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Exercise Problems - Chapter 2

1. For the system shown in Fig 5.10,determine the air pressure pA which will make the pressure at N one fourth of that at M.                                                [3.33 kPa] 

Exercise Problem - Fluid Statics - Mechanical Engineering

Fig 5.10

2. Consider the pipe and manometer system as shown in Fig 5.11. The pipe contains water. Find the value of manometer reading h, and the difference in pressure between A and B if there is no flow. If there is a flow from A towards B and the manometer reading is h = 60 mm, then determine the static pressure difference pA - pB

 

[0, 2.94 kPa; 3.53 kPa]   

Exercise Problem - Fluid Statics - Mechanical Engineering

Fig 5.11

3. Determine the air pressure above the water surface in the tank if a force of 8 kN is required to hold the hinged door in position as shown in Fig 5.12.

 

[10.76 kPa] 

Exercise Problem - Fluid Statics - Mechanical Engineering

4. The profile of the inner face of a dam takes the form of a parabola with the equation 18y = x2 , where y is the height above the base and x is the horizontal distance of the face from the vertical reference line. The water level is 27m above the base. Determine the thrust on the dam (per meter with) due to the water pressure, its inclination to the vertical and the point where the line of action of this force intersects the free water surface

 

[5.28 MN/m, 42o 33', 30.29 m from face ]  

5. A solid uniform cylinder of length 150 mm and diameter 75 mm is to float upright in water. Determine the limits within which its mass should lie.

 

[0.641 kg and 0.663 kg]

6. A long prism, the cross-section of which is an equilateral traingle of side a, floats in water with one side horizontal and submerged to a depth h. Find

      (a) h/a as a function of the specific gravity, S of the prism.
      (b) The metacentric height in terms of side a, for small angle of rotation if specific gravity, S=0.8.

7.  A metal sphere of volume Exercise Problem - Fluid Statics - Mechanical Engineeringm = 0.1m3 , specific gravity  sm = 2 and fully immersed in water is attached by a flexible wire to a buoy of volume   Exercise Problem - Fluid Statics - Mechanical EngineeringB = 1 mand specific gravity sB  = 0.1. Calculate the tension T in the wire and volume of the buoy that is submerged. Refer to Fig 5.13.

 

Exercise Problem - Fluid Statics - Mechanical Engineering

 

Recap

  In this course you have learnt the following 

  • Forces acting on a fluid element in isolation are of two types;
    • Body force : Body forces act over the entire volume of the fluid element and are caused by external agencies
    • Surface force. Surface forces, resulting from the action of surrounding mass on the fluid element, appear on its surfaces.
  • Normal stresses at any point in a fluid at rest, being directed towards the point from all directions, are of equal magnitude. The scalar magnitude of the stress is known as hydrostatic or thermodynamic pressure.
     
  • The fundamental equations of fluid statics are written as Exercise Problem - Fluid Statics - Mechanical Engineering with respect to a cartesian frame of reference with x - y plane as horizontal and axis z being directed vertically upwards. For an incompressible fluid, pressure P at a depth h below the free surface can be written as p = Po + ρ gh, where Po is the local atmospheric pressure.
  • At sea-level, the international standard atmospheric pressure has been chosen as Patm = 101.32 kN/m2. The pressure expressed as the difference between its value and the local atmospheric pressure is known as gauge pressure.
     
  • Piezometer tube measures the gauge pressure of a flowing liquid in terms of the height of liquid column. Manometers are devices in which columns of a suitable liquid are used to measure the difference in pressure between two points or between a certain point and the atmosphere. A simple U-tube manometer is modified as inclined tube manometer, inverted tube manometer and micro manometer to measure a small difference in pressure through a relatively large deflection of liquid columns.
     
  • The hydrostatic force on anyone side of a submerged plane surface is equal to the product of the area and the pressure at the centre of area. The force acts in a direction perpendicular to the surface and its point of action, known as pressure centre, is always at a higher depth than that at which the centre of area lies. The distance of centre of pressure from the centre of area along the axis of symmetry is given by     Exercise Problem - Fluid Statics - Mechanical Engineering
  • For a curved surface, the component of hydrostatic force in any horizontal direction is equal to the hydrostatic force on the projected plane surface on a vertical plane perpendicular to that direction and acts through the centre of pressure for the projected plane area. The vertical component of hydrostatic force on a submerged curved surface is equal to the weight of the liquid volume vertically above the submerged surface to the level of the free surface of liquid and acts through the centre of gravity of the liquid in that volume.
     
  • When a solid body is either wholly or partially immersed in a fluid, the hydrostatic lift due to net vertical component of the hydrostatic pressure forces experienced by the body is called the buoyant force. The buoyant force on a submerged or floating body is equal to the weight of liquid displaced by the body and acts vertically upward through the centroid of displaced volume known as centre of buoyancy.
     
  • The equilibrium of floating or submerged bodies requires that the weight of the body acting through its centre of gravity has to be colinear with an equal buoyant force acting through the centre of buoyancy. A submerged body will be in stable, unstable or neutral equilibrium if its centre of gravity is below, above or coincident with the centre of buoyancy respectively. Metacentre of a floating body is defined as the point of intersection of the centre line of cross-section containing the centre of gravity and centre of buoyancy with the vertical line through new centre of buoyancy due to any small angular displacement of the body. For stable equilibrium of floating bodies, metacentre M has to be above the centre of gravity G. M coinciding with G or lying below G refers to the situation of neutral and unstable equilibrium respectively. The distance of metacentre from centre of gravity along the centre line of cross-section is known as metacentric height and is given by.

 Exercise Problem - Fluid Statics - Mechanical Engineering

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FAQs on Exercise Problem - Fluid Statics - Mechanical Engineering

1. What is fluid statics?
Ans. Fluid statics is a branch of fluid mechanics that deals with the study of fluids at rest. It focuses on the behavior and equilibrium of fluids under various conditions without considering their motion or flow.
2. What are the principles of fluid statics?
Ans. The principles of fluid statics include Pascal's law, Archimedes' principle, and hydrostatic pressure. Pascal's law states that a change in pressure at any point in an enclosed fluid is transmitted equally in all directions. Archimedes' principle explains the buoyant force exerted on a submerged object, which is equal to the weight of the fluid displaced. Hydrostatic pressure refers to the pressure exerted by a fluid at rest and is dependent on the depth and density of the fluid.
3. How is hydrostatic pressure calculated?
Ans. Hydrostatic pressure can be calculated using the formula P = ρgh, where P is the hydrostatic pressure, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the depth of the fluid. This equation shows that the hydrostatic pressure increases with an increase in density, depth, or acceleration due to gravity.
4. What is the significance of fluid statics in mechanical engineering?
Ans. Fluid statics plays a crucial role in various aspects of mechanical engineering. It helps in designing and analyzing hydraulic systems, such as hydraulic lifts or jacks. The principles of fluid statics are also applied in the design and stability analysis of ships, submarines, and other floating structures. Additionally, fluid statics is essential in understanding the behavior of fluids in storage tanks, dams, and other structures that involve fluid containment.
5. How does fluid statics relate to atmospheric pressure?
Ans. Fluid statics is closely related to atmospheric pressure. Atmospheric pressure is the pressure exerted by the Earth's atmosphere on any surface within it. The concept of hydrostatic pressure can be applied to atmospheric pressure as well. The pressure at any point in a fluid, including the atmosphere, is influenced by the weight of the fluid above it. Therefore, changes in atmospheric pressure can affect the behavior of fluids in various applications like weather systems, barometers, and vacuum systems.
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