red,
   label(["(0.56,0.31)",x_C+del5+del2,h_D−del1]),
   label(["(0.45,0.44)",x_C+del5,h_D+del1]),
   color = black,
   label(["Px",x_C−0.3,h_D+0.1]),
   /* label_orientation = 'vertical,
color = "#654321",
label(["Vertical orientation",x_C-0.2,h_D+0.6]), */
   label_orientation = 'vertical,
label(["Pz",x_C−0.1,h_D+0.4]),
   line_type = solid,
   head_length = 0.2,
head_angle = 5,
   color = black,
   line_width=1.5,
vector([x_C,(h_D+P_zv)],[0,−(P_zv)]),
vector([x_C−P_xv,h_D],[P_xv,0]),
vector([(x_C−P_xv),(h_D+P_zv)],[P_xv,−P_zv]),
color = black, fill_color = "#eeeeee",
rectangle([2,0],[3,2]),
   color = black,
    fill_color = "#cccccc",
   line_width=4,
rectangle([2,0],[3,1]),
   /* Epura */
   line_width=1,color=blue,line_type = short_long_dashes,
   fill_color = "#ffffff",
poly:polygon([[0−0.1,0],[0−0.1,2],[−1−0.1,0],[0−0.1,0]]),
   line_width=1,color=blue,line_type = solid,
   fill_color = lightblue,
poly:polygon([[0−0.1,0],[0−0.1,1],[−0.5−0.1,1],[−1−0.1,0],[0−0.1,0]]),
   /*   vertical W
   Wall:polygon([[0,0],[0,2],[1,2],[1,1],[0.75,0.75**2],[0.5,0.5**2],[0.25,0.25**2],[0,0]]),
   W:polygon([[0,0],[0,2]]),
   W:polygon([[0.25,0.25**2],[0.25,2]]),
   W:polygon([[0.5,0.5**2],[0.5,2]]),
   W:polygon([[0.75,0.75**2],[0.75,2]]),
   W:polygon([[1,1**2],[1,2]]), */
   color = black, fill_color = "#cccccc",
rectangle([−0.1,0],[−0.05,1]),
   /* Epura vectors */
line_type = solid,
   head_length = 0.3,
head_angle = 5,
   color = blue,
   line_width=3,
   vector([−fv(0)−0.05,0],[fv(0)−0.05,0]),
   vector([−fv(1)−0.05,0.25],[fv(1)−0.05,0]),
   vector([−fv(2)−0.05,0.5],[fv(2)−0.05,0]),
   vector([−fv(3)−0.05,0.75],[fv(3)−0.05,0]),
   vector([−fv(4)−0.05,1],[fv(4)−0.05,0]),
     /* Angle "phi"
   color = black, fill_color = "#eeeeee",
    line_width=0.5,
    head_angle = 180,
   vector([h_D,x_C],[-P_x/45000,P_z/45000]),
   vector([h_D-P_x/45000,x_C+P_z/45000],[0.5,0]), */


           /* GPL */
font      = "Arial",
            font_size = 16,
   color = "#0e406e",
label_orientation = 'vertical,
label(["www.k123.org.ua ",3.5,1.3]),
   label(["Kopanytsia Y (c) 2025",3.8,1.3]),
   fill_color = white,
   rectangle([0,3.5−del5·2],[4,3.5]),
   label_orientation = 'vertical,
   head_both = true,
   line_type = solid,
   head_length = 0.2,
head_angle = 5,
label_orientation = 'horizontal,
label(["x_C = 0.45 m, P_z=49050.0 N;",2,3.5−del5]),
label(["h_D = 0.44 m, P_x=44145.0 N;",2,3.5−del2]),
   label(["x = 0.56 m, y=0.31 m, P=65990 N.",2,3.5−del3−del5]),
   label(["0.45",x_C,del1])
)$

\[\]\[; in: LAMBDA (0.5636428127589562) \]\[; (LAMBDA (0.5636428127589562d0) \]\[; (DECLARE (SPECIAL 0.5636428127589562d0)) \]\[; (LET ((MAXIMA::\$ RATPRINT NIL) (MAXIMA::\$ NUMER T) (MAXIMA::\ensuremath{\cdot}NOUNSFLAG\ensuremath{\cdot} T)) \]\[; (LET ((MAXIMA::RESULT \neq )) \]\[; (IF MAXIMA::RESULT \]\[; (CAR MAXIMA::RESULT) \]\[; T)))) \]\[; \]\[; caught ERROR: \]\[; Required argument is not a symbol: 0.5636428127589562 \]\[\] \texttt{%error Maxima encountered a Lisp error: }\[\]\[ Required argument is not a symbol: 0.5636428127589562 \]\[Automatically continuing. \]\[\] \texttt{%error To enable the Lisp debugger set \ensuremath{\cdot}debugger-hook\ensuremath{\cdot} to nil. }\[\]\[; \]\[; compilation unit aborted \]\[; caught 1 fatal ERROR condition \]\[; caught 1 ERROR condition\]

Figure 14:
Diagram

-->

\[\]\[\tag{%o212} 65990.02595089655\]

-->	h_D;x_C;x_C−P_x/40000;h_D+P_x/40000;

\[\]\[\tag{%o213} 0.44396301232922425\]

\[\]\[\tag{%o214} 0.45\]

\[\]\[\tag{%o215} \mathop{-}0.6536250000000001\]

\[\]\[\tag{%o216} 1.5475880123292243\]

-->	x_coord;h_coord;

\[\]\[\tag{%o217} {x_{\ensuremath{\mathrm{coord}}}}\]

\[\]\[\tag{%o218} {h_{\ensuremath{\mathrm{coord}}}}\]

-->

hh;h_D;

\[\]\[\tag{%o219} \ensuremath{\mathrm{hh}}\]

\[\]\[\tag{%o220} 0.44396301232922425\]

-----------------------------

Plot2d

-->	fh_D(x):=h_D_;

\[\]\[\tag{%o221} \mathop{fh\_ D}(x)\mathop{:=}\ensuremath{\mathrm{h\_ D\_ }}\]

-->

plot2d([fh(x),fHab(x),fH(x),fh_D(x),[discrete,[0.44,0.44],[0,1]],
       [discrete,[0.44],[0.45]],vector([0.44,0.45],[0,1])],[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]);

\[\]\[plotting: range must be of the form [variable, min, max]; found: \]\[ [0.5636428127589562,0,1.5] \]\[\] \texttt{%error -- an error. To debug this try: debugmode(true);}\[\]

Figure 15:
Diagram

Figure 16:D:\j_Iam_TEMP\_jh\_jh_ex2b_3a_v2_F8.svg
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;x_coord;h_coord;phi_grad;

\[\]\[\tag{%o223} 44145.0\]

\[\]\[\tag{%o224} 49050.0\]

\[\]\[\tag{%o225} 65990.02595089655\]

\[\]\[\tag{%o226} 0.4444444444444444\]

\[\]\[\tag{%o227} 0.45\]

\[\]\[\tag{%o228} {x_{\ensuremath{\mathrm{coord}}}}\]

\[\]\[\tag{%o229} {h_{\ensuremath{\mathrm{coord}}}}\]

\[\]\[\tag{%o230} 48.01278750418334\]

Figure 17:D:\j_Iam_TEMP\_jh\_jh_ex2.jpg
Diagram

Scale screen 300 px

-->	P_x/300;P_z/300;P/300;

\[\]\[\tag{%o231} 147.15\]

\[\]\[\tag{%o232} 163.5\]

\[\]\[\tag{%o233} 219.9667531696552\]

-->	P_:sqrt(P_x_··2+P_z_··2);

\[\]\[\tag{%o234} 66010.66445717201\]

REL_ERRORS

-->	Rel_ERROR_P_x:(100/P_x)·(P_x−P_x_),numer;

\[\]\[\tag{%o235} \mathop{-}0.033333333333308936\]

-->	Rel_ERROR_mP_x_:(100/mP_x_down)·(mP_x_down−mP_x_),numer;

\[\]\[\tag{%o236} 0.0750249999998791\]

-->	Rel_ERROR_mP_z:(100/mP_z)·(mP_z−mP_z_),numer;

\[\]\[\tag{%o237} 0.31335234633486214\]

-->	Rel_ERROR_P_z:(100/P_z)·(P_z−P_z_),numer;

\[\]\[\tag{%o238} \mathop{-}0.029608063620829527\]

-->	Rel_ERROR_P:(100/P)·(P−P_),numer;

\[\]\[\tag{%o239} \mathop{-}0.031275190421675786\]

Created with wxMaxima.

The source of this Maxima session can be downloaded here.