Lecture Notes
Engineering Sciences
DEC1013
TOPIC 11
HYDRAULICS/ HYDROSTATICS
Learning Outcomes
At the end of this chapter, student should be able to :
1. State Pascal’s law
2. understand that fluids are incompressible
3. recall and use the equation of pressure and force ratio
4. State Archimedes’ principle, density and relative density
5. Recall the instruments to measure specific gravity (or relative density)
11. HYDRAULICS AND HYDROSTATICS





Pascal’s law
incompressibility of fluids
pressure and force ratio
Archimedes principle
density and relative density
11.1 Definition:
Pascal's Law (also called Pascal's Principle) says that "changes in pressure at any point in
an enclosed fluid at rest are transmitted undiminished to all points in the fluid and act in
all directions."
This means,
1. if the weight of a fluid is neglected, the pressure throughout an enclosed volume
will be the same,
2. the static pressure in a fluid acts equally in all directions,
3. the static pressure acts at right angles to any surface in contact with the fluid.
DEC1013
Rev#:02
Date : 01 Feb 2014
Sem 1
Page 1
Lecture Notes
Engineering Sciences
DEC1013
11.2 Pascal’s law formula,
𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 =
𝑓𝑜𝑟𝑐𝑒
𝐹
→𝑃=
𝑎𝑟𝑒𝑎
𝐴
For the diagram above, use this formula:
𝑃=
𝐹1
𝐹2
=
𝐴1
𝐴2
In fluid mechanics, incompressible fluid is a fluid which is not reduced in volume by an
increase in pressure.
11.3 Pressure and force ratio
DEC1013
Rev#:02
Date : 01 Feb 2014
Sem 1
Page 2
Lecture Notes
Engineering Sciences
DEC1013
An open tube of length L = 1.8 m and cross-sectional area A = 4.6 cm2 is fixed to the top
of a cylindrical barrel of diameter D= 1.2m and height H=1.8 m
The barrel and the tube are filled with water (to the top of the tube). Calculate the ratio
of hydrostatic force on the bottom of the barrel to the gravitational force on the water
contained in the barrel. Ignore atmospheric pressure for this question.
2. Relevant equations
P= patm + ρgh
Weight=mg=ρVg
3. The attempt at a solution
So I tried calculating the volume of the cylinder, tube then substitute it into the equation
Weight=mg=ρVg
DEC1013
Rev#:02
Date : 01 Feb 2014
Sem 1
Page 3
Lecture Notes
Engineering Sciences
DEC1013
The answer is supposed to be 2.
Did i convert the Area to Radius wrong? Or is there another way I’m supposed to go
about this?
I set up my ratio as = Weight of Barrel + Tube / Weight of barrel only
V of barrel = π * (0.6)2 * 1.8 = 2.03575204
I substitute this into W=mg=ρVg = 1000 g/m3 * 2.03575204 * 9.8 m/s2
W=19950 N
DEC1013
Rev#:02
Date : 01 Feb 2014
Sem 1
Page 4
Lecture Notes
Engineering Sciences
DEC1013
11.4 Archimedes' principle
Archimedes' principle relates buoyancy to displacement. It is named after its discoverer,
Archimedes of Syracuse. Archimedes' treatise On floating bodies, proposition 5, states
that
Any floating object displaces its own weight of fluid.
For general objects, floating and sunken, and in gasses as well as liquids (i.e. a fluid),
Archimedes' principle may be stated thus in terms of forces:
Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the
weight of the fluid displaced by the object.
with the clarifications that for a sunken object the volume of displaced fluid is the
volume of the object, and for a floating object on a liquid, the weight of the displaced
liquid is the weight of the object.
In short, buoyancy = weight of displaced fluid.
Consequently, Archimedes’s principle give rise to the principle of buoyancy (apungan/
julangan).
Assuming Archimedes' principle to be reformulated as follows,
then inserted into the quotient of weights, which has been expanded by the mutual volume
DEC1013
Rev#:02
Date : 01 Feb 2014
Sem 1
Page 5
Lecture Notes
Engineering Sciences
DEC1013
yields the formula below. The density of the immersed object relative to the density of the
fluid can easily be calculated without measuring any volumes:
Example 1: If you drop wood into water, buoyancy will keep it afloat.
Example 2: A helium balloon in a moving car. In increasing speed or driving a curve, the
air moves in the opposite direction of the car's acceleration. The balloon however, is
pushed due to buoyancy "out of the way" by the air, and will actually drift in the same
direction as the car's acceleration.
11.5 Density, ρ and Relative Density, ρr (Specific Gravity, S.G.)
The relative density (or specific gravity) of a material is the ratio of the mass (or weight)
of a certain sample of it to the mass (or weight) of an equal volume of water, the
conventional reference material. In the metric system, the density of water is 1 g/cc,
which makes the specific gravity numerically equal to the density. Strictly speaking,
density has the dimensions g/cc, while specific gravity is a dimensionless ratio. Relative
density has no unit.
Things are complicated by the variation of the density of water with temperature, and
also by the confusion that gave us the distinction between cc and ml. The milliliter is the
volume of 1.0 g of water at 4°C, by definition. The actual volume of 1.0 g of water at 4°C
is 0.999973 cm3 by measurement. Since most densities are not known, or needed, to
more than three significant figures, it is clear that this difference is of no practical
importance, and the ml can be taken equal to the cc. The density of water at 0°C is
0.99987 g/ml, at 20° 0.99823, and at 100°C 0.95838. The temperature dependence of the
density may have to be taken into consideration in accurate work. Mercury, while we are
at it, has a density 13.5955 at 0°C, and 13.5461 at 20°C.
DEC1013
Rev#:02
Date : 01 Feb 2014
Sem 1
Page 6
Lecture Notes
Engineering Sciences
DEC1013
The basic idea in finding specific gravity is to weigh a sample in air, and then immersed in
water. Then the specific gravity is W/(W - W'), if W is the weight in air, and W' the weight
immersed. The denominator is just the buoyant force, the weight of a volume of water
equal to the volume of the sample. This can be carried out with an ordinary balance, but
special balances, such as the Jolly balance, have been created specifically for this
application. Adding an extra weight to the sample allows measurement of specific
gravities less than 1.
How do we find specific gravity?
The specific gravity of a liquid can be found with a collection of small weighted, hollow
spheres that will just float in certain specific gravities. The closest spheres that will just
float and just sink put limits on the specific gravity of the liquid. This method was once
used in Scotland to determine the amount of alcohol in distilled liquors. Since the density
of a liquid decreases as the temperature increases, the spheres that float are an
indication of the temperature of the liquid. Galileo's thermometer worked this way.
11.6 Instruments to measure specific gravity (or relative density)
1. The hydrometer, which consists of a weighted float and a calibrated stem that
protrudes from the liquid when the float is entirely immersed. A higher specific gravity
will result in a greater length of the stem above the surface, while a lower specific gravity
will cause the hydrometer to float lower.
2. The pycnometer.
A pycnometer (from Greek: πυκνός (puknos) meaning "dense"), also called pyknometer
or specific gravity bottle, is a device used to determine the density of a liquid. A
pycnometer is usually made of glass, with a close-fitting ground glass stopper with a
capillary tube through it, so that air bubbles may escape from the apparatus. A
pcynometer or specific gravity bottle is a flask with a stopper that has a capillary tube
DEC1013
Rev#:02
Date : 01 Feb 2014
Sem 1
Page 7
Lecture Notes
Engineering Sciences
DEC1013
through it, which allows air bubbles to escape. The pycnometer is used to obtain
accurate measurements of density.
THANK YOU
DEC1013
Rev#:02
Date : 01 Feb 2014
Sem 1
Page 8