3_hydrostatic-force_tutorial-solution(1)
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1. The document contains 14 problems involving calculation of hydrostatic forces on submerged objects and gates of various shapes. Forces are calculated using principles of pressure variation with depth, center of gravity, buoyancy and taking moments. 2. Problems involve determining total force, location of center of pressure, and reactions at hinges/supports for objects like rectangular/inclined gates, circular gates, cylinders, and dams of different cross-sections immersed in water or other liquids. 3. Additional considerations like fluid density, negative pressure, and imaginary water levels are incorporated based on problem details.
![= PAA Abottom+ [ ]
=
= 129.7N (down)
17. The cylinder in the fig. is 1.5m long and its radius is 1.25m. Compute the horizontal and vertical
components of the pressure force on the cylinder.
Solution:
AB = 1.25+1.25Sin45 = 1.25+0.88 = 2.13m
Horizontal force ̅ = 9810x(2.13x1.5)x(2.13/2) = 33380 N = 33.38 KN (right)
Vertical force (Fv) = Weight of volume of water vertically above ABC
=
[ ]
= 67029N = 670.29 KN (up)
18. The 1m diameter log (sp gr = 0.82) divides two shallow ponds as shown in the fig. Compute the net
horizontal and vertical reactions at point C, if the log is 3.7m.
Solution:
Horizontal force on ADC ̅̅̅̅ = 9810x3.7x1x1.2= 43556 N (right)
MN
B
D
A
0.88m
B
A
45
0
1 2
3
4
Water
1.25m
C
0.7m
1m
0.5mWater
C
O
Water](https://image.slidesharecdn.com/8c419466-d928-4664-ae96-0f2cc103ce9e-160306115146/85/3_hydrostatic-force_tutorial-solution-1-13-320.jpg)
![Additional problems on hydrostatic force
For the system shown in figure, calculate the height H of water at which the rectangular hinged gate will
just begin to rotate anticlockwise. The width of gate is 0.5m.
Solution:
Force due to water ̅
CP of F1
M.I. about CG = 0.072m4
Vertical distance of CP of F1 from free surface
̅ ̅
Force due to air pressure (F2)= PA = 40x1000x1.2x0.5 =24000N, which acts at a distance of H-0.6 from
the free surface.
Taking moment about hinge,
[ ]
* +
H = 1.6m
F1
F2
Hinge
Gate
Water
H
Air
40KPa 1.2m](https://image.slidesharecdn.com/8c419466-d928-4664-ae96-0f2cc103ce9e-160306115146/85/3_hydrostatic-force_tutorial-solution-1-23-320.jpg)
3_hydrostatic-force_tutorial-solution(1)
- 1. Tutorial 3 Hydrostatic force on submerged bodies 1. A vertical rectangular gate, 1.4m high and 2 m wide, contains water on one side. Determine the total resultant force acting on the gate and the location of c.p. Solution: Area (A) = 2x1.4 = 2.8 m2 Location of CG (̅) = (3+1.4/2) = 3.7m Resultant force on gate (F) = ? Cp ( ) = ? ̅ = 9810x2.8x3.7 = 101631 N = 101.631 KN M.I. about CG = 0.457 m4 ̅ ̅ =3.74m 2. An inclined rectangular gate (1.5m wide) contains water on one side. Determine the total resultant force acting on the gate and the location of c.p. Solution: Water 3m Gate 60 0 1.2m 30 0 Water Gate 2.4m
- 2. Area (A) = 1.5x1.2 = 1.8 m2 Location of CG (̅) = (2.4+1.2Sin30/2) = 2.7m Resultant force on gate (F) = ? Cp ( ) = ? ̅ = 9810x1.8x 2.7 = 47676N = 47.676 KN M.I. about CG = 0.216 m4 ̅ ̅ = 2.71m 3. An inclined circular with water on one side is shown in the fig. Determine the total resultant force acting on the gate and the location of c.p. Solution: Area (A) = = 0.785 m2 Location of CG (̅) = (1.8+1.0Sin60/2) = 2.23m Resultant force on gate (F) = ? Cp ( ) = ? ̅ = 9810x0.785x2.23 = 17173N = 17.173 KN M.I. about CG = 0.049 m4 ̅ ̅ = 2.25m 1m 60 0 Water Gate 1.8m
- 3. 4. Gate AB in the fig. is 1m long and 0.7m wide. Calculate force F on the gate and position X of c.p. Solution: Sp. wt of oil ( ) =0.81x9810 = 7946 N/m3 Area (A) = 0.7x1 = 0.7 m2 Location of CG (̅) = (3+1Sin50+1Sin50/2) = 4.15m Resultant force on gate (F) = ? x = ? ̅ = 7946x0.7x4.15 = 23083 N = 23.08 KN M.I. about CG = 0.058 m4 Vertical distance of CP from free surface ̅ ̅ = 4.161m Vertical distance between CP from CG = 4.161-(3+1Sin50)= 0.395m x = 0.395/sin50= 0.515m 5. The gate in the fig. is 1.2m wide, is hinged at point B, and rests against a smooth wall at A. Compute (a) the force on the gate due to sea water pressure, (b) the horizontal force exerted by the wall at point A, and (c) the reaction at hinge B. x 1m 1m 50 0 Oil (sp gr = 0.81) F 3m 7m
- 4. Solution: Sp wt of sea water ( ) = 1025x9.81 = 10055 N/m3 Area (A) = 1.2x3.6 = 4.32 m2 Location of CG (̅) = (5.1-2.2)+2.2/2 = 4.0m a) Resultant force on gate (F) = ? ̅ = 10055x4.32x4.0 = 173750 N = 173.75 KN b) Force P = ? M.I. about CG = 4.665 m4 Vertical distance of CP from free surface ̅ ̅ = 4.1m Vertical distance between B and CP = 5.1-4.1 = 1m Location of F from B = 1/sinθ = 1.636m Taking moment about B, Px2.2-173.75x1.636 = 0 P = 129.2KN c) Reactions at hinge, Bx and By = ? ∑ Bx – 129.2 + 173.75x2.2/3.6 = 0 Bx = 23.02 KN ∑ CP CG F P θ Bx A B P B A θ Patm Wall Gate Hinge Sea water Density = 1025kg/m 3 5.1m 2.85m 2.2m By 3.6m
- 5. By - 173.75x2.85/3.6 = 0 By = 137.55 KN 6. Gate AB in fig. is 4.8m long and 2.4m wide. Neglecting the weight of the gate, compute the water level h for which the gate will start to fall. Solution: Area (A) = 2.4xh/Sin60 = 2.77h m2 Location of CG (̅) = h/2 m Resultant force on gate (F) is ̅ = 9810x2.77hxh/2 = 13587h2 N M.I. about CG = 0.308h3 m4 Vertical distance of CP from free surface ̅ ̅ = 0.667h Distance of F from B = (h-0.667h)/Sin60=0.384h Taking moment about B, 5000x4.8-13587h2 x0.384h = 0 h = 1.66m CP A C 60 0 B A 4.8m h Water 5000N 5000N F B CG Zcp
- 6. 7. Find the net hydrostatic force per unit width on rectangular panel AB in the fig. and determine its line of action. Solution: Area (A) = 2x1 = 2 m2 Location of CG (̅) = 2+1+2/2 = 4 m for water side Location of CG (̅̅̅̅) = 1+2/2 = 2 m for glycerin side Resultant force on gate (F) = ? ̅ Force due to water (Fwater) = ̅ = 9.81x2x4 = 78.49KN Force due to glycerin (Fglyc) = ̅̅̅̅ =12.36x2x2 = 49.44KN Net force (F) = 78.49-49.44 = 29.04KN M.I. about CG = 0.666 m4 Distance of Fwater from CG ( ) ̅ ̅ = 4.083m Distance of Fglyc from CG ( ) ̅ ̅̅̅̅ = 2.166m Taking moment about B, 29.04y = 78.49x(5-4.083)-49.44x(3-2.166) y = 0.945m y FWate r F A CG B A 1m 2m 1m 2m Water Glycerin Sp. wt. = 12.36 KN/m 3 Fglycr B
- 7. 8. Circular gate ABC in the fig. is 4m in diameter and is hinged at B. Compute the force P just sufficient to keep the gate from opening when h is 8m. Solution: Area (A) = = 12.56 m2 Location of CG (̅) = 8m P = ? Resultant force on gate (F) ̅ = 9810x12.56x8 = 985708 N = 985.708KN M.I. about CG = 12.56 m4 Position of CP from free surface ̅ ̅ = 8.125 m Taking moment about B, 985.7x0.125-Px2=0 P = 61.6KN 9. The tank in the fig. contains oil (sp gr = 0.8) and water as shown. Find the resultant force on side ABC and its point of application. ABC is 1.2m wide. Solution: Sp wt of oil ( ) = 0.8*9810 N/m3 = 7848 N/m3 Area (A1) = 1.2x3 = 3.6 m2 2.4m P 2m 2m C B A Water h P F ycp C B A Water1.8m 3m Oil IWS
- 8. Area (A2) = 1.2x1.8 = 2.16 m2 Location of CG for AB (̅̅̅̅) = 3/2 =1.5m Resultant force on ABC = ? Force on AB ̅̅̅̅ =7848x3.6x1.5 = 42379N Pt. of application of FAB = (2/3)x3 = 2m below A Force on BC Water is acting on BC and any superimposed liquid can be converted to an equivalent depth of water. Equivalent depth of water for 3m of oil = = 2.4m Employ an imaginary water surface of 2.4m. Location of CG for IWS (̅̅̅̅) = 2.4+1.8/2 = 3.3m ̅̅̅̅ = 9810x2.16x3.3 = 69925N Point of application of FBC from A is ̅ ̅̅̅̅ ̅̅̅̅ =3.38 m i.e. 3.38+0.6 = 3.98m from A Total force on side ABC (F) = 42379+69925 = 112304N = 112.304 KN Taking moment about A, 112304 y = 42379x2+69925x3.98 y = 3.23m F acts at 3.23m below A. (Alternative method: Solve by drawing pressure diagram. Force = Area of pressure diagram x width. Take moment to find position of resultant force.) 10. Gate AB in the fig. is 1.25m wide and hinged at A. Gage G reads -12.5KN/m2 , while oil (sp gr = 0.75) is in the right tank. What horizontal force must be applied at B for equilibrium of gate AB? Solution: Sp wt of oil ( ) = 0.75*9810 N/m3 = 7357.5 N/m3 Area (A) = 1.25x1.8 = 2.25 m2 2.33m 1.27m B 2m FAB FBC A C 3.38m Air B A G Water 1.8m 5.4m Oil
- 9. Location of CG for right side (̅̅̅̅) = 1.8/2 =0.9m Force on AB at the right side ̅̅̅̅ =7357.5x2.25x0.9 = 14899N Pt. of application of Foil = (2/3)x1.8 = 1.2m from A Left side For the left side, convert the negative pressure due to air to equivalent head in water. Equivalent depth of water for -12.5KN/m2 pressure = = -1.27m This negative pressure head is equivalent to having 1.27m less water above A. Location of CG from imaginary water surface (̅̅̅̅) = 2.33 + 1.8/2 = 3.23m ̅̅̅̅ = 9810x2.25x3.23 = 71294N Point of application of Fwater from A is ̅ ̅ ̅ =3.31m from IWS i.e. 3.31-2.33 = 0.98m from A Taking moment about A Px1.8 + 14899x1.2-71294x0.98 = 0 P = 28883 N = 28.883 KN 11. The gate AB shown is hinged at A and is in the form of quarter-circle wall of radius 12m. If the width of the gate is 30m, calculate the force required P to hold the gate in position. Solution: Horizontal force ̅ = 9810x(30x12)x12/2 = 21189600 N = 21189.6 KN (right) FH acts at a distance of 12x1/3 =4m above the hinge A. Vertical force (Fv) = Weight of volume of water vertically above AB = * + = 33284546 N = 33284.546 KN (downward) FV acts at a distance of = 4x12/3x3.1416 = 5.1m from the vertical AO. Taking moment about A, Px12= 21189.6x4+33284.546x5.1 P = 21209 KN FH A FoilFwate r B P A BO Water P FV
- 10. 12. The water is on the right side of the curved surface AB, which is one quarter of a circle of radius 1.3m. The tank’s length is 2.1m. Find the horizontal and vertical component of the hydrostatic acting on the curved surface. Solution: Horizontal force ̅ = 9810x(1.3x2.1)x(2.5+1.3/2) = 84361 N = 84.361 KN (right) Vertical force (Fv) = Weight of imaginary volume of water vertically above AB = * + = 94297 N = 94.297KN (downward) 13. The 1.8m diameter cylinder in the fig. weighs 100000N and 1.5m long. Determine the reactions at A and B, neglecting friction. Solution: Horizontal force ̅ = 0.8x9810x(1.8x1.5)x(1.8/2) = 19071 N (right) Vertical force (Fv) = Weight of volume of water vertically above BDC Vertical force (Fv) =(Fv)DB - (Fv)DC = O D B C A Water 2.5m 1.3m E C B D O Oil Sp gr = 0.8 A
- 11. = * + = 14978 N (up) Reaction at A = FH = 19071 N (left) Reaction at B = Weight of cylinder – FV = 100000-14978 = 85022N (up) 14. In the fig., a 2.4m diameter cylinder plugs a rectangular hole in a tank that is 1.4m long. With what force is the cylinder pressed against the bottom of the tank due to the 2.7m depth of water? 1.2cos30 = 1.04m 0.16m Solution: Water is above the curve portion CDE, whereas it is below the curve portion AC and BE. For AC and BE, imaginary weight of water vertically above them is considered and the vertical force on theses part acts upwards. Net vertical force= (FV)CDE (down) – (FV)AC (up) – (FV)BE (up) = Weight of volume of water vertically above CDE- imaginary Weight of volume above arc AC- imaginary weight of volume above arc BE = = *( ) ( ) ( )+ = 27139N (down) C N P Q M A B D E O 0.6m 2.1m 300 SR
- 12. 15. A dam has a parabolic profile as shown in the fig. Compute the horizontal and vertical components of the force on the dam due to the water. The width of dam is 15m. (Parabolic area = 2/3(b*d) Solution: Horizontal force ̅ = 9810x(15x6.9)x6.9/2 = 3502906 N = 3502.906 KN (right) Vertical force (Fv) = Weight of volume of water vertically above AB = = 2030670 N = 2030.67 KN (down) 16. The bottled liquid (sp gr = 0.9) in the fig. is under pressure, as shown by the manometer reading. Compute the net force on the 50mm radius concavity in the bottom of the bottle. Solution: From symmetry, FH = 0 Manometeric equation for pressure, PAA + 0.9x9810x0.07 = 13.6x9810x0.12 PAA = 15392 N/m2 FV = PAA Abottom+ Weight of liquid below AA = PAA Abottom+ B A 3m Water 6.9m A A 7cm 12cm 16cm Hg 50mm radius
- 13. = PAA Abottom+ [ ] = = 129.7N (down) 17. The cylinder in the fig. is 1.5m long and its radius is 1.25m. Compute the horizontal and vertical components of the pressure force on the cylinder. Solution: AB = 1.25+1.25Sin45 = 1.25+0.88 = 2.13m Horizontal force ̅ = 9810x(2.13x1.5)x(2.13/2) = 33380 N = 33.38 KN (right) Vertical force (Fv) = Weight of volume of water vertically above ABC = [ ] = 67029N = 670.29 KN (up) 18. The 1m diameter log (sp gr = 0.82) divides two shallow ponds as shown in the fig. Compute the net horizontal and vertical reactions at point C, if the log is 3.7m. Solution: Horizontal force on ADC ̅̅̅̅ = 9810x3.7x1x1.2= 43556 N (right) MN B D A 0.88m B A 45 0 1 2 3 4 Water 1.25m C 0.7m 1m 0.5mWater C O Water
- 14. Horizontal force on BC ̅̅̅̅ = 9810x3.7x0.5x0.5/2= 4537 N (left) Vertical force on ADC (FV1) = Weight of volume of water vertically above ADC Vertical force (Fv1) =(Fv)MNDCOAM (up)- (Fv)MNDAM (down) = 14254N (up) Vertical force on BC (FV2) = Weight of volume of water (imaginary) vertically above BC = = 7127N (up) Weight of log (W) = = 23376N (down) Horizontal reaction at C (Rx) -RX +FH1- FH2 = 0 Rx = FH1- FH2 = 43556 – 4537 = 39019N (left) Vertical reaction at C (Ry) Ry + FV1 + FV2 – W = 0 Ry = 23376-14254-7127 = 1995N (up) 19. The 0.9m diameter cylinder in the fig. is 7m long and rests in static equilibrium against a frictionless wall at point B. Compute the specific gravity of the cylinder. Solution: Vertical force (FV) = Weight of volume of water vertically above ADBECA E C A D W FH1 FH2 Fv1 Fv2 Rx Ry B Water Wall BO Water Wall
- 15. = (Fv) on semi-circle ACE + (Fv) on quadrant BE For BE, imaginary weight of fluid vertically above it is considered = * + = 46670N (up) The reaction at B is purely horizontal. Weight of cylinder (W) = FV W = 46670N = 10480 N/m3 Sp gr of cylinder = = 1.07 20. Find the horizontal and vertical forces per m of width on the tainter gate shown in the fig. Solution: Horizontal force ̅ = 9810x(7.5x1)x7.5/2 = 275906 N = 275.906 KN (right) FH acts at a distance of 7.5x2/3 = 5m from water surface. Vertical force (Fv) = Weight of imaginary volume of water vertically above ABCA = * + = 49986N = 49.986KN (up) FV acts through the centroid of the segment ABCA. 30 0 30 0 A B C7.5m Water 7.5m radius O
- 16. 21. The tank whose cross section is shown in fig. is 1.2m long and full of water under pressure. Find the components of the force required to keep the cylinder in position, neglecting the weight of the cylinder. Solution: Pressure =14KPa Equivalent head of water = = 1.43m Apply 1.43m water above the cylinder. Horizontal force ̅ = 9810x(0.9x1.2)x(1.43+0.9/2) = 19918 N = 19.918 KN (right) Sin(OEF) = 0.3/0.6 <OEF = 300 = <EOD EF = 0.6Cos30 = 0.52m Vertical force (Fv) = (Fv)MABFOCDEM -(Fv)MAED = Weight of volume of water vertically above ABFOCDEA = * + = 14110 N = 14.11 KN (up) Forces required to keep the cylinder in positions are: 19.918KN to the right and 14.11KN to the up. M E D C BA 0.3m 0.6m 14Kpa 0.6m radius 0.9m Water F O 1.43m
- 17. 22. Each gate of a lock is 6m high and is supported by two hinges placed on the top and the bottom. When the gates are closed, they make an angle of 1200 . The width of the lock is 7m. If the water levels are 5m and 2m at upstream and downstream respectively, determine the magnitude of forces on the hinge due to the water pressure. Solution: F = Resultant water force, P = Reaction between gates, R = total reaction at hinge θ = 300 Width of lock = 3.5/cos30 = 4.04m Resolving forces along gate Pcosθ = Rcosθ i.e. P=R (a) Resolving forces normal to gate PSinθ +RSinθ = F (b) From a and b P = F/2Sinθ Horizontal force on upstream side ̅̅̅̅ = 9810x4.04x5x5/2= 495405 N F1 acts at 5/3m = 1.66 from bottom Horizontal force on downstream side ̅̅̅̅ = 9810x4.04x3x3/2= 178346N F2 acts at 3/3 = 1m from bottom F = F1– F2 = 495405-178346 = 317059 N Taking moment about the bottom to find the point of application of F, 317059y = 495405x1.66-178346x1 y = 2.03m P = F/2Sinθ = 317059/2sin30 =317059N R = P = 317059N Taking moment about bottom hinge Rb F2F1 y Rt θ P F B A C 3m 120 0 7m 6m 5m Water Water R F
- 18. Rt x6 =317059x2.03 Rt = 107272N = 107.272KN Rb = R – Rt = 317059-107272 = 209787N = 209.787KN 23. Find the net horizontal and vertical forces acting on the surface ABCDEF of width 5m as shown in the figure below. BCD is a half circle. Solution: AB = 2/Sin450 = 2.8284m EF = 2/Sin450 = 2.8284m Pressure force on inclined surface AB ̅̅̅ = 9810x(2.8284x5)x1 = 138733N which is perpendicular to AB F1x = F1 Cos450 = 138733 Cos450 = 98099N (right) F1y = F1 Sin450 = 138733 Sin450 = 98099N (up) For curved surface BCD ̅̅̅ = 9810x(2x5)x3 = 294300 N (right) ( ) = 77048N (down) Pressure force on EF due to water ̅̅̅ = 9810x(2.8284x5)x5 = 693665N which is perpendicular to EF F3x = F3 Cos450 = 693665 Cos450 = 490495N (right) F3y = F3 Sin450 = 693665 Sin450 = 490495N (up) Pressure force on EF due to oil ̅̅̅ = 0.8x9810x(2.8284x5)x1 = 110986N which is perpendicular to EF Oil of s.g. 0.8 2m 2m 2m 2m2m D B C E F A 2m Water
- 19. F4x = F4 Cos450 = 1109955 Cos450 = 78479N (left) F4y = F4 Sin450 1109955 Sin450 =78479N (down) Net horizontal force = 98099+294300+490495-78479N = 804415N (right) Net vertical force = 98099-77048+490495-78479N = 433067N (up) 24. Calculate the pressure force on the curved surface ABCD as shown in the figure below. AB is a quadrant of radius 1m and BCD is a semi-circle of radius 1m. Take width of curve = 5m. Solution: Horizontal force on AB ( ̅̅̅ = 9810x(1x5)x3.5 = 171675N (right) Vertical force on AB ( ) = 185674N (up) Horizontal force on BCD from the left side ( ̅̅̅ = 9810x(2x5)x5 = 490500N (right) Vertical force on BCD from the left side ( ) = 77048N (down) Horizontal force on BCD from the right side ( ̅̅̅ = 0.82x9810x(2x5)x1 = 80442N (left) Vertical force on BCD from the right side ( ) = 63179N (up) Net horizontal force = 171675+490500-80442 = 581733N = 581.733KN (right) Net vertical force = 185674-77048+63179 = 171805N = 171.805KN (up) 1m 2m A B C D 2m 3m Water Oil of sp gr 0.82
- 20. 25. Find the weight of the cylinder (dia. =2m) per m length if it supports water and oil (sp gr = 0.82) as shown in the figure. Assume contact with wall as frictionless. Solution: Downward force on AC due to oil (FVAC) = Weight of oil supported above curve AC = = ( ) = ( ) = 1726N Pressure at C due to 1m oil (P) = = 0.82x9810x1 = 8044.2 Pa Equivalent head of water due to 1m oil = = 0.82m Apply 0.82m water above EC. Upward vertical force on CBE (FVCBE) = Weight of water above CBE N A F E B C A 1m 1m D Oil Water E B C 1m 1m D Oil Water M E B C 0.82 m 1m D Water
- 21. 12m O AB x dy y ( ) = 31498N Weight of cylinder = FVCBE - FVAC = 31498-1726 = 29772 N 26. Find the magnitude and direction of the resultant pressure force on a curved face of a dam which is shaped according to the relation y = x2 /6. The height of water retained by the dam is 12m. Assume unit width of the dam. Solution: The equation of the dam y = x2 /6 √ Consider an element of thickness dy and length x at a distance y from the base. Area of element = xdy Area of OAB = ∫ ∫ √ √ | | = 67.882 m2 Horizontal force ̅ = 9810x(12X1)x6 = 706320N Vertical force (Fy) = Weight of water vertically above dam OA = = 9810x67.882x1 = 665922N Resultant force √ = 970742N = 970.742KN Direction of resultant force = = = 43.310 27. A cylinder, 2m in diameter and 3m long weighing 3KN rests on the floor of the tank. It has water to a depth of 0.6m on one side and liquid of sp gr 0.7 to a depth of 1.25m on the other side. Determine the magnitude and direction of the horizontal and vertical components of the force required to hold the cylinder in position.
- 22. Solution: OA= OB = OC = 1m, BD = 0.6m OD = 1-0.6 = 0.4m CD = = 0.9165m ( )= 66.40 OE = 1.25-1 = 0.25m ( )= 75.50 <AOB = 180-75.5 = 104.50 AE = 0.25 tan75.5 = 0.96m Weight of cylinder = 3KN = 3000N Net horizontal force ̅̅̅̅ ̅̅̅̅ = 0.7x9810x1.25x3x1.25/2-9810x0.6x3x0.6/2 = 10797N (left) Net vertical force (FV)= Weight of volume of oil vertically above AB + Weight of volume of water vertically above BC = FvAB (up) + FvBC (up) = = ( ) ( ) = 32917N (up) The components to hold the cylinder in place are 10797 N to the right and 32917-3000 = 29917N down. Oil Water C A B 0.6m E O D 1.25m
- 23. Additional problems on hydrostatic force For the system shown in figure, calculate the height H of water at which the rectangular hinged gate will just begin to rotate anticlockwise. The width of gate is 0.5m. Solution: Force due to water ̅ CP of F1 M.I. about CG = 0.072m4 Vertical distance of CP of F1 from free surface ̅ ̅ Force due to air pressure (F2)= PA = 40x1000x1.2x0.5 =24000N, which acts at a distance of H-0.6 from the free surface. Taking moment about hinge, [ ] * + H = 1.6m F1 F2 Hinge Gate Water H Air 40KPa 1.2m
- 24. A 3m square gate provided in an oil tank is hinged at its top edge. The tank contains gasoline (sp. gr. = 0.7) up to a height of 1.6m above the top edge of the plate. The space between the oil is subjected to a negative pressure of 8 Kpa. Determine the necessary vertical pull to be applied at the lower edge to open the gate. Solution: Head of oil equivalent to -8 Kpa pressure = = -1.16m This negative pressure will reduce the oil surface by 1.16m. Let AB = new level. Make calculation by taking AB as free surface. h = 1.6-1.16= 0.44m ̅ = 1.5m Hydrostatic force ̅ = 0.7x9810x(3x3)x1.5 = 92704.5N CP of F M.I. about CG = 6.75m4 Vertical distance of CP of F1 from free surface ̅ ̅ = 1.75m Vertical distance between the hinge and F = 1.75-0.44 = 1.31m Taking moment about the hinge P = 80962N h B A1.16m F P 45 0 -8Kpa pressure 1.6m Gasoline Gasoline surface Gate Hinge New level due to pressure yp
- 25. There is an opening in a container shown in the figure. Find the force P and the reaction at the hinge (R). Solution: Equivalent head of oil due to 24 Kpa pressure = = 2.9m of oil Apply 2.9m of oil above the hinge. ̅ = 3.2m Hydrostatic force ̅ = 0.85x9810x(1.2x1.2)x3.2= 38424N CP of F M.I. about CG 0.1728m4 Vertical distance of CP of F1 from free surface ̅ ̅ = 3.209m Vertical distance between the hinge and F = 3.209-2.9 = 0.309m Taking moment about the hinge P = 19788N R+P=F R = 38424-19788 = 18636N R CGCP F P 30 0 24Kpa Gate (1.2mx1.2m) P Hinge oil, sp. gr. = 0.85 Container 2.9m yp