Unit and measurements

The standard measure used for comparing (measurement) of a physical quantity is called a unit. To measure a physical quantity, we need some standard units of that quantity.

Measurement

means comparing a physical quantity with another homogenous quantity of the same kind taken as a standard to determine how many times the given standard is contained in the given physical quantity.

Fundamental and Derived Quantities

We can define a set of fundamental quantities as follows :

1.  The fundamental quantities should be independent of each other, and 
2. All the quantities may be expressed in terms of the fundamental quantities.

It turns out that the number of fundamental quantities is only seven. All the rest may be derived from these quantities by multiplication division. The units defined for the fundamental quantities are called fundamental units, and those obtained for the derived quantities are called the derived units.

SI Units 

In 1971, CGPM held its meeting and decided on a system of units known as the International System of Units. It is abbreviated as SI .

Fundamental or base quantities

Besides the seven fundamental units two supplementary units are defined . They are plane angles and solid angles. The unit for plane angle is radian with the symbol rad, and the unit for the solid angle is steradian with the symbol sr.

Definitions of base units

• Metre: It is the unit of length 
• Kilogram: The mass of a cylinder made of platinum-iridium alloy kept at the International Bureau of Weights and measurement is defined as 1kg
• Second: Cesium-133 atoms emit electromagnetic radiation of several wavelengths. a particular radiation is selected, which corresponds to the transition between the two hyperfine levels of the ground state of Cs-133. Each radiation has a period of repetition of certain characteristics.The time duration in 9,192,631,770 time periods of the selected transition is defined as 1s
• Ampere: 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 metre apart in vacuum, would produce between these conductors a force equal to 0.0000002 newton per metre of length
• Kelvin: The kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water 
• Mole: The mole is the amount of substance of a system, which contains as many elementary entities as there are atoms in 0.012 kilograms of carbon -12
• Candela: The SI unit of luminous intensity is 1 cd which is the luminous intensity of blackbody of surface area 1/600000 m2 placed at the temperature of freezing platinum and pressure of 101,325 N/m2, in the direction perpendicular to its surface.

Parallax Method 

The parallax method is used to measure the distance of planets and stars from earth.

If a distant object , a planet or star subtends parallax angle 𑁜 on an arc of radius b on earth, then distance of that distant object from the basis is given by 

         s=b/𑁜

Dimension

 All the physical quantities of interest can be derived from the base quantities. When a quantity is expressed in terms of the base quantities , it is written as a product of different powers of the base quantities. The exponent of a base quantity that enters into the expression is called the dimension of the quantity in that base

 Dimensional formula: The dimensional formula of a physical quantity is an expression telling us how and which of the fundamental quantities enter into the unit of that quantity.

 Eg. dimensional formula of force is equal to 

                    Force = mass x acceleration

                            M LT-2

Applications of dimensions

• To check the correctness of an equation
• To derive the relation between different physical quantities
• To convert one system of units into another system.

Significant figures 

 The number of significant figures in a result is simply the number of figures that are known with some degree of reliability. The number 13.2 is said to have 3 significant figures. The number is said to have 4 significant figures.

Rules for deciding the number of significant figures in a measured quantity :

1. All nonzero digits are significant. 

1.234 g  has 4 significant figures.

1. Zeros between nonzero digits are significant.

1002 kg has 4 significant figures.

    (c )    zeros to the left of the nonzero digits are not significant; such zeros merely indicate the position of the decimal point.

 0.001 g has only 1 significant figures

 (d)  Zeros to the right of a decimal point in a significant number.

     0.023 ml has 2 significant figures 

(e) When a number ends in zeros that are not to the right of a decimal point, the zeros are necessarily significant.

 5.06✖104 kg  has 3 significant figures

Error 

 The measured value of the physical quantity is usually different from its true value. The result of every measurement by any measuring instrument is an approximate number, which contains some uncertainty. This uncertainty is called an error. Every calculated quantity , which is based on measured values, also has an error.

1. Systematic Error : errors resulting from measuring devices being out of calibration. Such measurements will be consistently too small or too large. These errors can be eliminated by pre-calibrating against a known, trusted standard.

1. Random errors : errors resulting from the fluctuation of measurement of the same quantity about the average. The measurements are equally probable of being too large or too small. These errors generally result from the fitness of scale division of a measuring device.

 Conclusion

• The chosen reference standard of measurement in multiples of which a physical quantity is expressed is called the unit of that quantity. 
• Dimensions of a physical quantity are the powers or exponents to which the base quantities are raised to represent that quantity.
• The number of digits in the measured value includes certain digits plus one uncertain digit are called significant figures.