Lecture no 11 & 12

HYDROSTATIC FORCE AND PRESSURE ON PLATES

Prepared by

Engr.Sarafaraz Khan Turk

Lecturer at IBT LUMHS Jamshoro

HYDROSTATIC FORCE AND PRESSURE

Fluid statics or hydrostatics is the branch of

fluid mechanics that studies fluids at rest. It

embraces the study of the conditions under

which fluids are at rest in stable equilibrium; and

is contrasted with fluid dynamics, the study of

fluids in motion

HYDROSTATIC FORCE AND PRESSURE

The hydrostatic pressure can be determined from a control

volume analysis of an infinitesimally small cube of fluid. Since

pressure is defined as the force exerted on a test area (p = F/A,

with p: pressure, F: force normal to area A, A: area), and the

only force acting on any such small cube of fluid is the weight of

the fluid column above it, hydrostatic pressure can be calculated

according to the following formula:

HYDROSTATIC FORCE AND PRESSURE

where:

p is the hydrostatic pressure (Pa),

ρ is the fluid density (kg/m3),

g is gravitational acceleration (m/s2),

A is the test area (m2),

z is the height (parallel to the direction of gravity) of

the test area (m),

z0 is the height of the zero reference point of the

pressure (m).

Scuba Diving and Hydrostatic Pressure

Deep-sea divers realize that water

pressure increases as they dive deeper.

This is because the weight of the water above them increases.

HYDROSTATIC FORCE AND PRESSURE

Pressure on diver at

100 ft?

Danger of emergency

ascent?

,2 3 2

,2 ,2

1998 9.81 100

3.28

1298.5 2.95

101.325

2.95 1 3.95

gage

abs gage atm

kg m mP gz ft

m s ft

atmkPa atm

kPa

P P P atm atm atm

Scuba Diving and Hydrostatic Pressure

1 1 2 2

1 2

2 1

3.954

1

PV PV

V P atm

V P atm

100 ft

1

2

Boyle’s law

If you hold your breath on ascent, your lungvolume would increase by a factor of 4, which would result in embolism and/or death.

Suppose that a thin plate with area A m2 is

submerged in a fluid of density ρ kg/m3 at a

depth d meters below the surface of the fluid.

HYDROSTATIC FORCE AND PRESSURE

The fluid directly above the plate has volume

V = Ad

So, its mass is:

m = ρV = ρAd

HYDROSTATIC FORCE AND PRESSURE

Thus, the force exerted by the fluid on

the plate is

F = mg = ρgAd

where g is the acceleration due to gravity.

HYDROSTATIC FORCE

The pressure P on the plate is defined

to be the force per unit area:

HYDROSTATIC PRESSURE

FP gd

A

The SI unit for measuring pressure is newtons

per square meter—which is called a pascal

(abbreviation: 1 N/m2 = 1 Pa).

As this is a small unit, the kilopascal (kPa) is often used.

HYDROSTATIC PRESSURE

For instance, since the density of water is

ρ = 1000 kg/m3, the pressure at the bottom

of a swimming pool 2 m deep is:

HYDROSTATIC PRESSURE

3 21000kg/m 9.8m/s 2m

19,600Pa

19.6kPa

P gd

An important principle of fluid pressure is

the experimentally verified fact that, at any

point in a liquid, the pressure is the same in

all directions.

This is why a diver feels the same pressure on nose and both ears.

HYDROSTATIC PRESSURE

Thus, the pressure in any direction at

a depth d in a fluid with mass density ρ

is given by:

HYDROSTATIC PRESSURE

P gd d

Equation 1

This helps us determine the hydrostatic

force against a vertical plate or wall or dam

in a fluid.

This is not a straightforward problem.

The pressure is not constant, but increases as the depth increases.

HYDROSTATIC FORCE AND PRESSURE

A dam has the shape of the trapezoid

shown below.

The height is 20 m. The width is 50 m at the top and 30 m at the bottom.

HYDROSTATIC F AND P Example 1

Find the force on the dam due to

hydrostatic pressure if the water level

is 4 m from the top of the dam.

HYDROSTATIC F AND P Example 1

We choose a vertical x-axis with origin

at the surface of the water.

HYDROSTATIC F AND P Example 1

The depth of the water is 16 m.

So, we divide the interval [0, 16] into subintervals of equal length with endpoints xi.

We choose xi* [xi–1, xi].

HYDROSTATIC F AND P Example 1

The i th horizontal strip of the dam is

approximated by a rectangle with height Δx

and width wi

HYDROSTATIC F AND P Example 1

From similar triangles,

HYDROSTATIC F AND P

* *

*

1610or 8

16 20 2 2i i

i

x xaa

x

Example 1

Hence,

*12

*

2(15 )

2(15 8 )

46

i

i

i

w a

x

x

HYDROSTATIC F AND P Example 1

If Ai is the area of the strip, then

If Δx is small, then the pressure Pi on the i th

strip is almost constant, and we can use

Equation 1 to write:

HYDROSTATIC F AND P

*(46 )i i iA w x x x

*1000i iP gx

Example 1

The hydrostatic force Fi acting on the i th

strip is the product of the pressure and

the area:

HYDROSTATIC F AND P

* *1000 (46 )

i i i

i i

F PA

gx x x

Example 1

Adding these forces and taking the limit as

n → ∞, the total hydrostatic force on the dam

is:

HYDROSTATIC F AND P

* *

1

16

0

16 2

0

1632 7

0

lim 1000 (46 )

1000 (46 )

1000(9.8) (46 )

9800 23 4.43 10 N3

n

i ini

F gx x x

gx x dx

x x dx

xx

Example 1