Dimensional analysis

Dimensional Analysis (also called Factor-Label Method or the Unit Factor Method) is a problem-solving method that uses the fact that any number or expression can be multiplied by one without changing its value. It is a useful technique.
Unit factors may be made from any two terms that describe the same or equivalent "amounts" of what we are interested in. For example, we know that
1 inch = 2.54 centimeters Note: Unlike most English-Metric conversions, this one is exact. There are exactly 2.540000000... centimeters in 1 inch.We can make two unit factors from this information:

Now, we can solve some problems. Set up each problem by writing down what you need to find with a question mark. Then set it equal to the information that you are given. The problem is solved by multiplying the given data and its units by the appropriate unit factors so that only the desired units are present at the end.
(1) How many centimeters are in 6.00 inches?

(2) Express 24.0 cm in inches.
You can also string many unit factors together.

(3) How many seconds are in 2.0 years?

Scientists generally work in metric units. Common prefixes used are the following:

Prefix	Abbreviation	Meaning	Example
mega-	M	106	1 megameter (Mm) = 1 x 106 m
kilo-	k	103	1 kilogram (kg) = 1 x 103 g
centi-	c	10-2	1 centimeter (cm) = 1 x 10-2 m
milli-	m	10-3	1 milligram (mg) = 1 x 10-3 g
micro-		10-6	1 micrometer (g) = 1 x 10-6 g
nano-	n	10-9	1 nanogram (ng) = 1 x 10-9 g

(4) Convert 50.0 mL to liters. (This is a very common conversion.)

(5) What is the density of mercury (13.6 g/cm³) in units of kg/m³?
We also can use dimensional analysis for solving problems.

(6) How many atoms of hydrogen can be found in 45 g of ammonia, NH₃?
We will need three unit factors to do this calculation, derived from the following information:

1. 1 mole of NH₃ has a mass of 17 grams.
2. 1 mole of NH₃ contains 6.02 x 10²³ molecules of NH₃.
3. 1 molecule of NH₃ has 3 atoms of hydrogen in it.

Units of Measurement - Derived SI Units

Derived quantities are defined in terms of the seven base quantities via a system of quantity equations. The SI derived units for these derived quantities are obtained from these equations and the seven SI base units. Examples of such SI derived units are given below. 

Derived quantity	Name	Symbol
area	square meter	m2
volume	cubic meter	m3
speed, velocity	meter per second	m/s
acceleration	meter per second squared	m/s2
wave number	reciprocal meter	m-1
mass density	kilogram per cubic meter	kg/m3
specific volume	cubic meter per kilogram	m3/kg
current density	ampere per square meter	A/m2
magnetic field strength	ampere per meter	A/m
amount-of-substance concentration	mole per cubic meter	mol/m3
luminance	candela per square meter	cd/m2
mass fraction	kilogram per kilogram, which may be represented by the number 1	kg/kg = 1

For ease of understanding and convenience, 22 SI derived units have been given special names and symbols, as shown below.

Derived quantity	Name	Symbol	Expression  in terms of  other SI units	Expression in terms of SI base units
plane angle	radian (a)	rad	-	m·m-1 = 1 (b)
solid angle	steradian (a)	sr (c)	-	m2·m-2 = 1 (b)
frequency	hertz	Hz	-	s-1
force	newton	N	-	m·kg·s-2
pressure, stress	pascal	Pa	N/m2	m-1·kg·s-2
energy, work, quantity of heat	joule	J	N·m	m2·kg·s-2
power, radiant flux	watt	W	J/s	m2·kg·s-3
electric charge, quantity of electricity	coulomb	C	-	s·A
electric potential difference, electromotive force	volt	V	W/A	m2·kg·s-3·A-1
capacitance	farad	F	C/V	m-2·kg-1·s4·A2
electric resistance	ohm		V/A	m2·kg·s-3·A-2
electric conductance	siemens	S	A/V	m-2·kg-1·s3·A2
magnetic flux	weber	Wb	V·s	m2·kg·s-2·A-1
magnetic flux density	tesla	T	Wb/m2	kg·s-2·A-1
inductance	henry	H	Wb/A	m2·kg·s-2·A-2
Celsius temperature	degree Celsius	°C	-	K
luminous flux	lumen	lm	cd·sr (c)	m2·m-2·cd = cd
illuminance	lux	lx	lm/m2	m2·m-4·cd = m-2·cd
activity (of a radionuclide)	becquerel	Bq	-	s-1
absorbed dose, specific energy (imparted), kerma	gray	Gy	J/kg	m2·s-2
dose equivalent (d)	sievert	Sv	J/kg	m2·s-2
catalytic activity	katal	kat		s-1·mol

Units of Measurement - Fundamental SI Units

The SI base units are a choice of seven well-defined units which by convention are regarded as dimensionally independent: metre, m The metre is the length of the path traveled by light in vacuum during a time interval of 1/299 792 458 of a second. kilogram, kg The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. second, s The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. ampere, A The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force equal to 2 x 10–7 newton per metre of length. kelvin, K The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. mole, mol The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles. candela, cd The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

Spherical Polar Coordinates

Physical systems which have spherical symmetry are often most conveniently treated by using spherical polar coordinates.

Cylindrical Polar Coordinates

With the axis of the circular cylinder taken as the z-axis, the perpendicular distance from the cylinder axis is designated by r and the azimuthal angle taken to be Φ.

Physical systems which have cylindrical symmetry are often most conveniently treated by using cylindrical polar coordinates.

Rectangular Coordinates

The most common coordinate system for representing positions in space is one based on three perpendicular spatial axes generally designated x, y, and z. 

Any point P may be represented by three signed numbers, usually written (x, y, z) where the coordinate is the perpendicular distance from the plane formed by the other two axes.
Often positions are specified by a position vector r which can be expressed in terms of the coordinate values and associated unit vectors.

Although the entire coordinate system can be rotated, the relationship between the axes is fixed in what is called a right-handed coordinate system.

For the display of some kinds of data, it may be convenient to have different scales for the different axes, but for the purpose of mathematical operations with the coordinates, it is necessary for the axes to have the same scales. The term "Cartesian coordinates" is used to describe such systems, and the values of the three coordinates unambiguously locate a point in space. In such a coordinate system you can calculate the distance between two points and perform operations like axis rotations without altering this value.

The distance between any two points in rectangular coordinates can be found from the distance relationship.

Coordinate Systems

Rectangular Coordinates

Cylindrical Polar Coordinates

Spherical Polar Coordinates