Mechanical Engineering Principles

HYDROSTATICS 241

Hence, y′=

∫

ρgy^2 sinθdA

ρgsinθ

∫

ydA

=

ρgsinθ

∫

y^2 dA

ρgsinθ

∫

ydA

=

(Ak^2 )Ox

Ay

where(Ak^2 )Ox=the second moment of area
aboutOx
k=the radius of gyration fromO.

Now try the following exercise

Exercise 110 Further problems on hydro-

static pressure on sub-

merged surfaces

(Takeg= 9 .81 m/s^2 )

1. Determine the gauge pressure acting on
   the surface of a submarine that dives to
   a depth of 500 m. Take water density as
   1020 kg/m^3. [50.03 bar]


2. Solve Problem 1, when the submarine
   dives to a depth of 780 m. [78.05 bar]


3. If the gauge pressure measured on the sur-
   face of the submarine of Problem 1 were
   92 bar, at what depth has the submarine
   dived to? [919.4 m]


4. A tank has a flat rectangular end, which
   is of size 4 m depth by 3 m width. If
   the tank filled with water to its brim
   and the flat end is vertical, determine the
   thrust on this end and the position of its
   centre of pressure. Take water density as
   1000 kg/m^3. [0.235 MN; 2.668 m]


5. If another vertical flat rectangular end of
   the tank of Problem 4 is of size 6 m depth
   by 4 m width, determine the thrust on this
   end and position of the centre of pressure.
   The depth of water at this end may be
   assumed to be 6 m. [0.706 MN; 4 m]


6. A tank has a flat rectangular end, which
   is inclined to the horizontal surface, so
   thatθ = 30 °,whereθ is as defined in
   Figure 21.11, page 240. If this end is of
   size 6 m height and 4 m width, determine
   the thrust on this end and the position of

the centre of pressure from the top. The

tank may be assumed to be just full.

[0.353 MN; 2 m]

21.12 Hydrostatic thrust on curved

surfaces

As hydrostatic pressure acts perpendicularly to a

surface, the integration ofδFover the surface can be

complicated. One method of determining the thrust

on a curved surface is to project its area on flat

vertical and horizontal surfaces, as shown byAB

andDE, respectively, in Figure 21.12.

Free surface D E

G

A

B

FFx

−Fy

W

Figure 21.12

From equilibrium considerations,F =Fxand

W =Fyand these thrusts must act through the

centre of pressures of the respective vertical and hor-

izontal planes. The resultant thrust can be obtained

by addingFxandFyvectorially, where

W=weight of the fluid enclosed by the curved

surface and the vertical projection lines

to the free surface, and

G=centre of gravity ofW

21.13 Buoyancy

The upward force exerted by the fluid on a body

that is wholly or partially immersed in it is called

the buoyancy of the body.