O Q.3.1. Fill in the blanks
1. The point of action of resultant fluid forces is called _________.
2. The moment of area about any axis through the centre of gravity will be _________.
3. The second moment of area about an axis through the centre of gravity will _________ compared to any other axis.
4. The centre of pressure will generally be _________ the centroid.
5. The hydrostatic force on a submerged plane surface depends on the _________ of the centroid.
6. The force due to liquid pressure acts _________ to the surface.
7. The law for calculating hydrostatic pressure is _________.
8. The second moment about any axis differs from the second moment through a parallel axis through the centroid by _________.
9. The distance of centre of pressure from its centroid for a vertical area immersed in liquid is given by _________.
10. The vertical distance between the centriod and centre of pressure over a plane area immersed at an angle θ to the free surface is given by _________.
11. The pressure at the same horizontal level in a static liquid is _________.
Answers
(1) Centre of pressure, (2) zero, (3) lower, (4) below, (5) depth, (6) normal, (7) (dp/dy) = – γ, (8) A x2, x-distance between the axes, 9. IG / A h, 10. IG sin2 θ/ A h 11. the centre.
O Q.3.2 Fill in the blanks:
1. The horizontal force on a curved surface immersed in a liquid is equal to the force on _________.
2. The vertical force on a curved surface equals the _________.
3. The line of action of horizontal force on a curved surface immersed in a liquid is_________.
4. The line of action of vertical force on a curved surface immersed in a liquid is _________.
5. The force due to gas pressure on curved surface in any direction _________.
6. The resultant force on cylindrical or spherical surfaces immersed in a fluid passes through _______.
Answers
(1) the vertical projected area, (2) the weight of column of liquid above the surface, (3) the centre of pressure of the vertical projected area, (4) the centriod of the liquid column above the surface, (5) equals the product of gas pressure and projected area in that direction, (6) the centre.
O Q.3.3 Fill in the blanks using increases, decreases or remains constant : 1. The force due to liquid pressure _________ with depth of immersion.
2. The distance between the centriod and the center of pressure _________ with depth of immersion.
3. When a plane is tilted with respect to any centriodal axis the normal force on the plane due to liquid pressure _________.
4. The location of centre of pressure of a plane immersed in a liquid _________ with change in density of the liquid.
Chapter 3 Answers
Increases : 1, Decreases : 2, Remains Constant : 3, 4.
O Q.3.4 Indicate whether the statements are correct or incorrect :
1. The centre of pressure on a plane will be at a lower level with respect to the centroid.
2. In a plane immersed in a liquid the centre of perssure will be above the centroid.
3. The resultant force due to gas pressure will act at the centroid.
4. The vertical force on an immersed curved surface will be equal to the column of liquid above the surface.
5. The normal force on an immersed plane will not change as long as the depth of the centroid is not altered.
6. When a plane is tilted along its centroidal axis so that its angle with horizontal increases, the normal force on the plane will increase.
Answers (1) Correct : 1, 3, 5 (2) Incorrect : 2, 4, 6.
O Q.3.5 Choose the correct answer :
1. The pressure at a depth ‘d’ in a liquid, (above the surface pressure) is given by
(a) ρ g (b) γ d
(c) – γ d (d) (ρ/g)d (usual notations)
2. The density of a liquid is 1000 kg/m3. At location where g = 5 m/s2, the specific weight of the liquid will be
(a) 200 N/m3 (b) 5000 N/m3
(c) 5000 × 9.81 / 5 N/m3 (d) 5000 × 5 / 9.81 N/m3
3. The centre of pressure of a rectangular plane with height of liquid h m from base
(a) h/2 m from bottom (b) h/3 m from top
(c) h/3 m from bottom
(d) can be determined only if liquid specific weight is known.
4. The horizontal force on a curved surface immersed in a liquid equals (a) the weight of the column of liquid above the surface
(b) the pressure at the centroid multiplied by the area (c) the force on the vertical projection of the surface
(d) the pressure multiplied by the average height of the area.
5. The location of the centre of pressure over a surface immersed in a liquid is (a) always above the centroid
(b) will be at the centroid (c) will be below the centroid
(d) for higher densities it will be above the centroid and for lower densities it will be below the centroid.
6. The pressure at a point y m below a surface in a liquid of specific weight γ as compared to the surface pressure, P will be equal to
(a) P + (y/γ) (b) P + yγ
(c) P – (y g/γ) (d) P + (y . g/γ).
7. When the depth of immersion of a plane surface is increased, the centre of pressure will (a) come closer to the centroid
(b) move farther away from centroid
(c) will be at the same distance from centroid (d) depend on the specific weight of the liquid.
8. A sphere of R m radius is immersed in a fluid with its centre at a depth h m The vertical force on the sphere will be
(a) γ (4/3)π R3 (b) γπ R2 h
(c) γ (π R2 h + 8 π R2/3) (d) γ (π R2 h – 8 π R2/3).
Answers (1) b (2) b (3) c (4) c (5) c (6) b (7) a (8) a.
O Q.3.6 Match the sets A and B :
A B
(I)
1. Specific weight (a) m3
2. Density (b) m4
3. Second moment of area (c) N/m2
4. First moment of area (d) kg/m3
5. Pressure (e) N/m3
Answers 1 – e, 2 – d, 3 – b, 4 – a, 5 – c.
(II)
A B
1. Centroid (a) always positive
2. Centre of pressure (b) area moment zero
3. Free surface (c) resultant force
4. Second moment of area (d) constant pressure Answers
1 – b, 2 – c, 3 – d, 4 – a.
(III) Moment of inertia of various shapes :
1. Circle about centroidal axis (a) B h3 / 36 2. Rectangle about centroidal axis (b) D4 / 64 3. Triangle about centroidal axis (c) D4 / 128 4. Semicircle about base (d) B h3 / 12
Answers 1 – b, 2 – d, 3 – a, 4 – c.
Chapter 3
EXERCISE PROBLEMS
E.3.1. Determine the centroid of the following shapes shown in Fig E. 3.1 from the given reference lines.
X 3 m X
5 m
2 m
2 m
3 m
X X
0.5 0.5 1 m
X 2 m X
2 m
1 m
X 1.414 m X 1 m
1 m
X X
1 m 60°
X X
1 m 1 m
1 m
X 3 m X
1.5 m
Ellipse 1 m
1 m
Figure E. 3.1
E.3.2. For the shapes in Fig E. 3.1, determine the moment of inertia of the surfaces about the axis xx and also about the centroid.
E.3.3. From basics (by integration) determine the forces acting on one side of a surface kept vertical in water as shown in Fig. E. 3.3.
h
a
b
5 m 30°
5 m WL
y = 2
y = X /62
Figure E. 3.3
E.3.4. Determine the magnitude and location of the hydrostatic force on one side of annular surface of 2 m ID and 4 m OD kept vertical in water.
E.3.5. Determine the moment required to hold a circular gate of 4 m dia, in the vertical wall of a reservoir, if the gate is hinged at (i) the mid diameter (ii) at the top. The top of the gate is 8 m from the water surface.
E.3.6. Determine the compressive force on each of the two struts supporting the gate, 4 m wide, shown in Fig. E. 3.6.
4m8m
WL
Hinge i)
WL
1 m
3 m Gate
Strut
60° 30°
Figure E. 3.5 Figure E. 3.6
E.3.7. An annular plate of 4 m OD and 2 m ID is kept in water at an angle of 30° with the horizontal, the centre being at 4 m depth. Determine the hydrostatic force on one side of the plane. Also locate the centre of pressure.
E.3.8. A tank contains mercury upto a height of 0.3 m over which water stands to a depth of 1 m and oil of specific gravity 0.8 stands to a depth of 0.5 m over water. For a width of 1 m determine the total pressure and also the point of action of the same.
E.3.9. A trapezoidal gate of parallel sides 8 m and 4 m with a width of 3 m is at an angle of 60° to the horizontal as shown in Fig. E. 3.9 with 8 m length on the base level. Determine the net force on the gate due to the water. Also find the height above the base at which the resultant force acts.
4 m
8 m 60°
Gate
3 m 4 m WL
Figure E. 3.9
E.3.10. Show that the resultant force on a submerged plane remains unchanged if the area is rotated about an axis through the centroid.
E.3.11. A gate as shown in Fig. E. 3.11 weighing 9000 N with the centre of gravity 0.5 m to the right of the vertical face holds 3 m of water. What should be the value of counter weight W to hold the gate in the position shown.
Chapter 3 E.3.12. A rectangular gate of 2 m height and 1 m width is to be supported on hinges such that it will
tilt open when the water level is 5 m above the top. Determine the location of hinge from the base.
1.5 m 0.5 3 m 9 kN
1 m
W WL
5 m
2 m
Hinge level
y
Figure E. 3.11 Figure E. 3.12
E.3.13. Show that as the depth of immersion increases, the centre of pressure approaches the centroid.
E.3.14. Determine the magnitude and line of action of the hydrostatic force on the gate shown in Fig.
E. 3.14. Also determine the force at the edge required to lift the gate. The mass of the gate is 2500 kg and its section is uniform. The gate is 1 m wide.
WL
2 m WL
2 m F
60°
2.4 m
WL
8 m 4 m
2500 kg/m3
11 20 m
3.5
Figure E. 3.14 Figure E. 3.15
E.3.15. A dam section is shown in figure. Determine the location where the resultant hydrostatic force crosses the base. Also calculate the maximum and minimum compressive stress on the base.
E.3.16. An automatic flood gate 1.5 m high and 1 m wide is installed in a drainage channel as shown in Fig. E. 3.16. The gate weighs 6 kN. Determine height of water backing up which can lift the gate.
E.3.17. Compressed air is used to keep the gate shown in Fig. E. 3.17 closed. Determine the air pressure required.
W
Hinge WL
2m
Compressed air
0.8 msq.
WL
6 m
Figure E. 3.16 Figure E. 3.17
E.3.18. A spherical container of 6 m diameter is filled with oil of specific gravity 0.73. Determine the resultant force on one half of the sphere divided along the vertical plane. Also determine the direction of action of the force.
E.3.19. An inverted frustum of a cone of base dia 1 m and top dia 6 m and height 5 m is filled with water. Determine the force on one half of the wall. Also determine the line of action.
E.3.20. A conical stopper is used in a tank as shown in Fig. E. 3.20. Determine the force required to open the stopper.
E.3.21. Determine the total weight/m length of a gate made of a cylindrical drum and a plate as shown in Fig. E. 3.21, if it is in equilibrium when water level is at the top of the cylinder.
F
0.4 m
1 m
0.2 m 60°
60°
WL
1 m W
0.8 m 0.8 m
Plate
Figure E. 3.20 Figure E. 3.21
E.3.22. A gate 12 m long by 3 m wide is vertical and closes an opening in a water tank. The 3 m side is along horizontal. The water level is up to the top of the gate. Locate three horizontal posi-tions so that equal forces acting at these locaposi-tions will balance the water pressure.
[3.4641 m, 8.4452 m 10.9362 m]