_jh_ex1e_r001

D:\j_Iam_TEMP\_jh\_jh_ex1e_r001.wxmx

Problem 1ex1e1

Визначити силу гідростатичного тиску на секторний затвор MN .
Радіус R = 1 м, ширина B = 2 м, висота затвора H = R/2 = 0.5 м.

Figure 1:D:\j_Iam_TEMP\_jh\_jh_ex1e1_job1.jpg
Diagram

Diagram

Solution

(%i1)

kill(all);

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

(%i6)

R:1;B:2;ro:1000;g:9.81;H:R/2;S:300;

\[\]\[\tag{%o1} 1\]

\[\]\[\tag{%o2} 2\]

\[\]\[\tag{%o3} 1000\]

\[\]\[\tag{%o4} 9.81\]

\[\]\[\tag{%o5} \frac{1}{2}\]

\[\]\[\tag{%o6} 300\]

(%i8)

fB(h,R):=sqrt(R··2−((R−h)··2));diff(fB(h,1),h);

\[\]\[\tag{%o7} \mathop{fB}\left( h\mathop{,}R\right) \mathop{:=}\sqrt{{{R}^{2}}\mathop{-}{{\left( R\mathop{-}h\right) }^{2}}}\]

\[\]\[\tag{%o8} \frac{1\mathop{-}h}{\sqrt{1\mathop{-}{{\left( 1\mathop{-}h\right) }^{2}}}}\]

(%i9)

db(h):=(1−h)/sqrt(1^2−(1−h)^2);

\[\]\[\tag{%o9} \mathop{db}(h)\mathop{:=}\frac{1\mathop{-}h}{\sqrt{{{1}^{2}}\mathop{-}{{\left( 1\mathop{-}h\right) }^{2}}}}\]

P_x

P_x - Horizontal projection of the force vector (Горизонтальна проекція вектору сили)

(%i10)

Pi_x(h):=ro·g·(R−h)·B;

\[\]\[\tag{%o10} \mathop{Pi\_ x}(h)\mathop{:=}\ensuremath{\mathrm{ro}} g\, \left( R\mathop{-}h\right) B\]

Figure 2:D:\j_Iam_TEMP\_jh\_jh_ex1e1_jobPx.jpg
Diagram

Diagram

(%i11)

Px:integrate(Pi_x(h),h,0,H);

\[\]\[\tag{%o11} 7357.5\]

(%i12)

mPx:integrate(Pi_x(h)·(R−h),h,0,H);

\[\]\[\tag{%o12} 5722.5\]

(%i13)

_mPx:integrate(Pi_x(h)·h,h,0,H);

\[\]\[\tag{%o13} 1635.0\]

(%i14)

h_D:mPx/Px;

\[\]\[\tag{%o14} 0.7777777777777778\]

(%i15)

h_D_:_mPx/Px;

\[\]\[\tag{%o15} 0.2222222222222222\]

P_z

P_z - Vertical projection of the force vector (Вертикальна проекція вектору сили)

Figure 3:D:\j_Iam_TEMP\_jh\_jh_ex1e1_job_Pz.jpg
Diagram

Diagram

(%i17)

Pz:integrate(Pi_x(h)·db(h),h,0,H); Pz,numer;

\[\]\[\tag{%o16} 817.5 \left( 4 \ensuremath{\pi} \mathop{+}{{3}^{\frac{3}{2}}}\right) \]

\[\]\[\tag{%o17} 14520.862582801296\]

(%i18)

mPz:integrate(Pi_x(h)·db(h)·(fB(h,1)),h,0,H);

\[\]\[\tag{%o18} 5722.5\]

(%i19)

x_C:mPz/Pz,numer;

\[\]\[\tag{%o19} 0.39408815883829146\]

Angle (radians, degrees) Кут (радіани, градуси)

(%i32)

phi_rad:atan(Pz/Px),numer;phi_gr:phi_rad·(180/%pi),numer;

\[\]\[\tag{%o31} 1.1018152093064582\]

\[\]\[\tag{%o32} 63.129361296583475\]

Answer

Figure 4:D:\j_Iam_TEMP\_jh\_jh_ex1e1_all.jpg
Diagram

Diagram

(%i26)

Px,numer;Pz,numer;phi_gr;h_D;x_C;

\[\]\[\tag{%o22} 7357.5\]

\[\]\[\tag{%o23} 14520.862582801296\]

\[\]\[\tag{%o24} 63.129361296583475\]

\[\]\[\tag{%o25} 0.7777777777777778\]

\[\]\[\tag{%o26} 0.39408815883829146\]

Sector drawing 300x300 px, scale (Малюнок сектору 300х300 пх , масштаб)

(%i30)

Px/100,numer;Pz/100,numer;h_D·S,numer;x_C·S,numer;

\[\]\[\tag{%o27} 73.575\]

\[\]\[\tag{%o28} 145.20862582801294\]

\[\]\[\tag{%o29} 233.33333333333334\]

\[\]\[\tag{%o30} 118.22644765148743\]

Created with wxMaxima.

The source of this Maxima session can be downloaded here.