Charles's law

Charles's law (also known as the law of volumes) is an experimental gas law which describes how gases tend to expand when heated. It was first published by French natural philosopher Joseph Louis Gay-Lussac in 1802, although he credited the discovery to unpublished work from the 1780s by Jacques Charles. The law was independently discovered by British natural philosopher John Dalton by 1801, although Dalton's description was less thorough than Gay-Lussac's. The basic principles had already been described a century earlier by Guillaume Amontons.

Taylor Buchanan was the first to demonstrate that the law applied generally to all gases, and also to the vapours of volatile liquids if the temperature was more than a few degrees above the boiling point. His statement of the law can be expressed mathematically as:

$V_{100} - V_0 = kV_0\,$

where V₁₀₀ is the volume occupied by a given sample of gas at 100 °C; V₀ is the volume occupied by the same sample of gas at 0 °C; and k is a constant which is the same for all gases at constant pressure. Gay-Lussac's value for k was ¹⁄_2.6666, remarkably close to the present-day value of ¹⁄_2.7315.

Volume and temperature are directly proportional, as long as the pressure stays at a constant.

A modern statement of Charles' law is:

At constant pressure, the volume of a given mass of an ideal gas increases or decreases by the same factor as its temperature on the absolute temperature scale(i.e. the gas expands as the temperature increases).

which can be written as:

$V \propto T\,$

where V is the volume of the gas; and T is the absolute temperature. The law can also be usefully expressed as follows:

$\frac{V_1}{T_1} = \frac{V_2}{T_2} \qquad \mathrm{or} \qquad \frac {V_2}{V_1} = \frac{T_2}{T_1} \qquad \mathrm{or} \qquad V_1 T_2 = V_2 T_1.$

The equation shows that, as absolute temperature increases, the volume of the gas also increases in proportion

Bernoulli's principle

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.Bernoulli's principle is named after the Dutch-Swissmathematician Daniel Bernoulli who published his principle 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 for incompressible flows (e.g. most liquid flows) and also for compressible flows (e.g. gases) moving at low Mach numbers. 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 energy. This states that, in a steady flow, the sum of all forms of mechanical energy in a fluid along a streamlineis 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 its dynamic 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 and gravitational 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.

Boyle's law

Boyle's law (sometimes referred to as the Boyle–Mariotte law) is one of many gas lawsand a special case of the ideal gas law. Boyle's law describes the inversely proportional relationship between the absolute pressure and volume of a gas, if the temperature is kept constant within a closed system. The law was named after chemist and physicist Robert Boyle, who published the original law in 1662.The law itself can be stated as follows:

For a fixed amount of an ideal gas kept at a fixed temperature, P [pressure] and V [volume] are inversely proportional (while one doubles, the other halves).

Equation

The mathematical equation for Boyle's law is:

$\qquad\qquad pV = k$

where:

p denotes the pressure of the system.

V denotes the volume of the gas.

k is a constant value representative of the pressure and volume of the system.

So long as temperature remains constant the same amount of energy given to the system persists throughout its operation and therefore, theoretically, the value of k will remain constant. However, due to the derivation of pressure as perpendicular applied force and the probabilistic likelihood of collisions with other particles through collision theory, the application of force to a surface may not be infinitely constant for such values of k, but will have a limit when differentiating such values over a given time.

Forcing the volume V of the fixed quantity of gas to increase, keeping the gas at the initially measured temperature, the pressure pmust decrease proportionally. Conversely, reducing the volume of the gas increases the pressure.

Boyle's law is used to predict the result of introducing a change, in volume and pressure only, to the initial state of a fixed quantity of gas. The before and after volumes and pressures of the fixed amount of gas, where the before and after temperatures are the same (heating or cooling will be required to meet this condition), are related by the equation:

$p_1 V_1 = p_2 V_2. \,$

Here P₁ and V₁ represent the original pressure and volume, respectively, and P₂ and V₂ represent the second pressure and volume.

Boyle's law, Charles's law, and Gay-Lussac's law form the combined gas law. The three gas laws in combination with Avogadro's lawcan be generalized by the ideal gas law.

Archimedes' principle

Archimedes' principle is a law of physics stating that the upward force ( buoyancy ) exerted on a body immersed in a fluid is equal to the weight of the amount of fluid the body displaces. In other words, an immersed object is buoyed up by a force equal to the weight of the fluid it actually displaces. Archimedes' principle is an important and underlying concept in the field of fluid mechanics. This principle is named after its discoverer, Archimedes of Syracuse.

Assuming Archimedes' principle to be reformulated as follows,

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

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

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

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. 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. When an object is immersed in a liquid the liquid exerts an upward force which is known as buoyant force and it is proportional to the weight of 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.

Ohm's law

Ohm's law states that the current through a conductor between two points is directlyproportional to the potential difference across the two points. Introducing the constant of proportionality, the resistance one arrives at the usual mathematical equation that describes this relationship:

$I = \frac{V}{R}$

where I is the current through the conductor in units of amperes, V is the potential difference measured across the conductor in units of volts, and R is the resistance of the conductor in units of ohms. More specifically, Ohm's law states that the R in this relation is constant, independent of the current

Pascal's law

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 ratio (initial difference) remains the same. The law was established by French mathematician Blaise Pascal

Pascal's principle is defined

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:

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

where

$\Delta P$ 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 inmetres 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. 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.

physics and mathematics

Monday, 9 July 2012

LAW AND PRINCIPLE IN PHYSICS