Density, Distance, Time And Speed In Mathematics
Density , Distance Time And Speed
The density, or more accurately, the volumetric mass density, of a substance is defined as its mass per unit volume. The symbol most frequently used for density is ρ (the lower case Greek letter for rho). Mathematically, density is defined as mass divided by volume:]
ρ = mass / volume
where ρ is the density, m is the mass, and V is the volume. In a few instances like in the United States oil and gas industry, density is loosely defined as its weight per unit volume despite the fact that this is scientifically incorrect– this quantity is more particularly known as specific weight.
For a pure substance, the density has equivalent numerical value as its mass concentration. Different materials normally have different densities, and density may be relevant to buoyancy, purity and packaging. Osmium and iridium are the densest known elements at standard conditions for temperature and pressure but specific chemical compounds may be denser.
To simplify comparisons of density across different systems of units, it is often times replaced by the dimensionless quantity "relative density" or "specific gravity", which means the ratio of the density of the material to that of a standard material, normally water. Therefore, a relative density less than one means that the substance is less dense than water and thus floats in water.
The density of a material differs with temperature and pressure. This difference is normally small for solids and liquids but much more for gases. Increasing the pressure on an object decreases the volume of the object and therefore increases its density. Increasing the temperature of a substance (with a few exceptions) decreases its density by increasing its volume. In a majority of materials, heating the bottom of a fluid leads to convection of the heat from the bottom to the top, as a result of the decrease in the density of the heated fluid. This makes it to rise with respect to more dense unheated material.
The reciprocal of the density of a substance is on occasion referred to as its specific volume, a term occasionally made use of in thermodynamics. Density is an intensive property because increasing the amount of a substance does not increase its density; instead it increases its mass.
From the equation for density (ρ = m / V), mass density has units of mass divided by volume. As there are a lot of units of mass and volume covering a lot of different magnitudes, there are a large number of units for mass density in use. The SI unit of kilogram per cubic metre (kg/m3) and the cgs unit of gram per cubic centimetre (g/cm3) are likely the most frequently used units for density.1,000 kg/m3 equals 1 g/cm3. (The cubic centimeter can be on the other hand known as a millilitre or a cc.) In industry, other larger or smaller units of mass and or volume are often more practical and US customary units may be utilized. See below a list of a few of the most common units of density.
Measurement of density
Homogeneous materials
The density at all points of a homogeneous object equals its total mass divided by its total volume. The mass is usually measured with a scale or balance; the volume may be measured directly (from the geometry of the object) or by the displacement of a fluid. To determine the density of a liquid or a gas, a hydrometer, a dasymeter or a Coriolis flow meter might be utilized, correspondingly. In the same way, hydrostatic weighing makes use of the displacement of water as a result of a submerged object to determine the density of the object.
Heterogeneous materials
If the body is not homogeneous, then its density differs between various regions of the object. In that situation the density around any given place is obtained by computing the density of a small volume around that location. In the limit of an infinitesimal volume the density of an inhomogeneous object at a point turns into: ρ( ) = dm/dV, where dV is an elementary volume at position r. The mass of the body then can be calculated as
Non-compact materials
In practice, bulk materials like sugar, sand, or snow is made up of voids. A lot of materials exist naturally as flakes, pellets, or granules.
Voids are areas which are made up of something instead of the considered material. Most frequently the void is air, but it could as well be vacuum, liquid, solid, or a different gas or gaseous mixture.
The bulk volume of a material—inclusive of the void fraction—is frequently gotten by a simple measurement (like with a calibrated measuring cup) or geometrically from known dimensions.
Mass divided by bulk volume determines bulk density. This is not the same thing as volumetric mass density.
To obtain volumetric mass density, one ought to first discount the volume of the void fraction. Frequently, this can be determined by geometrical reasoning. For the close-packing of equal spheres the non-void fraction can be at most about 74%. It can as well be determined empirically. A few bulk materials, like sand though, is made up of a variable void fraction which depends on how the material is agitated or poured. It might be loose or compact, with more or less air space depending on the way they are handled.
Practical, the void fraction is not essentially air, or even gaseous. In the case of sand, it could be water, which can be advantageous for measurement as the void fraction for sand saturated in water—once any air bubbles are completely eliminated possibly more consistent than dry sand measured with an air void.
In the case of non-compact materials, one ought to as well take care in determining the mass of the material sample. If the material is under pressure (normally ambient air pressure at the earth's surface) the determination of mass from a measured sample weight may be required to account for buoyancy effects as a result of the density of the void constituent, depending on the way the measurement was carried out. In the case of dry sand, sand is so much denser than air that the buoyancy effect is mainly neglected (less than one part in one thousand).
Mass change upon displacing one void material with another while maintaining constant volume can be utilized to estimate the void fraction, if the difference in density of the two voids materials is reliably known.
Changes of density:
Compressibility and Thermal expansivity
In general, density can be altered by changing either the pressure or the temperature. Increasing the pressure at all times increases the density of a material. Increasing the temperature in general decreases the density, but there are noteworthy exceptions to this generalization. For instance, the density of water increases between its melting point at 0 °C and 4 °C; similar behavior is seen in silicon at low temperatures.
The effect of pressure and temperature on the densities of liquids and solids is small. The compressibility for a typical liquid or solid is 10−6 bar−1 (1 bar = 0.1 MPa) and a typical thermal expansivity is 10−5 K−1. This more or less translates into requiring around ten thousand times atmospheric pressure to reduce the volume of a substance by one percent despite the fact that the pressures required may be nearly a thousand times smaller for sandy soil and a few clays. A one percent expansion of volume characteristically needs a temperature increase on the order of thousands of degrees Celsius.
In contrast, the density of gases is strongly affected by pressure. The density of an ideal gas is
ρ=MP ⁄RT
where M is the molar mass, P is the pressure, R is the universal gas constant, and T is the absolute temperature. This means that the density of an ideal gas can be doubled by doubling the pressure, or by halving the absolute temperature
In the case of volumic thermal expansion at constant pressure and small intervals of temperature the temperature dependence of density is :
where ρT10 is the density at a particular temperature, is the thermal expansion coefficient of the material at temperatures close to T0 .
Density of solutions
The density of a solution is the sum of mass (massic) concentrations of the components of that solution.
Mass (massic) concentration of each given component ρi in a solution sums to density of the solution.
Expressed as a function of the densities of pure constituents of the mixture and their volume participation, it permits the determination of excess molar volumes:
provided that there is no interaction between the constituents.
Being aware of the relation between excess volumes and activity coefficients of the constituent, one can determine the activity coefficients.
Speed is the distance covered in 1 unit of time. For example, a speed of 70 km/h means that 70 km are covered in 1 hour
Example 1. A car travelled 80 minutes at a speed of 120 km/h. What distance did it cover?. Answer. 80 minutes = 1.33 hours. The big plus is
?= 120×1.33⁄1=160KM hours
Example 2.
A man takes 2 hours and 15 minutes to run over 12 km. What was his (average) speed?
km=12; hours=2.15; hpors=1
Answer
The time (in hours) is 2.25 hours. The big plus is
12 ×1 ⁄2.25=5.33. The speed was 53KM/h
Density, mass and volume
You work with these things precisely as you do with speed, distance and time
If in a question you can read the word density, mass and volume (or equivalent,) it is very likely that: The question can be solved with the big plus
Make use of the fact that density is the mass of 1 unit of volume
Example 1.
A piece of metal of density 5 g/cm3weights 3 grams. What is its volume? Answer. 1 cm3 of the metal weight 5 grams.
grams cm3 e=?3×1⁄5=0.6cm3
Example 2.
If 700 cm3 of a liquid weight 500 grams, what is the density of the liquid?
Answer.
e=?500×1⁄700=0.71
The density of the liquid is 0.71 g/cm3
Foreign exchange means the exchange of one currency for another, or the conversion of one currency into another currency. Foreign exchange as well refers to the global market where currencies are traded more or less for 24 hours every day. The term foreign exchange is normally abbreviated as "forex" and infrequently as "FX."
Foreign exchange transactions include everything from the conversion of currencies by a traveler at an airport kiosk to billion-dollar payments made by corporate organizations and governments for goods and services bought overseas. Increasing globalization has resulted to a massive increase in the number of foreign exchange transactions in recent decades. The global foreign exchange market is by far the biggest financial market, with average daily volumes in the trillions of dollars.
Time, Distance, & Mass
Time, distance, and mass are the three basic physical quantities that most of mechanics is built upon. While there are a lot of units that can be used to describe the magnitude of these fundamental quantities, for simplicity we will make use of seconds to measure time, meters to measure distance, and kilograms to measure mass.
We can create a lot of slightly more composite ideas out of the simple quantities of time (s) distance (m) mass (kg). Speed is the ratio of distance to time, velocity is as well the ratio of distance to time, but unlike speed which is one number, velocity is a vector and is represented by a magnitude and a direction. Velocity Is speed in a specific straight line direction.
speed = distance/time = m/s
velocity = distance/time = m/s
Acceleration is the ratio of velocity to time, and just like velocity is a vector and is specified by a magnitude and a direction.
acceleration = velocity/time = distance/time/time = m/s²
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