\( \DeclareMathOperator{\abs}{abs} \newcommand{\ensuremath}[1]{\mbox{$#1$}} \)
Задача. Визначити силу тиску на плоску прямокутну поверхню шириною B=3 м та висотою H=7 м. На вільній поверхні рідини атмосферний тиск .
| --> | kill(all); |
\[\]\[\tag{%o0} \ensuremath{\mathrm{done}}\]
| --> | ro:1000;g:9.81;H:7;B:3;p_m:10000; |
\[\]\[\tag{%o1} 1000\]
\[\]\[\tag{%o2} 9.81\]
\[\]\[\tag{%o3} 7\]
\[\]\[\tag{%o4} 3\]
\[\]\[\tag{%o5} 10000\]
| --> | del1:H/15,numer;del2:H/10,numer;del3:H/7,numer; del4:H/5,numer;del5:H/3,numer; |
\[\]\[\tag{%o6} 0.4666666666666667\]
\[\]\[\tag{%o7} 0.7\]
\[\]\[\tag{%o8} 1\]
\[\]\[\tag{%o9} 1.4\]
\[\]\[\tag{%o10} 2.3333333333333335\]
| --> | dh:H/5,numer; |
\[\]\[\tag{%o11} 1.4\]
| --> | 'integrate((ro·g·(H−h))·B,h,0,H);integrate((ro·g·(H−h))·B,h); |
\[\]\[\tag{%o12} 29430.0 \int 7\mathop{-}h{\, dh}\]
\[\]\[\tag{%o13} 29430.0 \left( 7 h\mathop{-}\frac{{{h}^{2}}}{2}\right) \]
| --> | pi(h):=9810.0·(H−h); |
\[\]\[\tag{%o14} \mathop{pi}(h)\mathop{:=}9810.0 \left( H\mathop{-}h\right) \]
| --> | P:integrate(ro·g·(H−h)·B,h,0,H); |
\[\]\[\tag{%o15} 721035.0\]
| --> |
if P<10 then scale:3 elseif P<100 then scale:20 elseif P<1000 then scale:200 elseif P<10000 then scale:2000 elseif P<100000 then scale:20000 elseif P<1000000 then scale:200000 elseif P<10000000 then scale:2000000 ; |
\[\]\[\tag{%o16} 200000\]
| --> | pi(0);pi(0)/scale; |
\[\]\[\tag{%o17} 68670.0\]
\[\]\[\tag{%o18} 0.34335\]
| --> | 'integrate(ro·g·((H−h)··2)·B,h,0,H); |
\[\]\[\tag{%o19} 29430.0 \int {{\left( 7\mathop{-}h\right) }^{2}}{\, dh}\]
| --> | mP:integrate(ro·g·((H−h)··2)·B,h,0,H); |
\[\]\[\tag{%o20} 3364830.0\]
| --> | h_D:mP/P;h_D_:H−h_D; |
\[\]\[\tag{%o21} 4.666666666666667\]
\[\]\[\tag{%o22} 2.333333333333333\]
| --> | x_draw:3·B;y_draw:H+2; |
\[\]\[\tag{%o23} 9\]
\[\]\[\tag{%o24} 9\]
| --> | n: h_D_;mod(n,1); |
\[\]\[\tag{%o25} 2.333333333333333\]
\[\]\[\tag{%o26} 0.33333333333333304\]
| --> | /* make float to string */; |
| --> | if floor(n·100)/100>0 then n:floor(n·100)/100 elseif floor(n·1000)/1000>0 then n:floor(n·1000)/1000; |
\[\]\[\tag{%o27} \frac{233}{100}\]
| --> | n:n,numer; |
\[\]\[\tag{%o28} 2.33\]
| --> | hh:sconcat(n); |
\[\]\[\tag{%o29} "2.33"\]
| --> | hh; |
\[\]\[\tag{%o30} "2.33"\]
| --> | pp:sconcat(P); |
\[\]\[\tag{%o31} "721035.0"\]
| --> | pp; |
\[\]\[\tag{%o32} "721035.0"\]
| --> | /* p_:printf(false, "~d", n);stringp(pp);p_;n; */; |
| --> | /* https://sourceforge.net/p/maxima/mailman/maxima-discuss/thread/34939E59-E2C1-484B-A0CF-C6D70CB9E835@peterpall.de/ */; |
| --> | scale_p:pi(0); |
\[\]\[\tag{%o33} 68670.0\]
| --> |
draw2d(xrange = [−5,x_draw], yrange = [0,y_draw], font = "Arial", font_size = 16, title="Rectangle", xlabel = "Presure, Pa", ylabel = "h,m", grid = true, proportional_axes=xy, line_type = solid, color = black, fill_color = "#cccccc", line_width=1, rectangle([0,0],[B,H]), line_width=2,color=blue, water:polygon([[−3,H],[−1,H]]), water:polygon([[0,H],[B,H]]), color=black,line_width=1, head_both = true, head_length = 0.2, head_angle = 10, vector([0,H+del1],[B,0]), label(["B",B/2,H+del3]), points_joined = true, points([[0,H],[0,H+del2]]), points([[B,H],[B,H+del2]]), points([[B,0],[B+del4,0]]), points([[B,H],[B+del4,H]]), vector([B+del4,0],[0,H]), points_joined = false, label_orientation = 'vertical, label(["H",B+del2,H/2]), /* p_a */ color = black, label_orientation = 'horizontal, head_both = false, line_type = solid, head_length = 0.2, head_angle = 10, color = black, vector([−2.5,(H+del3)],[0,−del3]), label(["p_a",−2.5−0.5+del3,H+del3]), /* Epura */ line_width=1,color=blue,line_type = solid, fill_color = lightblue, poly:polygon([[−1,0],[−1,H],[−pi(0)/scale_p−1,0],[0−1,0]]), head_length = 0.3, head_angle = 10, color = blue, line_width=2, head_both = false, vector([−pi(dh·0)/scale_p−1,H−5·dh],[pi(dh·0)/scale_p,0]), vector([−pi(dh·1)/scale_p−1,H−4·dh],[pi(dh·1)/scale_p,0]), vector([−pi(dh·2)/scale_p−1,H−3·dh],[pi(dh·2)/scale_p,0]), vector([−pi(dh·3)/scale_p−1,H−2·dh],[pi(dh·3)/scale_p,0]), vector([−pi(dh·4)/scale_p−1,H−1·dh],[pi(dh·4)/scale_p,0]), transparent = true, line_width=1, color = black, points_joined = true, point_size=0.1, points([[−2,0],[−1,0],[−1,H+del2]]), line_width=2,color=blue, water:points([[−2,H],[−1,H]]), color = red, label(["Px",−1−del4,h_D_+del2]), label([hh,−1+del2,h_D_+del1]), line_type = solid, head_length = 0.3, head_angle = 15, line_width=2, point_type = filled_circle, point_size=1, vector([−1−P/(scale·2),h_D_],[P/(scale·2),0]), points([[−1,h_D_]]), points([[B/2,h_D_]]), label_orientation = 'horizontal, label(["D",B/2+del1,h_D_]), /* h_D */ color=black,line_width=1, head_both = true, head_length = 0.2, head_angle = 10, vector([−1+del2,0],[0,h_D_]), points_joined = false, label_orientation = 'vertical, label(["h_D_",−del2,h_D_/2]), points_joined = true, point_size=0.1, points([[−1,h_D_],[−1+del3,h_D_]]), /* copy right */ color = "#0e406e", label_orientation = 'vertical, label(["www.k123.org.ua ",3·B−1,H/2]), label(["Kopanytsia Y (c) 2025",3·B−0.5,H/2]), /* Results */ color = "#0e406e", /* rectangle([0,y_draw-del4],[x_draw,y_draw]), */ label_orientation = 'horizontal, label([hh,x_draw/2,y_draw−del3]), label(["h_D [m]",x_draw/2,y_draw−del1/2]), label([pp,x_draw−del4,y_draw−del3]), label(["P [N]",x_draw−del4,y_draw−del1/2]) /* key = "Force", xtics_secondary = true, ytics_secondary = true, xaxis_secondary = true, yaxis_secondary = true, xlabel_secondary="P[N]", ylabel_secondary="p[Pa]", yrange_secondary=[-49050.0,2/5*49050.0], xrange_secondary=[-P,P] */ )$ |
| --> | P;P/scale; |
\[\]\[\tag{%o36} 44145.0\]
\[\]\[\tag{%o37} 4.4145\]
| --> | H−2·dh;H;dh; |
\[\]\[\tag{%o40} 4.2\]
\[\]\[\tag{%o41} 7\]
\[\]\[\tag{%o42} 1.4\]
| --> | 'integrate(ro·g·((H−h)·h)·B,h,0,H); |
\[\]\[\tag{%o43} 29430.0 \int \left( 7\mathop{-}h\right) h{\, dh}\]
| --> | mP_down:integrate(ro·g·((H−h)·h)·B,h,0,H); |
\[\]\[\tag{%o44} 1682415.0\]
| --> | h_D_down:mP_down/P; |
\[\]\[\tag{%o45} 2.3333333333333335\]
Задача. Визначити силу тиску на плоску
прямокутну поверхню шириною B=3 м та
висотою H=7 м. На вільній поверхні рідини
атмосферний тиск . Прошарок води h1=1 м.
| (%i37) | kill(all); |
\[\]\[\tag{%o0} \ensuremath{\mathrm{done}}\]
| (%i5) | ro:1000;g:9.81;H:7;B:3;p_m:10000; |
\[\]\[\tag{%o1} 1000\]
\[\]\[\tag{%o2} 9.81\]
\[\]\[\tag{%o3} 7\]
\[\]\[\tag{%o4} 3\]
\[\]\[\tag{%o5} 10000\]
| (%i6) | h1:1; |
\[\]\[\tag{%o6} 1\]
| (%i11) | del1:H/15,numer;del2:H/10,numer;del3:H/7,numer; del4:H/5,numer;del5:H/3,numer; |
\[\]\[\tag{%o7} 0.4666666666666667\]
\[\]\[\tag{%o8} 0.7\]
\[\]\[\tag{%o9} 1\]
\[\]\[\tag{%o10} 1.4\]
\[\]\[\tag{%o11} 2.3333333333333335\]
| (%i12) | dh:H/5,numer; |
\[\]\[\tag{%o12} 1.4\]
| (%i14) | 'integrate((ro·g·(H−h+h1))·B,h,0,H);integrate((ro·g·(H−h+h1))·B,h); |
\[\]\[\tag{%o13} 49050.0 \int_{0}^{7}9\mathop{-}h{\, dh}\]
\[\]\[\tag{%o14} 49050.0 \left( 9 h\mathop{-}\frac{{{h}^{2}}}{2}\right) \]
| (%i15) | pi(h):=9810.0·(H−h+h1); |
\[\]\[\tag{%o15} \mathop{pi}(h)\mathop{:=}9810.0 \left( H\mathop{-}h\mathop{+}\ensuremath{\mathrm{h1}}\right) \]
| (%i16) | P_h1:integrate(ro·g·(H−h+h1)·B,h,0,H); |
\[\]\[\tag{%o16} 1888425.0\]
| (%i17) |
if P_h1<10 then scale:3 elseif P_h1<100 then scale:20 elseif P_h1<1000 then scale:200 elseif P_h1<10000 then scale:2000 elseif P_h1<100000 then scale:20000 elseif P_h1<1000000 then scale:200000 elseif P_h1<10000000 then scale:2000000 ; |
\[\]\[\tag{%o17} 2000000\]
| (%i20) | pi(0);pi(0)/scale;pi(h1)/scale; |
\[\]\[\tag{%o18} 88290.0\]
\[\]\[\tag{%o19} 0.044145\]
\[\]\[\tag{%o20} 0.034335\]
| (%i21) | mP_h1:integrate(ro·g·((H−h+h1)··2)·B,h,0,H); |
\[\]\[\tag{%o21} 1.178835 {{10}^{7}}\]
| (%i22) | h_D_h1:mP_h1/P_h1; |
\[\]\[\tag{%o22} 6.242424242424242\]
| (%i23) | mP_h1_down:integrate(ro·g·((H−h+h1)·h)·B,h,0,H); |
\[\]\[\tag{%o23} 5207475.0\]
| (%i24) | h_D_h1_down:mP_h1_down/P_h1; |
\[\]\[\tag{%o24} 2.757575757575758\]
| (%i25) | h_D_h1+h_D_h1_down; |
\[\]\[\tag{%o25} 9.0\]
| (%i27) | x_draw:3·B;y_draw:H+h1+2; |
\[\]\[\tag{%o26} 15\]
\[\]\[\tag{%o27} 11\]
| (%i29) | n: h_D_h1_down;mod(n,1); |
\[\]\[\tag{%o28} 2.757575757575758\]
\[\]\[\tag{%o29} 0.7575757575757578\]
| (%i30) | if floor(n·100)/100>0 then n:floor(n·100)/100 elseif floor(n·1000)/1000>0 then n:floor(n·1000)/1000; |
\[\]\[\tag{%o30} \frac{11}{4}\]
| (%i31) | n:n,numer; |
\[\]\[\tag{%o31} 2.75\]
| (%i32) | hh:sconcat(n); |
\[\]\[\tag{%o32} "2.75"\]
| (%i33) | pp:sconcat(P_h1); |
\[\]\[\tag{%o33} "1888425.0"\]
| (%i35) | scale_p:pi(0)/2;scale; |
\[\]\[\tag{%o34} 44145.0\]
\[\]\[\tag{%o35} 2000000\]
| (%i36) |
draw2d(xrange = [−5,x_draw], yrange = [0,y_draw], font = "Arial", font_size = 16, title="Rectangle", xlabel = "Presure, Pa", ylabel = "h,m", grid = true, proportional_axes=xy, line_type = solid, color = black, fill_color = "#cccccc", line_width=2, rectangle([0,0],[B,H]), line_width=2,color=blue, water:polygon([[−3,H+h1],[−1,H+h1]]), water:polygon([[0,H+h1],[B,H+h1]]), color=black,line_width=1, head_both = true, head_length = 0.2, head_angle = 10, vector([0,H+h1+del1],[B,0]), label(["B",B/2,H+h1+del3]), points_joined = true, point_size=0.1, points([[0,H+h1],[0,H+h1+del2]]), points([[B,H+h1],[B,H+h1+del2]]), points([[B,0],[B+del4,0]]), points([[B,H],[B+del4,H]]), vector([B+del4,0],[0,H]), points([[B,H+h1],[B+del4,H+h1]]), vector([B+del4,H],[0,h1]), line_width=2, points([[0,0],[0,H+del1/2+h1]]), points([[B,0],[B,H+del1/2+h1]]), points_joined = false, label_orientation = 'vertical, label(["H",B+del2,H/2]), label(["h1",B+del2,(H+h1/2)]), /* p_a */ color = black, label_orientation = 'horizontal, head_both = false, line_type = solid, head_length = 0.2, head_angle = 10, color = black, vector([−2.5,(H+h1+del3)],[0,−del3]), label(["p_a",−2.5−0.5+del3,H+h1+del3]), /* Epura */ line_width=1,color=blue,line_type = dashes, fill_color = white, poly:polygon([[−1,0],[−1,H+h1],[−pi(0)/scale_p−1,0],[0−1,0]]), line_type = solid, fill_color = lightblue, poly:polygon([[−1,0],[−1,H],[−pi(H)/scale_p−1,H],[−pi(0)/scale_p−1,0],[0−1,0]]), head_length = 0.3, head_angle = 10, color = blue, line_width=2, head_both = false, vector([−pi(dh·0)/scale_p−1,H−5·dh],[pi(dh·0)/scale_p,0]), vector([−pi(dh·1)/scale_p−1,H−4·dh],[pi(dh·1)/scale_p,0]), vector([−pi(dh·2)/scale_p−1,H−3·dh],[pi(dh·2)/scale_p,0]), vector([−pi(dh·3)/scale_p−1,H−2·dh],[pi(dh·3)/scale_p,0]), vector([−pi(dh·4)/scale_p−1,H−1·dh],[pi(dh·4)/scale_p,0]), transparent = true, line_width=2, color = black, points_joined = true, point_size=0.1, points([[−2,0],[−1,0],[−1,H+h1+del1/2]]), line_width=2,color=blue, color = red, label(["Px",−1−del4,h_D_h1_down+del2]), label([hh,−1+del2,h_D_h1_down+del1]), line_type = solid, head_length = 0.3, head_angle = 15, line_width=2, point_type = filled_circle, point_size=1, vector([−1−P_h1/(scale·2),h_D_h1_down],[P_h1/(scale·2),0]), points([[−1,h_D_h1_down]]), points([[B/2,h_D_h1_down]]), label_orientation = 'horizontal, label(["D",B/2+del1,h_D_h1_down]), /* h_D_h1_down */ color=black,line_width=1, head_both = true, head_length = 0.2, head_angle = 10, vector([−1+del2,0],[0,h_D_h1_down]), points_joined = false, label_orientation = 'vertical, label(["h_D_",−del2,h_D_h1_down/2]), points_joined = true, point_size=0.1, points([[−1,h_D_h1_down],[−1+del3,h_D_h1_down]]), /* copy right */ color = "#0e406e", label_orientation = 'vertical, label(["www.k123.org.ua ",3·B−1,H/2]), label(["Kopanytsia Y (c) 2025",3·B−0.5,H/2]), /* Results */ color = "#0e406e", rectangle([x_draw/3,y_draw−del4],[x_draw,y_draw]), label_orientation = 'horizontal, label([hh,x_draw/2,y_draw−del3]), label(["h_D [m]",x_draw/2,y_draw−del1/2]), label([pp,x_draw−del4,y_draw−del3]), label(["P [N]",x_draw−del4,y_draw−del1/2]) /* key = "Force", xtics_secondary = true, ytics_secondary = true, xaxis_secondary = true, yaxis_secondary = true, xlabel_secondary="P[N]", ylabel_secondary="p[Pa]", yrange_secondary=[-49050.0,2/5*49050.0], xrange_secondary=[-P,P] */ )$; |
Triangle
| (%i13) | H:4; |
\[\]\[\tag{%o13} 4\]
| (%i14) | fB(h):=(B/H)·(H−h); |
\[\]\[\tag{%o14} \mathop{fB}(h)\mathop{:=}\frac{B}{H} \left( H\mathop{-}h\right) \]
| (%i16) | fB(0);fB(H); |
\[\]\[\tag{%o15} 3\]
\[\]\[\tag{%o16} 0\]
| (%i17) | P_tri:integrate(ro·g·(H−h)·fB(h),h,0,H); |
\[\]\[\tag{%o17} 156960.0\]
| (%i18) | mP_tri:integrate(ro·g·((H−h)··2)·fB(h),h,0,H); |
\[\]\[\tag{%o18} 470880.0\]
| (%i19) | h_D_tri:mP_tri/P_tri; |
\[\]\[\tag{%o19} 3.0\]
| (%i20) | h_D_tri_down:H−h_D_tri; |
\[\]\[\tag{%o20} 1.0\]
| --> | P_trap:integrate(ro·g·(H−h)·fB(h),h,0,H/2); |
\[\]\[\tag{%o59} 91560.0\]
| --> | mP_trap:integrate(ro·g·((H−h)··2)·fB(h),h,0,H/2); |
\[\]\[\tag{%o60} 294300.0\]
| --> | h_D_trap:mP_trap/P_trap; |
\[\]\[\tag{%o61} 3.2142857142857144\]
| --> | mP_trap_down:integrate(ro·g·((H−h)·h)·fB(h),h,0,H/2); |
\[\]\[\tag{%o62} 71940.0\]
| --> | h_D_trap_down:mP_trap_down/P_trap; |
\[\]\[\tag{%o63} 0.7857142857142857\]
| --> | R:1; |
\[\]\[\tag{%o64} 1\]
| --> | fB_circle(h):=2·sqrt(R··2−h··2); |
\[\]\[\tag{%o65} \mathop{fB\_ circle}(h)\mathop{:=}2 \sqrt{{{R}^{2}}\mathop{-}{{h}^{2}}}\]
| --> | fB_circle(0);fB_circle(R); |
\[\]\[\tag{%o66} 2\]
\[\]\[\tag{%o67} 0\]
| --> | P_cir_up:integrate(ro·g·(R−h)·fB_circle(h),h,0,R); |
\[\]\[\tag{%o68} 19620.0 \left( \frac{\pi }{4}\mathop{-}\frac{1}{3}\right) \]
| --> | P_cir_up,numer; |
\[\]\[\tag{%o69} 8869.511965857935\]
| --> | mP_cir_up:integrate(ro·g·((R−h)··2)·fB_circle(h),h,0,R); |
\[\]\[\tag{%o70} 19620.0 \left( \frac{5 \pi }{16}\mathop{-}\frac{2}{3}\right) \]
| --> | h_D_cir_up:mP_cir_up/P_cir_up,numer; |
\[\]\[\tag{%o71} 0.6969819738807302\]
| --> | h_c:(1−0.4244)·R;p:ro·g·h_c;w:(%pi·(R··2))/2;P:p·w,numer;I:0.1098·R··4; |
\[\]\[\tag{%o72} 0.5756\]
\[\]\[\tag{%o73} 5646.636\]
\[\]\[\tag{%o74} \frac{\pi }{2}\]
\[\]\[\tag{%o75} 8869.715087547827\]
\[\]\[\tag{%o76} 0.1098\]
| --> | h_D:h_c+I/(h_c·w),numer; |
\[\]\[\tag{%o77} 0.6970399774252266\]
Created with wxMaxima.
The source of this Maxima session can be downloaded here.