# Physical Quantity and SI Units

We always measure in Physics. We measure a physical quantity by comparing with the unit of the physical quantity. A unit is the standard form of a physical quantity that allows you to measure the quantity. Anyone is free to make any standards for measurement, so there must be a universal system of measurement accepted worldwide and anywhere in the universe called metric system or SI units.

## Physical Quantity

You have a book and you try to find how long or how wide the book is. The measure of how long is length and that of how wide is breadth (width). These two things length and breadth are the physical quantities of the book. An object can be described by its physical quantities. If you have a ball, the ball is an object and you can describe the ball by its weight, radius, mass etc.

If you are measuring the radius of a ball, the ball is an object and its radius is a physical quantity. Mass of the ball is another physical quantity. You can quantitatively describe the properties of an object by physical quantities.

PHYSICAL QUANTITY: A number which can describe the physical property of a material or a physical phenomenon quantitatively is called physical quantity.

The most fundamental physical quantities can only be defined if you know how to measure them. Such a definition is called *operational definition*. If you want to define the fundamental physical quantities such as length, mass and time, you must know how to measure them.

When you measure any physical quantity you must compare it with a particular standard of that physical quantity called unit. You compare with the unit of a physical quantity to measure that physical quantity.

The base quantities such as length, mass and time are the fundamental ones which do not depend on other quantities. The derived quantities are derived from the base quantities such as area, volume speed etc.

In Physics we often need to describe physical quantities by numerical value or by both the numerical value and direction. The physical quantities with direction are called vectors and those without direction are called scalars. The vectors and scalars distinguish all physical quantities by their distinguishing factor called direction.

## SI Units

How do you measure the length of a book? You need to make the standards of length to say the book has some length. And the standards you use to give the physical quantities a measurable value are the units. Now you can say the book has a certain length using your length standard.

For example, you made a stick of a certain length and measured other lengths with that stick and found the length of your bed to be five times that stick. So here the length of the stick you made recently is your temporary length standard (you used it to measure other lengths).

When we make a standard to define a physical quantity, that becomes a unit. Notice that you can give your own standard to any physical quantity, for example if you made the period of a particular pendulum to be your time standard and calculated other time periods like day, month or year, that time period will be based on your own time standard.

Anyone is free to make any standards for units, so it's very important to define universal standards (the same standards in the entire world) for units called SI units (*system international* units). Now we learn how some SI units of some fundamental physical quantities like length, mass and time are defined with a little bit of their history. You may not be interested in the history of SI units but a little bit of skim on it can provide you a way how they approached to their current operational definition.

### SI Unit of Time

The SI unit of time is **second** denoted by letter **s**. The fundamental unit of time was used to be defined in terms of the *mean solar day*. A solar day is the time interval between successive arrivals of the Sun at the highest point in the sky and the mean solar day is the average time between successive arrivals of the sun at the highest point in the sky. The second was the certain fraction of this mean solar day, that is $(\frac{1}{24})(\frac{1}{60})(\frac{1}{60})$ of the mean solar day.

This definition is not precise and not universal because it is based on the Earth's rotation only. So, this operational definition must be changed to something more precise and universal which is exactly done by the new definition based on *atomic clock*. The more precise definition of second in terms of atomic clock is

SECOND: Second is the time period equal to 9192631770 times the period of vibration of radiation from caesium-133 atom.

You may need to dive into atomic physics to understand how atomic clock works. For now we do not go into its details in this article. We use this standard to measure other time intervals or periods. For example, a particular time period is 1s, 21s, 3s, 15s and so on.

### SI Unit of Length

The SI unit of length is **meter** and denoted by the letter **m**. The standard of length was defined as meter by the French Academy of Sciences in 1790s legalized in 1799 and defined as the one ten-millionth of distance from the equator of Earth to the North pole. A bar of platinum was made to represent this length. This standard is not universal as it depends on the Earth's measurement.

The previous standard was not applicable and a new definition was introduced in 1889 and defined as the length between two lines in a bar of platinum-iridium alloy. It was again redefined in 1960 as the 1,650,763.73 wavelengths of orange light emitted by krypton-86 gas lamp. The more accuracy was required and it was again redefined in 1983 in terms of the speed of light and it is the current definition of meter.

METER: Meter is the distance the light travels in 1/2299792458 second.

This definition of meter entirely depends on the speed of light, that is this definition of meter gives the speed of light the exact value of 2299792458 m/s. If you require more information, you may read more about the history on meter. This standard of length is used to measure other lengths. For example the length of a particular rod is 1m, 5m, 2m and so on.

### SI Unit of Mass

The SI unit of mass has the name **kilogram** and denoted by **kg**. The standard of mass has not been changed since its establishment which was defined as the mass of a particular alloy of platinum and iridium kept at the International Bureau of Weights and Measures at Severs, near Paris, France. This definition has its own problems as this is based on an object with mass and it is not universal either and is indeed contaminated easily. It is true that platinum-iridium alloy is unusually stable alloy but it has lost some mass from the date it was created which makes this definition unreliable. We can not think about making our standard that changes.

A change on the standard definition of kilogram was filed and a new definition is established just now to be effective from 2019 may 20. It has been taken as the big achievement on the standard of mass which is now universal and can never change or contaminated. The new definition is based on the fundamental constant called *Planck's constant*.

The SI unit of Planck's constant is $\text{Js}$ which is $\text{kg}\cdot\text{m}^2/\text{s}$. The Plank's constant is the universal constant and it contains second, meter and kilogram. We already have the standard definitions of second and meter and we can use those definitions to define the kilogram. Since we are defining the kilogram in terms of Planck's constant, we must determine the most accurate value of this constant and it's new value is forever taken to be $6.626 070 15 \times 10^{-34} \text{Js}$.

KILOGRAM: The kilogram is the quantity of matter that gives the fixed value of Planck's constant to be $6.626 070 15 \times 10^{-34} \text{Js}$.

After this definition we we never need to rely on a piece of material anymore which is prone to contamination. When we need to measure very precisely, the old definition was not enough. The new definition of kilogram took a long time as the Plank's constant must be calculated to a very accurate value. It's value is measured with a machine called Kibble balance.

Now we have the SI unit of mass and we can calculate the mass using this standard such as 1kg, 2kg, 32kg etc.

Interesting thing you can notice here in the definition of these SI units is that just look at the SI unit of length which depends on the SI unit of time. And the SI unit of mass depends on the SI units of time and length.

In addition of to the three SI base units we recently described, there are other four SI base units which are kelvin (K) for temperature, ampere (A) for electric current, candela (cd) for luminocity and mole (mol) for amount of substance. For the recent redefinition of SI base units you may also check 2019 redefinition of the SI base units

## SI Base Units

There are also other units other than the SI units used in specific regions of the world. We will only consider the SI units here. There are two types of SI units - one type is SI base units and another is SI derived units. The following table shows 7 SI base units and their corresponding physical quantities.

SI Unit | Symbol | Physical Quantity |
---|---|---|

meter | m | length |

second | s | time |

kilogram | kg | mass |

kelvin | K | temperature |

ampere | A | current |

candela | cd | luminous intensity |

mole | mol | amount of substance |

The 7 SI base units in the above table can be used to derive all other SI units called SI derived units. For example the SI unit of speed is the SI derived unit using two base units meter and second, that is meters per second (m/s).

The two units radian for plane angle and steradian for solid angle are neither SI base units nor SI derived units. These two were expressed as SI supplementary units and don't hold the agreement to be SI base units yet. They are more likely considered as SI derived units.

## SI Derived Units List

The following table shows some commonly used SI derived units. You can also download the PDF of this table of the most commonly used SI derived units list.

Qunatity | Symbol | SI Unit Name | SI Unit | Conversion to Base Units | Remarks |
---|---|---|---|---|---|

Area | $A$ | - | $\text{m}^2$ | $\text{m}^2$ | - |

Volume | $V$ | - | $\text{m}^3$ | $\text{m}^3$ | - |

Force | $F$ | Newton | $\text{N}$ | $\text{kg} \cdot \text{m/s}^2$ | - |

Weight | $w$ | Newton | $\text{N}$ | $\text{kg} \cdot \text{m/s}^2$ | - |

Work | $W$ | Joule | $\text{J}$ or $\text{N}\cdot\text{m}$ | $\text{kg} \cdot \text{m}^2/\text{s}^2$ | - |

Kinetic Energy | $K$ | Joule | $\text{J}$ or $\text{N}\cdot\text{m}$ | $\text{kg} \cdot \text{m}^2/\text{s}^2$ | - |

Potential Energy | $U$ | Joule | $\text{J}$ or $\text{N}\cdot\text{m}$ | $\text{kg} \cdot \text{m}^2/\text{s}^2$ | - |

Energy | $E$ | Joule | $\text{J}$ or $\text{N}\cdot\text{m}$ | $\text{kg} \cdot \text{m}^2/\text{s}^2$ | - |

Power | $P$ | Watt | $\text{W}$ or $\text{J/s}$ | $\text{kg} \cdot \text{m}^2/\text{s}^3$ | - |

Impulse | $J$ | - | $\text{N}\cdot\text{s}$ | $\text{kg} \cdot \text{m/s}$ | - |

Momentum (Linear) | $p$ | - | $\text{kg} \cdot \text{m/s}$ | $\text{kg} \cdot \text{m/s}$ | - |

Angular Momentum | $L$ | - | $\text{kg} \cdot \text{m}^2/{s}$ | $\text{kg} \cdot \text{m}^2/{s}$ | - |

Moment of Inertia | $I$ | - | $\text{kg} \cdot \text{m}^2$ | $\text{kg} \cdot \text{m}^2$ | - |

Torque | $\tau$ | - | $\text{N} \cdot \text{m}$ | $\text{kg} \cdot \text{m}^2/\text{s}^2$ | The SI unit of torque is not Joule($\text{J}$) even though we used Newton-meter as Joule for work and energy. Torque is not the work or energy. |

Pressure | $p$ | Pascal | $\text{Pa}$ or $\text{N}/\text{m}^2$ | $\text{kg}/(\text{m} \cdot \text{s}^2)$ | - |

Stress | - | Pascal | $\text{Pa}$ or $\text{N}/\text{m}^2$ | $\text{kg}/(\text{m} \cdot \text{s}^2)$ | - |

Density | $\rho$ | - | $\text{kg}/\text{m}^3$ | $\text{kg}/\text{m}^3$ | - |

Frequency | $f$ | Hertz | $\text{Hz}$ | $\text{s}^{-1}$ | - |

Heat | $Q$ | Joule | $\text{J}$ | $\text{kg} \cdot \text{m}^2/\text{s}^2$ | - |

Heat Current | $H$ | Watt | $\text{W}$ or $\text{J/s}$ | $\text{kg} \cdot \text{m}^2/\text{s}^3$ | - |

Entropy | $S$ | - | $\text{J/K}$ | $(\text{kg} \cdot \text{m}^2)/(\text{K} \cdot \text{s}^2)$ | - |

Electric Charge | $q$ or $Q$ | Coulomb | $\text{C}$ | $\text{A} \cdot \text{s}$ | - |

Electric Flux | $\Phi$ | - | $\text{N} \cdot \text{m}^2 / \text{C}$ | $(\text{kg}\cdot\text{m}^3)/(\text{A}\cdot\text{s}^3)$ | - |

Electric Potential or Potential Difference | $V$ | Volt | $\text{J} / \text{C}$ | $(\text{kg}\cdot\text{m}^2)/(\text{A}\cdot\text{s}^3)$ | - |

Emf | $\mathcal{E}$ | Volt | $\text{J} / \text{C}$ | $(\text{kg}\cdot\text{m}^2)/(\text{A}\cdot\text{s}^3)$ | - |

Electric Field | $E$ | - | $\text{N} / \text{C}$ or $\text{V} /\text{m}$ | $(\text{kg}\cdot\text{m})/(\text{A}\cdot\text{s}^3)$ | - |

Capacitance | $C$ | Farad | $\text{F}$ or $\text{C} /\text{V}$ | $(\text{A}^2\cdot\text{s}^4)/(\text{kg}\cdot\text{m}^2)$ | - |

Resistance | $R$ | Ohm | $\Omega$ or $\text{V} /\text{A}$ | $(\text{kg}\cdot\text{m}^2)/(\text{A}^2\cdot\text{s}^3)$ | - |

Magnetic Field | $B$ | Tesla | $\text{T}$ or $\text{N} /\text{A}\cdot\text{m}$ | $\text{kg}/(\text{A}\cdot\text{s}^2)$ | - |

Magnetic Flux | $\Phi_B$ | Weber | $\text{Wb}$ or $\text{T}\cdot\text{m}^2$ | $\text{kg}\cdot\text{m}^2/(\text{A}\cdot\text{s}^2)$ | - |

Magnetic Dipole Moment | $\mu$ | - | $\text{A}\cdot\text{m}^2$ | $\text{A}\cdot\text{m}^2$ | - |

Inductance | $L$ | Henry | $\text{H}$ or $\text{Wb}/\text{A}$ or $\text{V}\cdot\text{s}/\text{A}$ or $\Omega \cdot\text{s}$ | $(\text{kg}\cdot\text{m}^2)/(\text{A}^2\cdot\text{s}^2)$ | - |

Reactance | $X$ | Ohm | $\Omega$ | $(\text{kg}\cdot\text{m}^2)/(\text{A}^2\cdot\text{s}^3)$ | - |

Impedance | $Z$ | Ohm | $\Omega$ | $(\text{kg}\cdot\text{m}^2)/(\text{A}^2\cdot\text{s}^3)$ | - |

These SI base units and the all the SI derived units from those SI base units are used to measure all physical quantities in Physics and you'll use them for our measurement. You can further increase or decrease size of these units using unit prefixes. We also need to convert one form of unit to another form, more generally from other types of units to the SI units. While doing so, we carry the units and cancel them as needed. A good habit in calculations in Physics is to carry them in the calculation and cancel them as needed. It will help to keep track of whether the calculation is going in the right direction or not.