Physical World and Measurement
Chapter 01: A guide to fundamental concepts, units, and dimensions.
Physics and Technology
Every natural occurrence around us like the sun, the wind, the planets, atmosphere, human body etc, follow some basic law. To understand by observing natural occurrence is called Physics.
These laws of Physics are related and applicable to every aspects of life, thus understanding them leads to their applications in several for further development of society, which is also as technology.
Measurement
Measurement is a process of determining how large or small a physical quantity is as compared to a basic reference standard. This reference standard is called the unit of the particular physical quantity.
Physical quantities and their units are described as below.
Physical Quantities
The quantities which can be measured directly or indirectly and by means of which we can describe the laws of Physics are called physical quantities. e.g. Length, mass, volume, etc. There are of different types of physical quantities:
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Fundamental Quantities: The quantities which do not depend upon other quantities for their complete definition are known as fundamental or base quantities. There are seven fundamental quantities:
Length, Mass, Time, Electric current, Thermodynamic temperature, Luminous intensity, Amount of substance.
- Derived Quantities: The quantities which can be expressed in terms of the fundamental quantities are called derived quantities. e.g. Speed, volume, acceleration, force, etc.
- Supplementary Quantities: Other than fundamental and derived quantities, there are two more quantities called as supplementary quantities. e.g. Plane angle and solid angle, etc.
Unit
It is a standard measurement in which the magnitude of a physical quantity is expressed, e.g. kilogram, metre, second, etc.
There are different types of units as below:
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Fundamental Units: The units of fundamental quantities are called fundamental or base units. Fundamental units of fundamental quantities are as follows:
Name of Quantities Name of Unit Name of Quantities Name of Unit Length metre Thermodynamics temperature kelvin Mass kilogram Luminous intensity candela Time second Amount of substance mole Electric current ampere -
Derived Units: The units of Physical quantities which can be obtained from fundamental units are called derived units. For example, unit of speed is a derived unit.
Speed = $\frac{\text{Distance travelled}}{\text{Time taken}}$
$\therefore$ Unit of speed = $\frac{\text{Unit of distance}}{\text{Unit of time}} = \frac{\text{metre}}{\text{second}} = \text{ms}^{-1}$
Thus, the unit of speed, ms⁻¹, is derived from fundamental units of length and time.
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Supplementary Units: The unit of supplementary quantities is called supplementary units. Supplementary units are given below in table:
S.No. Supplementary Quantities Units Symbol 1. Plane angle radian rad 2. Solid angle steradian sr
Some Practical Units
- 1 fermi = $10^{-15}$ m
- 1 X-ray unit = $10^{-13}$ m
- 1 astronomical unit = $1.49 \times 10^{11}$ m (average distance between sun and earth)
- 1 light year = $9.46 \times 10^{15}$ m
- 1 parsec = $3.08 \times 10^{16}$ m = 3.26 light year
System of Unit
A system of unit is the complete set of units, both fundamental and derived, for all kinds of physical quantities. The common system of units which is used in mechanics are given below:
- CGS System: CGS stands for centimetre, gram and second.
- FPS System: FPS stands for foot, pound and second.
- MKS System: It stands for metre, kilogram and second.
- SI System: SI is the abbreviation of System International unit, which is French equivalent of International system of units. It is based on the seven basic units and two supplementary units described above.
Dimensions
The dimension of a physical quantity are the powers to which the fundamental units are raised in order to obtain the derived unit of that quantity. To express the dimensions of physical quantities in mechanics, the mass, length and time are denoted by [M] [L] and [T], respectively. If the dimensions of a physical quantity are $a$ in mass, $b$ in length, and $c$ in time, then the dimensional formula of that physical quantity shall be written in the following manner: $[M^a L^b T^c]$.
To make it more clear, consider the physical quantity force:
force = mass $\times$ acceleration
= mass $\times \frac{\text{velocity}}{\text{time}}$
= mass $\times \frac{\text{length / time}}{\text{time}}$
= (mass) $\times$ (length) $\times$ (time)$^{-2}$
Thus, the dimensions of force are 1 in mass, 1 in length and -2 in time. Hence, the dimensional formula of force can be written as $[MLT^{-2}]$.
Dimensional Formulae and SI Units of Some Physical Quantities
| Physical Quantity with Formula | Dimensional Formula | SI Units |
|---|---|---|
| Velocity = Displacement/Time | $[M^0 L^1 T^{-1}]$ | m/s |
| Acceleration = Velocity/Time | $[M^0 L^1 T^{-2}]$ | m/s$^2$ |
| Force = Mass $\times$ Acceleration | $[MLT^{-2}]$ | kg-m/s$^2$ $\rightarrow$ Newton $\rightarrow$ N |
| Work = $Fs \cos \theta$ | $[ML^2T^{-2}]$ | kg-m$^2$/s$^2$ $\rightarrow$ joule $\rightarrow$ J |
| Kinetic energy = $\frac{1}{2}mv^2$ | $[ML^2T^{-2}]$ | joule |
| Potential energy = $mgh$ | $[ML^2T^{-2}]$ | J |
| Torque = $Fr \sin\theta$ | $[ML^2T^{-2}]$ | N-m |
| Power = Work/Time | $[ML^2T^{-3}]$ | kg-m$^2$/s$^3$ $\rightarrow$ J/s $\rightarrow$ watt $\rightarrow$ W |
| Momentum = Mass $\times$ Velocity | $[MLT^{-1}]$ | kg-m/s or N-s |
| Impulse = $F \Delta t$ | $[MLT^{-1}]$ | N-s |
| Angle = Arc/Radius | $[M^0 L^0 T^0]$ | radian $\rightarrow$ rad |
| Strain = $\Delta L / L$ or $\Delta V / V$ | Dimensionless | no unit |
| Frequency = 1/Time period | $[M^0 L^0 T^{-1}]$ | hertz $\rightarrow$ Hz |
| Angular velocity = Angle/Time | $[M^0 L^0 T^{-1}]$ | rad/s |
| Moment of inertia = $\Sigma mr^2$ | $[ML^2 T^0]$ | kg-m$^2$ |
| Angular momentum = $I\omega$ | $[ML^2T^{-1}]$ | kg-m$^2$/s or J-s |
| Surface tension = Force/Length | $[ML^0 T^{-2}]$ | N/m |
| Spring constant = $F/x$ | $[ML^0 T^{-2}]$ | N/m |
| Surface energy = Energy/Area | $[ML^0 T^{-2}]$ | J/m$^2$ |
| Intensity = $\frac{\text{Energy}}{\text{Area} \times \text{Time}}$ | $[ML^0 T^{-3}]$ | J/m$^2$-s $\rightarrow$ W/m$^2$ |
| Planck’s constant = $E/\nu$ | $[ML^2 T^{-1}]$ | J-s |
| Coefficient of viscosity = $\frac{\text{Force} \times \text{Distance}}{\text{Area} \times \text{Velocity}}$ | $[ML^{-1} T^{-1}]$ | Nm$^{-2}$s or Pa-s |
| Charge = Current $\times$ Time | $[M^0 L^0 T^1 A^1]$ | C |
| Electric potential = $\frac{\text{Work}}{\text{Charge}}$ | $[ML^2 T^{-3} A^{-1}]$ | N/C |
| Magnetic flux = $\frac{W}{q} dt$ | $[ML^2 T^{-2} A^{-1}]$ | Wb |
| Magnetic dipole moment = $IA$ | $[M^0 L^2 T^0 A^1]$ | A-m$^2$ |
| Electric flux = $E \times A$ | $[ML^3 T^{-3} A^{-1}]$ | Nm$^2$/C |
| Resistance = $V/I$ | $[ML^2 T^{-3} A^{-2}]$ | $\Omega$ |
Significant Figure
Significant figures in the measured value of a physical quantity tell the number of digits in which we have confidence. Following rules are observed in counting number of significant figures in a given measured quantity:
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All non-zero digits are significant.
e.g. 42.3 has three significant figures.
e.g. 243.4 has four significant figures.
e.g. 24.123 has five significant figures.
e.g. 1.987 has four significant figures.
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A zero becomes significant figure, if it appears between two non-zero digits.
e.g. 5.03 has three significant figures.
e.g. 5.604 has four significant figures.
e.g. 4.004 has four significant figures.
e.g. 2001.5 has five significant figures.
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Leading zeros or zeros placed to the left of the number are never significant.
e.g. 0.543 has three significant figures.
e.g. 0.045 has two significant figures.
e.g. 0.006 has one significant figure.
e.g. 0.00025 has two significant figures.
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Trailing zeros or zeros placed to the right of the number are significant.
e.g. 4.330 has four significant figures.
e.g. 433.00 has five significant figures.
e.g. 343.000 has six significant figures.
e.g. 1.0 has two significant figures.
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In exponential notation, the numerical portion gives the number of significant figures.
e.g. $1.32 \times 10^{-2}$ has three significant figures.
e.g. $1.32 \times 10^{4}$ has three significant figures.
e.g. $6.022 \times 10^{23}$ has four significant figures.
Rounding Off
The result of a calculation with numbers containing more than one uncertain digit should be rounded off. The rules for rounding off are as follows:
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If the digit to be dropped is less than 5, then the preceding digit is left unchanged.
e.g. 7.82 is rounded off to 7.8 (rounding to one decimal place).
e.g. 3.94 is rounded off to 3.9 (rounding to one decimal place).
e.g. 12.234 is rounded off to 12.23 (rounding to two decimal places).
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If the digit to be dropped is more than 5, then the preceding digit is raised by one.
e.g. 6.87 is rounded off to 6.9 (rounding to one decimal place).
e.g. 12.78 is rounded off to 12.8 (rounding to one decimal place).
e.g. 5.679 is rounded off to 5.68 (rounding to two decimal places).
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If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit is raised by one.
e.g. 16.351 is rounded off to 16.4 (rounding to one decimal place).
e.g. 6.758 is rounded off to 6.8 (rounding to one decimal place).
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If the digit to be dropped is 5 and is followed only by zeros (or nothing), then the preceding digit is left unchanged if it is even.
e.g. 3.250 is rounded off to 3.2 (rounding to one decimal place).
e.g. 12.65 is rounded off to 12.6 (rounding to one decimal place).
e.g. 8.4500 is rounded off to 8.4 (rounding to one decimal place).
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If the digit to be dropped is 5 and is followed only by zeros (or nothing), then the preceding digit is raised by one if it is odd.
e.g. 3.750 is rounded off to 3.8 (rounding to one decimal place).
e.g. 16.15 is rounded off to 16.2 (rounding to one decimal place).
e.g. 9.35 is rounded off to 9.4 (rounding to one decimal place).