Physics by Yixun

Wednesday 20 August 2014

Archimedes's Principle

Archimedes' principle indicates that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces. Archimedes' principle is a law of physics fundamental to fluid mechanics. Archimedes of Syracuse formulated this principle, which bears his name.

Explanation

In his treatise on hydrostatics, On Floating Bodies, Archimedes states:

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.

— Archimedes of Syracuse

Practically seen, the Archimedes principle allows the volume of an object to be measured by measuring the volume of the liquid it displaces after submerging, and the buoyancy of an object immersed in a liquid to be calculated.

For any immersed object, the volume of the submerged portion equals the volume of fluid it displaces. E.g., by submerging in water half of a sealed 1-liter container, we displace a half-liter volume of fluid, regardless of the container's contents. If we fully submerge the same container, we then displace one liter of liquid, which exactly equals the volume of the 1-liter container.

An empty 1-litre plastic bottle released in the air will fall down due to the gravitational force of the Earth acting on it. If the same bottle is released under water, the same gravitational force acts on it, but it will be pushed upwards towards the surface of the water. The extra force that pushes the bottle upwards comes from the upthrust or Archimedes force.

Formula

Cube immersed in a fluid, with its sides parallel to the direction of gravity. The fluid will exert a normal force on each face, and therefore only the forces on the top and bottom faces will contribute to buoyancy. The pressure difference between the bottom and the top face is directly proportional to the height (difference in depth). Multiplying the pressure difference by the area of a face gives the net force on the cube - the buoyancy, or the weight of the fluid displaced. By extending this reasoning to irregular shapes, we can see that, whatever the shape of the submerged body, the buoyant force is equal to the weight of the fluid displaced.

The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density). The weight of the object in the fluid is reduced, because of the force acting on it, which is called upthrust. In simple terms, the principle states that the buoyant force on an object is equal to the weight of the fluid displaced by the object, or the density of the fluid multiplied by the submerged volume times the gravitational constant, g. Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy.

Suppose a rock's weight is measured as 10 Newtons when suspended by a string in a vacuum with gravity acting on it. Suppose that when the rock is lowered into water, it displaces water of weight 3 Newtons. The force it then exerts on the string from which it hangs would be 10 Newtons minus the 3 Newtons of buoyant force: 10 − 3 = 7 Newtons. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea floor. It is generally easier to lift an object up through the water than it is to pull it out of the water.

For a fully submerged object, Archimedes' principle can be reformulated as follows,

\text{apparent immersed weight} = \text{weight of object} - \text{weight of displaced fluid}\,

then inserted into the quotient of weights, which has been expanded by the mutual volume

\frac { \text{density of object}} { \text{density of fluid} } = \frac { \text{weight}} { \text{weight of displaced fluid} }

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:

\frac { \text {density of object}} { \text{density of fluid} } = \frac { \text{weight}} { \text{weight} - \text{apparent immersed weight}}.\,

(This formula is used for example in describing the measuring principle of a dasymeter and of hydrostatic weighing.)

Example: If you drop wood into water, buoyancy will keep it afloat.

Example: A helium balloon in a moving car. When increasing speed or driving in a curve, the air moves in the opposite direction to the car's acceleration. However, due to buoyancy, the balloon is pushed "out of the way" by the air, and will actually drift in the same direction as the car's acceleration.

When an object is immersed in a liquid, the liquid exerts an upward force, which is known as the buoyant force, that is proportional to the weight of the displaced liquid. The sum force acting on the object, then, is proportional to the difference between the weight of the object ('down' force) and the weight of displaced liquid ('up' force), hence equilibrium buoyancy is achieved when these two weights (and thus forces) are equal consider a ball immersed in a liquid.the liquid experiences an upthrust which is the buoyant force.in otherwise is proportional to the weight of the liquid displaced.the total force acting on the object at thatb point in time is proportional to the difference between the weight exacted by the object and the weight of the displaced liquid hence equilibrium is attend.

Here's some interesting website to visit!

http://en.wikipedia.org/wiki/Archimedes'_principle

https://www.grc.nasa.gov/www/k-12/WindTunnel/Activities/buoy_Archimedes.html

http://global.britannica.com/EBchecked/topic/32827/Archimedes-principle

Bernoulli's Principle(Compressible flow equation and applications)

Bernoulli developed his principle from his observations on liquids, and his equation is applicable only to incompressible fluids, and compressible fluids up to approximately Mach number 0.3. It is possible to use the fundamental principles of physics to develop similar equations applicable to compressible fluids. There are numerous equations, each tailored for a particular application, but all are analogous to Bernoulli's equation and all rely on nothing more than the fundamental principles of physics such as Newton's laws of motion or the first law of thermodynamics.

Compressible flow in fluid dynamics

For a compressible fluid, with a barotropic equation of state, and under the action of conservative forces,

\frac {v^2}{2}+ \int_{p_1}^p \frac {d\tilde{p}}{\rho(\tilde{p})}\ + \Psi = \text{constant}

(constant along a streamline)

where:

p is the pressure

ρ is the density

v is the flow speed

Ψ is the potential associated with the conservative force field, often the gravitational potential

In engineering situations, elevations are generally small compared to the size of the Earth, and the time scales of fluid flow are small enough to consider the equation of state asadiabatic. In this case, the above equation becomes

\frac {v^2}{2}+ gz+\left(\frac {\gamma}{\gamma-1}\right)\frac {p}{\rho} = \text{constant}

(constant along a streamline)

where, in addition to the terms listed above:

γ is the ratio of the specific heats of the fluid

g is the acceleration due to gravity

z is the elevation of the point above a reference plane

In many applications of compressible flow, changes in elevation are negligible compared to the other terms, so the term gz can be omitted. A very useful form of the equation is then:

\frac {v^2}{2}+\left( \frac {\gamma}{\gamma-1}\right)\frac {p}{\rho} = \left(\frac {\gamma}{\gamma-1}\right)\frac {p_0}{\rho_0}

where:

p₀ is the total pressure

ρ₀ is the total density

Compressible flow in thermodynamics

Another useful form of the equation, suitable for use in thermodynamics and for (quasi) steady flow, is:

{v^2 \over 2} + \Psi + w =\text{constant}

Here w is the enthalpy per unit mass, which is also often written as h (not to be confused with "head" or "height").

Note that

w = \epsilon + \frac{p}{\rho}

where ε is the thermodynamic energy per unit mass, also known as the specific internal energy.

The constant on the right hand side is often called the Bernoulli constant and denoted b. For steady inviscid adiabatic flow with no additional sources or sinks of energy, b is constant along any given streamline. More generally, when b may vary along streamlines, it still proves a useful parameter, related to the "head" of the fluid (see below).

When the change in Ψ can be ignored, a very useful form of this equation is:

{v^2 \over 2}+ w = w_0

where w₀ is total enthalpy. For a calorically perfect gas such as an ideal gas, the enthalpy is directly proportional to the temperature, and this leads to the concept of the total (or stagnation) temperature.

When shock waves are present, in a reference frame in which the shock is stationary and the flow is steady, many of the parameters in the Bernoulli equation suffer abrupt changes in passing through the shock. The Bernoulli parameter itself, however, remains unaffected. An exception to this rule is radiative shocks, which violate the assumptions leading to the Bernoulli equation, namely the lack of additional sinks or sources of energy.

Applications

In modern everyday life there are many observations that can be successfully explained by application of Bernoulli's principle, even though no real fluid is entirely inviscid and a small viscosity often has a large effect on the flow.

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. To understand why, it is helpful to understand circulation, the Kutta condition, and theKutta–Joukowski theorem.

The carburetor used in many reciprocating engines contains a venturi to create a region of low pressure to draw fuel into the carburetor 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.

The Pitot tube and static port on an aircraft are used to determine the airspeed of the aircraft. These two devices are connected to the airspeed indicator, which determines the dynamic pressure of the airflow past the aircraft. Dynamic pressure is the difference between stagnation pressure and static pressure. Bernoulli's principle is used to calibrate the airspeed indicator so that it displays the indicated airspeed appropriate to the dynamic pressure.

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.

The maximum possible drain rate for a tank with a hole or tap at the base can be calculated directly from Bernoulli's equation, and is found to be proportional to the square root of the height of the fluid in the tank. This is Torricelli's law, showing that Torricelli's law is compatible with Bernoulli's principle. Viscosity lowers this drain rate. This is reflected in the discharge coefficient, which is a function of the Reynolds number and the shape of the orifice.
In open-channel hydraulics, a detailed analysis of the Bernoulli theorem and its extension were recently (2009) developed. It was proved that the depth-averaged specific energy reaches a minimum in converging accelerating free-surface flow over weirs and flumes . Further, in general, a channel control with minimum specific energy in curvilinear flow is not isolated from water waves, as customary state in open-channel hydraulics.

The Bernoulli grip relies on this principle to create a non-contact adhesive force between a surface and the gripper.

Flutes and fipple flutes

For futhermore details,please visit the following pages.

http://en.wikipedia.org/wiki/Bernoulli's_principle

http://www.grc.nasa.gov/WWW/k-12/airplane/bern.html

Bernoulli's Principle(Incompressible flow equation)

In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.The principle is named after Daniel Bernoulli who published it in his book Hydrodynamica in 1738.

Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation. In fact, there are different forms of the Bernoulli equation for different types of flow. The simple form of Bernoulli's principle is valid forincompressible flows (e.g. most liquid flows) and also for compressible flows (e.g. gases) moving at low Mach numbers (usually less than 0.3). More advanced forms may in some cases be applied to compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation).

Bernoulli's principle can be derived from the principle of conservation of mechanical energy. This states that, in a steady flow, the sum of all forms of mechanical energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy and potential energy remain constant. Thus an increase in the speed of the fluid occurs proportionately with an increase in both itsdynamic pressure and kinetic energy, and a decrease in its static pressure and potential energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same on all streamlines because in a reservoir the energy per unit volume (the sum of pressure andgravitational potential ρ g h) is the same everywhere.

Bernoulli's principle can also be derived directly from Newton's 2nd law. If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline.

Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.

Incompressible flow equation

In most flows of liquids, and of gases at low Mach number, the density of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow. Therefore, the fluid can be considered to be incompressible and these flows are called incompressible flow. Bernoulli performed his experiments on liquids, so his equation in its original form is valid only for incompressible flow. A common form of Bernoulli's equation, valid at any arbitrary point along a streamline, is:

{v^2 \over 2}+gz+{p\over\rho}=\text{constant}

(A)

where:

v\,

is the fluid flow speed at a point on a streamline,

g\,

is the acceleration due to gravity,

z\,

is the elevation of the point above a reference plane, with the positive z-direction pointing upward – so in the direction opposite to the gravitational acceleration,

p\,

is the pressure at the chosen point, and

\rho\,

is the density of the fluid at all points in the fluid.

For conservative force fields, Bernoulli's equation can be generalized as:

{v^2 \over 2}+\Psi+{p\over\rho}=\text{constant}

where Ψ is the force potential at the point considered on the streamline. E.g. for the Earth's gravity Ψ = gz.

The following two assumptions must be met for this Bernoulli equation to apply:

the flow must be incompressible – even though pressure varies, the density must remain constant along a streamline;
friction by viscous forces has to be negligible. In long lines mechanical energy dissipation as heat will occur. This loss can be estimated e.g. using Darcy–Weisbach equation.

By multiplying with the fluid density

\rho

, equation (A) can be rewritten as:

\tfrac12\, \rho\, v^2\, +\, \rho\, g\, z\, +\, p\, =\, \text{constant}\,

or:

q\, +\, \rho\, g\, h\, =\, p_0\, +\, \rho\, g\, z\, =\, \text{constant}\,

where:

q\, =\, \tfrac12\, \rho\, v^2

is dynamic pressure,

h\, =\, z\, +\, \frac{p}{\rho g}

is the piezometric head or hydraulic head (the sum of the elevation z and the pressure head) and

p_0\, =\, p\, +\, q\,

is the total pressure (the sum of the static pressure p and dynamic pressure q).

The constant in the Bernoulli equation can be normalised. A common approach is in terms of total head or energy head H:

H\, =\, z\, +\, \frac{p}{\rho g}\, +\, \frac{v^2}{2\,g}\, =\, h\, +\, \frac{v^2}{2\,g},

The above equations suggest there is a flow speed at which pressure is zero, and at even higher speeds the pressure is negative. Most often, gases and liquids are not capable of negative absolute pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be valid before zero pressure is reached. In liquids – when the pressure becomes too low – cavitation occurs. The above equations use a linear relationship between flow speed squared and pressure. At higher flow speeds in gases, or for sound waves in liquid, the changes in mass density become significant so that the assumption of constant density is invalid.

Pascal Pressure

Pascal's law or the principle of transmission of fluid-pressure is a principle in fluid mechanics that states that pressure exerted anywhere in a confined incompressible fluid is transmitted equally in all directions throughout the fluid such that the pressure variations (initial differences) remain the same. The law was established by French mathematician Blaise Pascal.

Definition

Pascal's principle is defined as

A change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid

This principle is stated mathematically as:

\Delta P =\rho g (\Delta h)\,

\Delta P

is the hydrostatic pressure (given in pascals in the SI system), or the difference in pressure at two points within a fluid column, due to the weight of the fluid;

ρ is the fluid density (in kilograms per cubic meter in the SI system);

g is acceleration due to gravity (normally using the sea level acceleration due to Earth's gravity in metres per second squared);

\Delta h

is the height of fluid above the point of measurement, or the difference in elevation between the two points within the fluid column (in metres in SI).

The intuitive explanation of this formula is that the change in pressure between two elevations is due to the weight of the fluid between the elevations. A more correct interpretation, though, is that the pressure change is caused by the change of potential energy per unit volume of the liquid due to the existence of the gravitational field. Note that the variation with height does not depend on any additional pressures. Therefore Pascal's law can be interpreted as saying that any change in pressure applied at any given point of the fluid is transmitted undiminished throughout the fluid.

Explanation

If a U-tube is filled with water and pistons are placed at each end, pressure exerted against the left piston will be transmitted throughout the liquid and against the bottom of the right piston. (The pistons are simply "plugs" that can slide freely but snugly inside the tube.) The pressure that the left piston exerts against the water will be exactly equal to the pressure the water exerts against the right piston. Suppose the tube on the right side is made wider and a piston of a larger area is used; for example, the piston on the right has 50 times the area of the piston on the left. If a 1 N load is placed on the left piston, an additional pressure due to the weight of the load is transmitted throughout the liquid and up against the larger piston. The difference between force and pressure is important: the additional pressure is exerted against the entire area of the larger piston. Since there is 50 times the area, 50 times as much force is exerted on the larger piston. Thus, the larger piston will support a 50 N load - fifty times the load on the smaller piston.

Forces can be multiplied using such a device. One newton input produces 50 newtons output. By further increasing the area of the larger piston (or reducing the area of the smaller piston), forces can be multiplied, in principle, by any amount. Pascal's principle underlies the operation of thehydraulic press. The hydraulic press does not violate energy conservation, because a decrease in distance moved compensates for the increase in force. When the small piston is moved downward 10 centimeters, the large piston will be raised only one-fiftieth of this, or 0.2 centimeters. The input force multiplied by the distance moved by the smaller piston is equal to the output force multiplied by the distance moved by the larger piston; this is one more example of a simple machine operating on the same principle as a mechanical lever.

Pascal's principle applies to all fluids, whether gases or liquids. A typical application of Pascal's principle for gases and liquids is the automobile lift seen in many service stations (the hydraulic jack). Increased air pressure produced by an air compressor is transmitted through the air to the surface of oil in an underground reservoir. The oil, in turn, transmits the pressure to a piston, which lifts the automobile. The relatively low pressure that exerts the lifting force against the piston is about the same as the air pressure in automobile tires. Hydraulics is employed by modern devices ranging from very small to enormous. For example, there are hydraulic pistons in almost all construction machines where heavy loads are involved.

You may visit the following pages for more informations.

http://en.wikipedia.org/wiki/Pascal's_law

http://www.grc.nasa.gov/WWW/k-12/WindTunnel/Activities/Pascals_principle.html