Units and Measurements:
1. Physical Quantity:
The quantities which can be measured by an instrument and by means of which we can describe the laws of physics are called physical quantities.
Fig: Physical quantitiesTypes of physical quantities:
1.1 Fundamental:
Although the number of physical quantities that we measure is very large, we need only a limited number of units for expressing all the physical quantities since they are interrelated with one another. So, certain physical quantities have been chosen arbitrarily and their units are used for expressing all the physical quantities, such quantities are known as Fundamental, Absolute or Base Quantities (such as length, time and mass in mechanics)
(i) All other quantities may be expressed in terms of fundamental quantities.
(ii) They are independent of each other and cannot be obtained from one another.
An international body named General Conference on Weights and Measures chose seven physical quantities as fundamental:
(1) length
(2) mass
(3) time
(4) electric current
(5) thermodynamic temperature
(6) amount of substance
(7) luminous intensity
Fig: New SI: Dependence of base unit definitions on physical constants with fixed numerical values and on other base units that are derived from the same set of constantsNote: These are also called as absolute or base quantities.
In mechanics, we treat length, mass and time as the three basic or fundamental quantities.
1.2 Derived: Physical quantities which can be expressed as combination of base quantities are called as derived quantities.
For example: Speed, velocity, acceleration, force, momentum, pressure, energy etc.
1.3 Supplementary: Beside the seven fundamental physical quantities two supplementary quantities are also defined, they are :
(1) Plane angle
(2) Solid angle
Note : The supplementary quantities have only units but no dimensions.
2. Magnitude:
Magnitude of physical quantity = (numerical value) x (unit)
Magnitude of a physical quantity is always constant. It is independent of the type of unit.
Ex. 2: Length of a metal rod bar is unchanged whether it is measured as 2 metre or 200 cm. Observe the change in the Numerical value (from 2 to 200) as unit is changed from metre to cm.
3. Unit:
Measurement of any physical quantity is expressed in terms of an internationally accepted certain basic reference standard called unit.
The units for the fundamental or base quantities are called fundamental or base unit. Other physical quantities are expressed as combination of these base units and hence, called derived units.
A complete set of units, both fundamental and derived is called a system of unit.
3.1. Principle systems of Unit:
There are various system in use over the world: CGS, FPS, SI (MKS) etc.
Fig: Silicon sphere for the Avogadro project used for measuring the Avogadro constant to a relative standard uncertainty of 2×10^{−8} or less, held by Achim Leistner
Table 1 : Units of some physical quantities in different systems.
| Physical Quantity | System | ||
CGS (Gaussian) | MKS (SI) | FPS (British) | ||
| Length | centimeter | meter | foot |
Fundamental | Mass | gram | kilogram | pound |
| Time | second | second | second |
| Force | dyne | newton → N | poundal |
Derived | Work or Energy | erg | joule → J | ft-poundal |
| Power | erg/s | watt → W | ft-poundal/s |
3.2 Supplementary units:
(1) Plane angle: radian (rad)
(2) Solid angle: steradian (sr)
* The SI system is at present widely used throughout the world. In IIT JEE only SI system is followed.
3.3 Definitions of some important SI Units:
(i) Metre: 1 m = 1,650, 763.73 wavelengths in vaccum, of radiation corresponding to organ-red light of krypton-86.
(ii) Second: 1 s = 9,192, 631,770 time periods of a particular from Ceasium - 133 atom.
(iii) Kilogram: 1 kg = mass of 1 litre volume of water at 4°C.
(iv) Ampere: It is the current which when flows through two infinitely long straight conductors of negligible cross-section placed at a distance of one metre in vacuum produces a force of 2 × 10^{-7} N/m between them.
(v) Kelvin: 1 K = 1/273.16 part of the thermodynamic temperature of triple point of water.
(vi) Mole: It is the amount of substance of a system which contains as many elementary particles (atoms, molecules, ions etc.) as there are atoms in 12 g of carbon - 12.
(vii) Candela: It is luminous intensity in a perpendicular direction of a surface of of a black body at the temperature of freezing point under a pressure of 1.013 × 10^{5} N/m^{2}.
(viii) Radian: It is the plane angle between two radii of a circle which cut-off on the circumference, an arc equal in length to the radius.
(ix) Steradian: The steradian is the solid angle which having its vertex at the centre of the sphere, cut-off an area of the surface of sphere equal to that of a square with sides of length equal to the radius of the sphere.
Ex. 3: Fin d the SI unit of speed\ acceleration
Sol.
(called as meter per second)
(called as meter per second square)
4. S.I Prefixes:
The magnitudes of physical quantities vary over a wide range. The CGPM recommended standard prefixes for magnitude too large or too small to be expressed more compactly for certain power of 10.
Power of 10 | Prefix | Symbol | Power of 10 | Prefix | Symbol |
10^{18} | exa | E | 10^{-1} | deci | d |
10^{15} | peta | P | 10^{-}^{2} | centi | c |
10^{12} | tera | T | 10^{-}^{3} | milli | m |
10^{9} | giga | G | 10^{-6} | micro | μ |
10^{8} | mega | M | 10^{-9} | nano | n |
10^{3} | kilo | k | 10^{-12} | pico | P |
10^{2} | hecto | h | 10^{-15} | femto | f |
10^{1} | deca | da | 10^{-18} | atto | a |
5. General Guidelines for using Symbols for SI Units, Some other Units, and SI prefixes:
(a) Symbols for units of physical quantities are printed/written in Roman (upright type), and not in italics.
For example: 1 N is correct but 1 N is incorrect.
(b) (i) Unit is never written with capital initial letter even if it is named after a scientist.
For example: SI unit of force is newton (correct) Newton (incorrect)
(ii) For a unit named after a scientist, the symbol is a capital letter. But for other units, the symbol is NOT a capital letter.
For example:
force → newton (N)
energy → joule (J)
electric current → ampere (A)
temperature → kelvin (K)
frequency → hertz (Hz)
For example:
length → meter (m)
mass → kilogram (kg)
luminous intensity → candela (cd)
time → second (s)
Note: The single exception is L, for the unit litre.
(c) Symbols for units do not contain any final full stop at the end of recommended letter and remain unaltered in the plural, using only singular form of the unit.
For example:
Quantity | Correct | Incorrect |
25 centimeters | 25 cm | 25 cm. 25 cms. |
(d) Use of slash (/) is recommended only for indicating a division of one letter unit symbol by another unit symbol. Not more than one slash is used.
For example:
Correct | Incorrect |
m/s^{2} | m / s / s |
N s/m^{2} | Ns/m/m |
J/K mol | J/K/ mol |
kg/m s | kg / m / s |
(e) Prefix symbols are printed in roman (upright) type without spacing between the prefix symbol and the unit symbol. Thus certain approved prefixes written very close to the unit symbol are used to indicate decimal fractions or multiples of a SI unit, when it is inconveniently small or large.
For example:
megawatt | 1 MW = 10^{6} W |
centimetre | 1 cm = 10^{-2} m |
kilometre | 1 km = 10^{3} m |
millivolt | 1 mV = 10^{-3} V |
kilowatt-hour | 1 kW h = 10^{3} W h = 3.6 M J = 3.6 * 10^{6} J |
microampere | 1μA=10^{-6} A |
angstrom | 1 Å = 0.1 nm = 10 ^{-10} m |
nanosecond | 1 ns = 10^{-9} s |
picofarad | 1 pF = 10^{-12} F |
microsecond | 1μs =10^{-6}s |
gigahertz | 1 GHz = 10^{9} Hz |
micron | 1 μm = 10^{-6} m |
The unit 'fermi', equal to a femtometre or 10^{-15} m has been used as the convenient length unit in nuclear studies.
(f) When a prefix is placed before the symbol of a unit, the combination of prefix and symbol is considered as a new symbol, for the unit, which can be raised to a positive or negative power without using brackets. These can be combined with other unit symbols to form compound unit.
For example:
Quantity | Correct | Incorrect |
cm^{3} | (cm)^{3} = (0.01 m)^{3} = (10-^{2}m)3 = 10-6 m^{3} | 0.01 m^{3} or 10^{-2} m^{3} |
mA^{2} | (mA)^{2} = (0.001 A)^{2} = (10-3A)^{2} = 10^{-6} A^{2} | 0.001 A^{2} |
(g) A prefix is never used alone. It is always attached to a unit symbol and written or fixed before the unit symbol.
For example :
10^{3}/m^{3} = 1000/m^{3} or 1000 m^{-3}, but not k/m^{3} or k m^{-3}.
(h) Prefix symbol is written very close to the unit symbol without spacing between them, while unit symbols are written separately with spacing with units are multiplied together.
For example:
Quantity | Correct | Incorrect |
1 ms^{-1} | 1 metre per second | 1 milli per second |
1 ms | 1 millisecond | 1 metre second |
1 C m | 1 coulomb metre | 1 centimetre |
1 cm | 1 centimetre | 1 coulomb metre |
(i) The use of double prefixes is avoided when single prefixes are available.
For example:
Quantity | Correct | Incorrect |
10^{-9} m | 1 nm (nanometre) | 1 mu m (milli micrometre) |
10^{-6} m | 1pm (micron) | 1 m m m (milli millimetre) |
10^{-12}F | 1 pF (picofarad) | 1 μ ,μ F (micro microfarad) |
10^{9}F | 1 GW (gigawatt) | 1 KM W (Kilo megawatt) |
(j) The use of a combination of unit and the symbols for unit is avoided when the physical quantity is expressed by combining two or more units.
Quantity | Correct | Incorrect |
joule per mole Kelvin | J/mol K | Joule / mole K |
newton metre second | N m s | newton m second |
5.1. Characteristics of base units or standards:
(A) Well defined
(B) Accessibility
(C) Invariability
(D) Convenience in use.
5.2 Some special types of units:
1. 1 Micron (1μ) = 10^{-4} cm = 10^{-6} m (length)
2. 1 Angstrom (1 Å) = 10^{-8} cm = 10^{-10}m (length)
3. 1 fermi (1f) = 10^{-13} cm = 10^{-15} m (length)
4. 1 inch = 2.54 cm (length)
5. 1 mile = 5280 feet = 1.609 km (length)
6. 1 atmosphere = 10^{5} N/m^{2} = 76 torr = 76 mm of Hg pressure (pressure)
7. 1 litre = 10^{-3} m^{3} = 1000 cm^{3 }(volume)
8. 1 carat = 0.0002 kg (weight)
9. 1 pound (Ib) = 0.4536 kg (weight)
Accuracy And Precision
The closeness of a measured value to the actual value of the object being measured is called as accuracy of a substance. For instance, if in lab you obtain a weight measurement of 3.2 kg for a given substance, but the actual or known weight is 10 kg, then your measurement is not accurate.
The closeness of two or more measurements to each other is known as the precision of a substance. From the above given example we can figure out that, if you weigh a given substance five times, and get 3.2 kg each time, then your measurement is very precise. Precision is independent of accuracy. The below example will tell you about how you can be precise but not accurate and vice versa.
Precision is sometimes separated into:
Repeatability
The variation arising when the conditions are kept identical and repeated measurements are taken during a short time period.
Reproducibility
The variation arising using the same measurement process among different instruments and operators, and over longer time periods.
In other words, accuracy is the degree of closeness between a measurement and the measurement’s true value. Precision is the degree to which repeated measurements under the same conditions are unchanged.
A good analogy for understanding accuracy and precision is to imagine a football player shooting at the goal. If the player shoots into the goal, he is said to be accurate. A football player who keeps striking the same goal post is precise but not accurate.
Therefore a football player can be accurate without being precise if he hits the ball all over the place but still scores. A precise player will hit the ball to the same spot repeatedly, irrespective of whether he scores or not.
A precise and accurate football player will not only aim at a single spot but also score the goal.