# Units and units conversion

#### Units:

The International System of units or simply SI system is the most widely used units system. In the SI system, there are seven base units and numerous derived units. Derived units are just the combination of the base units.Physical quantity | Unit name | Unit symbol |
---|---|---|

length | meter | $m$ |

mass | kilogram | $kg$ |

time | second | $s$ |

electric current | ampere | $A$ |

temperature | kelvin | $K$ |

amount of substance | mole | $mol$ |

luimnous intensity | candela | $cd$ |

#### SI common derived units:

Physical quantity | Unit |
---|---|

area | $m^2$ |

volume | $m^3$ |

speed | $m/s$ |

velocity | $m/s$ |

acceleration | $m/s^2$ |

force | $N$ |

work | $J$ |

energy | $J$ |

power | $W$ |

momentum | $kg.m/s$ or $N.s$ |

impulse | $kg.m/s$ or $N.s$ |

spring constant | $N/m$ |

angle | $rad.$ |

angular velocity | $rad./s$ |

angular acceleration | $rad./s^2$ |

torque | $N.m$ |

moment of inertia | $kg.m^2$ |

angular momentum | $kg.m^2/s$ |

stress | $N/m^2$ |

strain | $no\:unit$ |

Young's modulus | $N/m^2$ |

shear modulus | $N/m^2$ |

bulk modulus | $N/m^2$ or $Pa$ |

strength of materials | $N/m^2$ or $Pa$ |

density | $kg/m^3$ |

pressure | $N/m^2$ or $Pa$ |

period | $s$ |

frequency | $Hz$ or $s^{-1}$ |

wavelength | $m$ |

intensity | $W/m^2$ |

coefficient of linear expansion | $no\: unit$ |

coefficient of volume expansion | $no\: unit$ |

heat | $J$ |

specific heat | $J/kg.K$ |

latent heat | $J/kg$ |

thermal conductivity | $Wm^{-1}K^{-1}$ |

emissivity | $no\: unit$ |

entropy | $J/K$ |

#### Unit conversion

Sometimes you may need to convert a unit from one system of units to another. That is from another system of units to SI, or from SI to another system. Converting unit is same as finding the price of certain number of mangoes from the price of one mango. To do unit conversion, you need a conversion factor, that is like the price of 1 unit.Conversion is nothing but dividing out the unit that you don't want and multiplying the unit that you wanted.

For example, you know the price of 1 mango that is 2 dollars, and you need to find the price of 50 mangoes.

Here the mangoes and the dollars are the units, and you need to convert from mangoes to dollars.

To find the price of 50 mangoes, what you need to do is cancel out the unit mangoes and multiply the unit dollars.

As a first step, write,

$50\: mangoes = 50\: mangoes \times 1$

I multiplied a 1 as it does not change anything.

We have the conversion factor,

$1\: mango = 2\: dollars$

Since we are dividing out the unit mango, and in the conversion factor that is on the left hand side, so you divide everything on the left hand side on both sides, you will get

$1=\dfrac{2 \:dollars}{1\:mango}$

Now, replace 1 in the previous equation with this, you will get

$50\: mangoes = 50\: mangoes \times \dfrac{2\: dollars}{1\: mango}$

Now, cancel out the mangoe(s)

$50\: mangoes = 50\: \cancel{mangoes} \times \dfrac{2\: dollars}{1\: \cancel{mango}}$

And simplify, you will get,

$50\: mangoes = 100\: dollars$

**Question 1:**What is height of a 48 ft building in $m$?.

**Solution:**You are converting from ft to m. The conversion factor for that is

$1\: ft=0.3048\: m$

The unit that you want is m and you do not want is ft, which you should cancel out. Since ft on the left hand side, divide everything on the left (1 ft) on both sides of the conversion factor:

$1\:=\dfrac{0.3048\:m}{1 ft}$

Now, take the original number and multiply with this

$\begin{aligned}48ft &= 48ft\times 1\\ &= 48ft\times \dfrac{0.3048m}{1 ft} \\ &= 48 ft \times \dfrac{0.3048m}{1 {ft}}=14.6304 m \end{aligned}$.

**Conversion and significant figures:**When you are doing conversion, round the result to have same number of significant figures as the original number. That is you need to assume that the conversion factor is exact and has infinite number of significant figures.

In the problem above, the original number is 48, it has two significant figures. So rounding 14.6304 to two significant figures, you will get, $48\:ft=15\:m$.

Now you know how to convert, from one unit to another. The same procedure apply if you want to convert a quantity that has more than 1 unit ( a combination of units).

As an example, you have the speed of a car that is 75 km/h, and you want to know what is this speed in m/s.

In this problem, there are two units: km and h. And you need to convert km to m and h to s. So, you need two conversion factors:

$1\:km=1000\:m$ and $1\:h=3600\:s$

Since there are two units in the quantity, you multiply two 1's and replace those with the conversion factor.

$75\:km/h=75\:km/h\times 1 \times 1$

To cancel the km, you need to divide that out but to cancel the h, you need to multiply that as the h is in the denominator:

$75\:km/h=75\:km/h\times \dfrac{1000\:m}{1\:km} \times \dfrac{1\:h}{3600\:s}$

Now, cancel out the km and h:

$75\:km/h=75\:\cancel{km}/\cancel{h}\times \dfrac{1000\:m}{1\:\cancel{km}} \times \dfrac{1\:\cancel{h}}{3600\:s}$

Simplifying, and rounding to correct number of significant figures:

$75\:km/h=21\: m/s$

#### A table of conversion factor is given below,

Non-SI | SI unit | Conversion factor |
---|---|---|

$cm$ (centimeter) | $m$ | 100 $cm = $1 $m$ |

$1km$ | $m$ | $1\:km=1000\:m$ |

$in.$ (inch) |
$m$ | $1\:in.=0.0254\:m$ |

$ft$ (feet) |
$m$ | $1\:ft = 0.3048\: m$ |

$\mu m$ (micrometer) |
$m$ | $1\: \mu m = 1\times 10^{-6}\: m$ |

$mi$ (mile) |
$m$ | $1\: mi = 1609\: m$ |

$nm$ (nanometer) |
$m$ | $1\: nm = 10^{-9}\: m$ |

$L$ (liter) |
$m^3$ | $1\:L = 10^{-3}\: m^3$ |

$g$ (gram) |
$Kg$ | $1\: g=10^{-3}\:kg$ |

$lb$ (pound) |
$kg$ | $1\: lb = 0.4536\: kg$ |

$°$ (degree) |
$rad.$ | $1°=\dfrac{2\pi}{180}\:rad.$ |

$g/cm^3$ | $kg/m^3$ | $1\: g/cm^3 = 10^{3}\: kg/m^3$ |

$h$ (hour) |
$s$ | $1\: h = 3600\: s$ |

$mph$ (miles per hour) |
$m/s$ | $1\: mph = 0.447\: m/s$ |

$lbf$ (pound-force) |
$N$ | $1\: lbf = 4.4482\: N$ |

$atm$ |
$Pa$ | $1\: atm = 1.01325 \times 10^5\: Pa$ |

$bar$ | $Pa$ | $1\: bar = 10^5\: Pa$ |

$torr$ | $Pa$ | $1\: torr = 133.32 \:Pa$ |

$mmHg$ |
$Pa$ | $1\: mmHg = 133.32\: Pa$ |

$BTU$ |
$J$ | $1 \:BTU = 1.055\times 10^3\: J$ |

$P$ (poise) |
$Pa.s$ | $1\:P=0.1\:Pa.s$ |

$G$ (gauss) |
$T(tesla)$ | $1\:G= 10^{-4}\: T$ |

$°C$ (degree celsius) |
$K$ | $K=C+273.15$ To convert from degree C to K, just add 273.15 |