\r\n The quantities that can be measured and which can describe various physical happenings or laws of physics are called physical quantities.\r\n\r\n The unit of a physical quantity is a widely accepted standard to measure and compare the physical\r\n quantity.\r\n
\r\n\n A physical quantity that cannot be expressed in terms of any other physical quantity is called a fundamental\r\n quantity. The unit of such a quantity is known as a fundamental unit. For example, mass is a fundamental quantity\r\n as it cannot be expressed or described in terms of any other physical quantity. Accordingly, the unit of mass, \\(kg\\), is a fundamental unit.\r\n
\r\n\n Quantities that are expressed in terms of fundamental quantities are known as derived quantities and their\r\n units are called derived units. For example, density is a derived quantity as it is defined as mass per unit volume.\r\n Accordingly, the unit of density, \\(kg/m^3\\), is a derived unit.\r\n
\r\n\n A complete set of units comprising both fundamental and derived units is called a system of units. Every\r\n system of unit has a different set of fundamental and derived quantities. However, in most of the system of\r\n units in physics mass, length and time are chosen as the fundamental quantities. Some of the common system\r\n of\r\n units and the units accepted for length, mass and time in that system of units are mentioned below.\r\n
\r\n\r\n| \n Fundamental Quantity\n | \r\nUnit | \n
|---|---|
| Length | \r\nMetre | \n
| Mass | \r\nKilogram | \n
| Time | \r\nSecond | \n
| Electric current | \r\nAmpere | \r\n\r\n \r\n
| Thermodydnamic temperature | \r\nKelvin | \n
| Amount of substance | \r\nMole | \r\n
| Luminous intensity | \r\nCandela | \r\n
\n Besides fundamental quantities and derived quantities there are two other quantities known as supplementary\r\n quantities. The units of supplementary quantities are called supplementary units. The two supplementary\r\n quantities are plane angle and solid angle.\n
\r\n\n While expressing a derived quantity in terms of fundamental quantities, it is written as a product of\r\n fundamental quantities raised to different powers. These powers to which a base quantity is raised to\r\n express it in terms of fundamental quantities are called its dimensions.\r\n
The dimensions of mass length and time are described as [M], [L] and [T] respectively. For example, velocity as\r\n we know is the ratio of distance travelled by time i.e.,\r\n $$\\text{Velocity}= \\frac{\\text{Distance travelled}}{\\text{Time}}$$\r\n Thus, the dimensions of velocity is given as \\([L]/[T]\\) i.e., \\([M^0LT^{-1}]\\).\r\n
\n Dimensional analysis helps us to check the correct correctness of a formula and also to derive new formulas\r\n for research purpose, although it has some limitations as no work related to the magnitude of the quantities\r\n can be done.\n
\r\n \r\n\r\n\r\n\n Let us have a look on different methods that we use for measuring lengths. Depending on the range of length,\r\n there are different methods to measure it. For example we use a metre scale to measure length in the range\r\n of \\(10^{-3}\\)m to\r\n \\(10 ^{2}\\)m , a vernier calipers can measure lengths upto an accuracy of\r\n \\(10^{-4} \\)m, a screw gauge can be used to measure lengths up to \\(10^{-5}\\)m, but to\r\n measure lengths not in these ranges, we need some special methods for example to measure large distances such\r\n as the distance of planet or star from earth we use parallax method and to measure extremely small\r\n distances, for example the size of a molecule we use electron beams.\r\n
\r\n\n For measuring different ranges of lengths, there are some special units which help us in measuring\r\n very large and short lengths. Some of them are given below\r\n
\r\n\n While mass of the commonly available objects can be measured by a normal balance, we need special methods to\r\n measure the mass of extremely large and small objects. For example, mass of very large objects such as\r\n planets, stars etc. are measured with the help of gravitational method in which their gravitational effect\r\n of other objects are analyzed. However, for measuring mass of very small objects such as atomic of subatomic\r\n particles, mass spectrograph is used.\r\n
The mass of of atomic particles is expressed in a unit called\r\n atomic mass unit (a.m.u or u) which is equal to \\(\\frac{1}{12}^{th}\\) the mass of one carbon-12 atom and its\r\n value is given as,\r\n $$1 \\; a.m.u=1.66 \\times 10^{-27} kg $$\r\n
\n Any time interval is measured with the help of a clock. Now a days, a cesium clock is used in the national\r\n standards which is based on the periodic vibrations produced in a cesium atom. One second is the time\r\n interval\r\n for 91,926,311,770 vibrations of radiation in a Cs-133 atom. The radiation corresponds to the transition\r\n between the two hyperfine levels of a Cs-133 atom in ground state.\n
\r\n\r\n\n Whenever we do a measurement there is always a slight difference between the true and measured value of the\r\n quantity. The\r\n result of every experiment contains some uncertainty. These uncertainties in the measured value of the\r\n experiment are called errors.\r\n
Errors are broadly classified into systematic errors and random errors.\r\n
\n To minimise errors, same experiment is repeated a large number of times, then the arithmetic mean of all the\r\n observations is calculated. This arithmetic mean is the most accurate value of the measured quantity and is\r\n considered as the true value.\r\n If a physical quantity is measured n times and \\(a_1, a_2,...,a_n\\) are the measured values, its mean value\r\n \\(a_m\\) is given as,\n
\r\n $$ a_m=\\frac{a_1+a_2+...+a_n}{n} $$\r\n\n Ratio of mean absolute error to the mean value is called relative error.\r\n $$\\text{Relative error}= \\frac{∆a_{mean}}{a_m}$$\r\n Relative expressed in the form of percentage is called percentage error.\r\n $$ \\text{Percentage error}= \\frac{∆a_{mean}}{a_m} \\times 100 \\% $$\n
\r\n\r\n\n Since the results of measurements contains some errors, the results are reported in a way that determines the\r\n accuracy of the measured value and this is done with the help of significant figures.\r\n In a reported vaue, significant figures are the digits that are absolutely correct plus the first uncertain\r\n digit. More number of significant figures implies more accuracy of the measured value.\n
\r\n Rules for determining significant figures are given below.\r\n