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Hydrostatics is the branch of fluid mechanics that studies incompressible fluids at rest.

Density

The density of a substance is its mass per unit volume.

Relative density

the ratio of the density of a substance to the density of a standard, usually water for a liquid.

Pressure in a static fluid When a fluid is at rest , a force only acts at right angles.This is because a fluid cannot sustain a tangental force .When a fluid at rest forces simply act normal to the surface .

p={\frac {F}{A}}

where $p$ is the pressure, $F$ is the normal force, $A$ is the area of the surface on contact.

Pressure is a scalar quantity ,it has no fixed directions.The direction of the force exerted by the liquid is decided by the orientation of area present to the liquid.

Expression for fluid pressure

If we consider atmospheric pressure , say it’s P_o , then the pressure at depth h is given by P = P_o + hρg

In a static liquid, pressures at same level in a same liquid are equal.

Pressure of atmosphere

Our Earth is surrounded by air upto a considerable height.This envelope of air surrounding the Earth is called atmosphere.Since air has weight, a column of air is capable of exerting pressure.It is first measured by Toricelli.

Image result for simple barometer Atmospheric pressure P_o = hρg

1 atm=1.014 bar= 760 mmHg 1 torr = 1mmHg

Manometer Manometer consist of U tube, it is used to find the difference in pressures between the gas enclosed in a vessel and atmospheric pressure.

Pressure at x = Pressure at y
Unknown Pressure P = P_o + hρg

Determination of density of coconut oil using U tube

First take the water in the U tube then add coconut oil to one of the arm and give some time to settle.After it settles measure the height of coconut oil(h2) and water(h1) from common interface. To take several measurements add coconut oil and measure the heights from the common interface.

First take water and then take coconut oil.Otherwise coconut oil floats in both arms of U tube.
To take several readings we have to add less dense liquid for corresponding arm otherwise always h1 and h2 will be same.
For miscible liquids we cannot use the above method.

Hare’s apparatus

P0 – atmospheric pressure

Arrange the above setup as shown in the figure .Suck the air from the tube,then the liquids in both arm rises upto maximum level.Then measure the heights hw and hl.Then reduce heights and take several sets of reading.

Pascal’s law

Pascal’s law states that a pressure change occurring anywhere in a confined incompressible fluid is transmitted throughout the fluid such that the same change occurs everywhere.

Incompressible fluid means due to the pressure the density of the fluid

The pressure acting on both pistons in a hydraulic jack is equal.

hydraulic lift

The force equation for the small cylinder F_s = p A_s

where F_s = force acting on the piston in the small cylinder (N) , A_s = area of small cylinder (m²) , p = pressure in small and large cylinder (Pa, N/m²)

The force equation for the large cylinder F_l = p A_l

where F_l = force acting on the piston in the large cylinder (N) , A_l = area of large cylinder (m²) , p = pressure in small and large cylinder (Pa, N/m²)

F_s / A_s= F_l / A_l

F_s = F_l A_s/ A_l

Upthrust

A fluid will exert a force upward on a body if it is partly or wholly submerged within it. This is because the deeper into a fluid you go, the greater the weight of it and so the greater the pressure. This difference in pressure between the top and the bottom of the object produces an upward force on it. This is called Upthrust.
- According to Archimedes’ Principle, the upthrust on an object in a fluid is equal to the weight of the fluid displaced.
Upthrust = Weight of Fluid Displaced = mg = vρg

Upthrust = vρg

Archimede’s principle

It states that when a body is fully or partially emerged in an incompressible fluid at rest, it experiences an upward force equal to the weight of the fluid displaced.

When an object is placed on the surface of a liquid it will either float or sink. This depends upon two forces:

The weight W₁ acting vertically downwards, which is due to the gravitational pull of the earth

Upthrust acting vertically upwards, which is equal to the weight of the liquid displaced by the object W₂.

You know that the weight of an object is the product of its volume and its density. Hence,

W₁ = Volume of the object (V) x Density of the object (d₁)

W₂ = Volume of the liquid displaced (V) x Density of the liquid (d₂)

Let us see what are the different situations under which a body floats, sinks or remains submerged completely at any level in the liquid.

Case 1:

When the weight of the object W₁ is equal to the weight of the liquid displaced W₂, the object will stay in the position of rest completely immersed in the liquid. Here,

W₁ = W₂ or

V x d₁ = V x d₂ or d₁ = d₂, then the object will be floating completely immersed in a liquid as shown in figure below.

Weight of the object is equal to the weight of the liquid

Case 2:

When the weight of the object is greater than the weight of the displaced liquid, the object will sink. Here,

Thus, when the density of the object is greater than the density of the liquid, the object will sink.

Weight of the object is greater than the weight of the liquid displaced.

Case 3:-

When the weight of the object is less than the buoyant force (upthrust), the object will float on the surface of the liquid.

When the object floats, only a part of it is submerged in the liquid and volume of the liquid displaced will be less than its own volume. Let V’ be the volume of the liquid displaced by the submerged part of the object. Here

Density of the solid is less than the density of the liquid and hence the object will float.

Weight of the object is less than the weight of the liquid displaced

Hydrometer

Hydrometer is an instrument used for determining the density of a liquid. It usually consists of a glass float with a long thin stem which is graduated. The glass float is a large hollow bulb which increases the buoyancy so that the hydrometer floats. The narrow stem increases the sensitivity of the hydrometer. The bottom of the hydrometer is made heavier by loading it with lead shots so that it floats vertically.

Steady flow All the fluid particles that pass any given point, follow the same path at same speed. In steady flow streamlines never cross each other.

Laminar flow In laminar flow, the velocities of all the particles on any given streamline are equal.

Turbulent flow Above a certain critical speed,fluid become turbulent .It is an irregular flow ,we can’t predict the motion.

Incompressible fluid

In incompressible fluid change in pressure produces no change in density of the fluid. Liquids can be considered incompressible and gases can be considered for small pressure differences.

Viscous force

When two layers of liquid are moving with different velocities they experience tangental forces which tend to retard the faster layer and accelerate the slower layer.These forces are called viscous forces.

Ideal fluid flow means a fluid incompressible,non viscous at streamline flow.

Equation of continuity

If a fluid is undergoing streamline flow then the mass of fluid which enters one end of a tube of flow must be equal to the mass that leaves at the other end during same time.

Example – Equation of Continuity

10 m³/h of water flows through a pipe with 100 mm inside diameter. The pipe is reduced to an inside dimension of 80 mm.

Using equation (2) the velocity in the 100 mm pipe can be calculated

(10 m³/h) (1 / 3600 h/s) = v₁₀₀ (3.14 (0.1 m)² / 4)

or

v₁₀₀ = (10 m³/h) (1 / 3600 h/s) / (3.14 (0.1 m)² / 4)

= 0.35 m/s

Using equation (2) the velocity in the 80 mm pipe can be calculated

(10 m³/h) (1 / 3600 h/s) = v₈₀ (3.14 (0.08 m)² / 4)

or

v₈₀ = (10 m³/h) (1 / 3600 h/s) / (3.14 (0.08 m)² / 4)

= 0.55 m/s

Energy possessed by a fluid

Fluid Kinetic Energy

The kinetic energy of a moving fluid is more useful in applications like the Bernoulli equation when it is expressed as kinetic energy per unit volume

Fluid Potential Energy

The potential energy of a moving fluid is more useful in applications like the Bernoulli equation when is expressed as potential energy per unit volume

Pressure as Energy Density

Pressure in a fluid may be considered to be a measure of energy per unit volume or energy density. For a force exerted on a fluid, this can be seen from the definition of pressure:

Bernoulli’s principle

The Bernoulli Equation can be considered to be a statement of the conservation of energy principle appropriate for flowing fluids.

Applications

Bernoulli’s principle can be used to calculate the lift force on an airfoil, if the behaviour of the fluid flow in the vicinity of the foil is known. For example, if the air flowing past the top surface of an aircraft wing is moving faster than the air flowing past the bottom surface, then Bernoulli’s principle implies that the pressure on the surfaces of the wing will be lower above than below. This pressure difference results in an upwards lifting force. Whenever the distribution of speed past the top and bottom surfaces of a wing is known, the lift forces can be calculated (to a good approximation) using Bernoulli’s equations established by Bernoulli over a century before the first man-made wings were used for the purpose of flight. Bernoulli’s principle does not explain why the air flows faster past the top of the wing and slower past the underside. See the article on aerodynamic lift for more info.

The carburettor used in many reciprocating engines contains a venturi to create a region of low pressure to draw fuel into the carburettor and mix it thoroughly with the incoming air. The low pressure in the throat of a venturi can be explained by Bernoulli’s principle; in the narrow throat, the air is moving at its fastest speed and therefore it is at its lowest pressure.
An injector on a steam locomotive (or static boiler).
The pitot tube and static port on an aircraft are used to determine the airspeed of the aircraft.
The flow speed of a fluid can be measured using a device such as a Venturi meter or an orifice plate, which can be placed into a pipeline to reduce the diameter of the flow. For a horizontal device, the continuity equation shows that for an incompressible fluid, the reduction in diameter will cause an increase in the fluid flow speed. Subsequently, Bernoulli’s principle then shows that there must be a decrease in the pressure in the reduced diameter region. This phenomenon is known as the Venturi effect.

Airfoil

One of the most common everyday applications of Bernoulli’s principle is in air flight. The main way that Bernoulli’s principle works in air flight has to do with the architecture of the wings of the plane. In an airplane wing, the top of the wing is somewhat curved, while the bottom of the wing is totally flat. While in the sky, air travels across both the top and the bottom concurrently. Because both the top part and the bottom part of the plane are designed differently, this allows for the air on the bottom to move slower, which creates more pressure on the bottom, and allows for the air on the top to move faster, which creates less pressure. This is what creates lift, which allows planes to fly. An airplane is also acted upon by a pull of gravity in which opposes the lift, drag and thrust. Thrust is the force that enables the airplane to move forward while drag is air resistance that opposes the thrust force.

Venturi meter

The pressure in the first measuring tube (1) is higher than at the second (2), and the fluid speed at “1” is lower than at “2”, because the cross-sectional area at “1” is greater than at “2”.

A flow of air through a venturi meter, showing the columns connected in a manometer and partially filled with water. The meter is “read” as a differential pressure head in cm or inches of water

The Venturi effect is the reduction in fluid pressure that results when a fluid flows through a constricted section of a pipe.

In fluid dynamics, a fluid’s velocity must increase as it passes through a constriction in accord with the principle of mass continuity, while its static pressure must decrease in accord with the principle of conservation of mechanical energy. Thus any gain in kinetic energy a fluid may accrue due to its increased velocity through a constriction is balanced by a drop in pressure.

By measuring the change in pressure, the flow rate can be determined, as in various flow measurement devices such as venturi meters, venturi nozzles and orifice plates.

Using Bernoulli’s equation in the special case of steady, incompressible, non viscous flows along a streamline, the theoretical pressure drop at the constriction is given by:

p_{1}-p_{2}={\frac {\rho }{2}}\left(v_{2}^{2}-v_{1}^{2}\right)

where $\scriptstyle \rho \,$ is the density of the fluid, $\scriptstyle v_{1}$ is the (slower) fluid velocity where the pipe is wider, $\scriptstyle v_{2}$ is the (faster) fluid velocity where the pipe is narrower (as seen in the figure)

Just before collision
During collision
After collision

Law of conservation of linear momentum

The total linear momentum of a system of interacting bodies on which no external forces are acting remains constant.

Elastic collisions

During collisions, total kinetic energy just before the collision is equal to total kinetic energy just after the collision.

Inelastic collisions

Total kinetic energy is not conserved just before or after collision

Complete inelastic collisions

After the collision both objects move together.

Linear momentum

Linear momentum is a vector quantity defined as the product of an object’s mass m and its velocity v. Linear momentum is denoted by the letter p .

Note that a body’s momentum is always in the same direction as its velocity vector. The units of momentum are kg m/s.

p=mv

Law of conservation of linear momentum

The total linear momentum of a system of interacting bodies on which no external forces are acting remains constant.

Collisions

Let a constant force (f) act in the same direction in which the particle is moving . Initial and final velocities are u and v,time taken – t. Ft is an important quantity in dynamics.It has been named as impulse.

A large force(F) acting in a very small time interval(t) Then Impulse I = F×t

Impulse is a vector quantity.Direction of impulse is the direction of force.

I=Ft=Δ(mv) Unit – kgms^-1

Applications

(i) A cricket player lowers has hands while catching the ball : by doing so the time of impact increases and hence the effect of force decreases.

(ii) When a person falls from a certain height on floor, he receives more injuries as compared to falling on a heap of sand. It is because the Cemented floor does not yield whereas the sand yield there by increasing the time of impact hence decreasing the impact of force.

(iii) The shock absorbers provided in the vehicle helps to travel smoothly on an uneven road. It is because the shockers increases the time of impulse which reduces the force.

In a force – time graph area between the graph and the time axis gives the impulse.

Area=F×t=impulse

Force due to water flow

Conveyor belt

Force on helicopter blades

Length – Meter ruler , Vernier caliper , Micrometer screw guage , Spectrometer , Travelling microscope

Mass – Electronic balance , Triple beam balance

Time – Stop watch , Digital clock

Least count

It is the least value of physical quantity which can be measured using a measuring instrument.

Vernier caliper

Internal jaws – for measuring inner dimensions

External jaws – for measuring outer dimensions

Depth bar – for measuring depths

Least count = 0.1 mm

Reading =main scale reading + vernier scale coincide × least count = 100+2×0.1=100.2 mm

Zero error and correction

The instrument is said to have if the zero of the main scale doesn’t coincide with the zero of the vernier scale when the two jaws of vernier caliper are brought into contact.

Zero error = 0.3 mm

Correction = -0.3 mm

Zero error = 0.8 mm

Correction = +0.8 mm

Micrometer screw guage

Pitch – It is the linear distance moved by thimble along the main scale when the thimble is given one rotation.

Least count = Pitch/Number of divisions in the circular scale

Main scale – 0.5 mm divisions Least count=0.01mm

Reading = 2.5 + 38 ×0.01 = 2.88 mm

Zero error and correction

Spherometer

Pitch = It’s the linear distance moved along main scale when circular scale is given one completed rotation.

Least count = Pitch/Number of divisions in the circular scale.thDetermination of the radius of curvature of a spherical surface

Take a plain glass block and keep the spherometer on it and turn the screw until tip of the screw just touch the glass block and take the reading(h₁)
Then keep the curved surface on the glass block
Keep the spherometer on the curved surface and turn the screw until it just touch the curved surface and take the reading(h₂)
The height moved by the screw h = h₂-h₁

R = a²/6h + h/2

a = distance between two legs

Travelling microscope

The principle component of a travelling microscope is it’s microscope. It enlarges the diameter so measurement can be done easily.

To find the internal diameter of the capillary tube

Place the capillary tube horizontally on the adjustable stand.
Focus the microscope on the end dipped in water.
Make the horizontal cross- wire touch the inner circle at A (fig i). Note microscope reading on the vertical scale.
Raise the microscope to make the horizontal cross wire touch the circle at B (fig ii). Note the vertical scale reading.
The difference between the two readings will give the vertical internal diameter (AB) of the tube.
Move the microscope on the horizontal scale and make the vertical cross wire touch the inner circle at C (fig iii). Note microscope reading on the horizontal scale.
Move the microscope to the right to make the vertical cross wire touch the circle at D (fig iv). Note the horizontal scale reading.
The difference between the two readings will give the horizontal internal diameter (CD) of the tube.

We can calculate the diameter of the tube by calculating the mean of the vertical and horizontal internal diameters. Half of the diameter will give the radius of the capillary tube.

There are three types of physical quantities.

Fundamental physical quantities
Supplementary physical quantities
Derived physical quantities

The internationally accepted standard units and dimensions of the physical quantities are given below.

Fundamental physical quantities SI unit Dimensions

Mass kg M
Time s T
Length m L

Temperature K
Electric current A
Amuont of substance mol
Luminous intensity Cd

Supplementary physical quantities

Angle in a plane rd
Solid angle Sr

Derived physical quantities

Physical quantity	Unit	Dimesion
Speed	ms^-1	LT^-1
Velocity	ms^-1	LT^-1
Acceleration	ms^-2	LT^-2
Pressure	kgm^-1s^-2 =Pa	ML^-1T^-2
Work	kgm²s^-2=J	ML²T^-2
Energy	kgm²s^-2=J	ML²T^-2
Power	kgm²s^-3=W	ML²T^-3
Frequency	s^-1=Hz	T^-1
Area	m²	L²
Volume	m³	L³
Density	kgm^-3	ML^-3

Pysical quantities which can be defined in terms of fundamental quantities are called derived quantities.

Scalar quantities

Physical quantities which has magnitude are called scalars.

Time
Mass
Length
Distance
Work

Vector quantities

Physical quantities which have magnitude and direction are called vectors.

Displacement
Velocity
Force

Any vector can be represented by a segment of straight line.Length of the straight line represents the magnitude of the vector and direction of the straight line represents the direction of vector.

Vector addition

If we consider atmospheric pressure , say it’s Po , then the pressure at depth h is given by P = Po + hρg