| Dimension | Quantity | |
| \[[{{M}^{0}}{{L}^{0}}{{T}^{-1}}]\] | Frequency, angular frequency, angular velocity, velocity gradient and decay constant | |
| \[[{{M}^{1}}{{L}^{2}}{{T}^{-2}}]\] | Work, internal energy, potential energy, kinetic energy, torque, moment of force | |
| \[[{{M}^{1}}{{L}^{-1}}{{T}^{-2}}]\] | Pressure, stress, Young?s modulus, bulk modulus, modulus of rigidity, energy density | |
| \[[{{M}^{1}}{{L}^{1}}{{T}^{-1}}]\] | Momentum, impulse | |
| \[[{{M}^{0}}{{L}^{1}}{{T}^{-2}}]\] | Acceleration due to gravity, gravitational field intensity | |
| \[[{{M}^{1}}{{L}^{1}}{{T}^{-2}}]\] | Thrust, force, weight, energy gradient | |
| \[[{{M}^{1}}{{L}^{2}}{{T}^{-1}}]\] | Angular momentum and Planck?s constant | |
| \[[{{M}^{1}}{{L}^{0}}{{T}^{-2}}]\] | Surface tension, Surface energy (energy per unit area) | |
| \[[{{M}^{0}}{{L}^{0}}{{T}^{0}}]\] | Strain, refractive index, relative density, angle, solid angle, distance gradient, relative permittivity (dielectric constant), relative permeability etc. |
| Quantity | Unit | Dimension |
| Temperature (T) | Kelvin | \[[{{M}^{0}}{{L}^{0}}{{T}^{0}}{{\theta }^{1}}]\] |
| Heat (Q) | Joule | \[[M{{L}^{2}}{{T}^{\text{ }2}}]\] |
| Specific Heat (c) | Joule/kg-K | \[[{{M}^{0}}{{L}^{2}}{{T}^{\text{ }2}}\theta {{~}^{1}}]\] |
| Thermal capacity | Joule/K | \[[{{M}^{1}}{{L}^{2}}T{{~}^{\text{ }2}}\theta {{~}^{1}}]\] |
| Latent heat (L) | Joule/kg | \[[{{M}^{0}}{{L}^{2}}{{T}^{2}}]\] |
| Gas constant (R) | Joule/mol-K | \[[{{M}^{1}}{{L}^{2}}{{T}^{2}}{{\theta }^{1}}]\] |
| Boltzmann constant (k) | Joule/K | more...
(1) To find the unit of a physical quantity in a given system of units : To write the definition or formula for the physical quantity we find its dimensions. Now in the dimensional formula replacing M, L and T by the fundamental units of the required system we get the unit of physical quantity. However, sometimes to this unit we further assign a specific name,
e.g., Work = Force ´ Displacement
So \[[W]=[ML{{T}^{-2}}]\times [L]=[M{{L}^{2}}{{T}^{-2}}]\]
So its unit in C.G.S. system will be g \[c{{m}^{2}}/{{s}^{2}}\] which is called erg while in M.K.S. system will be \[kg-{{m}^{2}}/{{s}^{2}}\] which is called joule.
(2) To find dimensions of physical constant or coefficients : As dimensions of a physical quantity are unique, we write any formula or equation incorporating the given constant and then by substituting the dimensional formulae of all other quantities, we can find the dimensions of the required constant or coefficient.
(i) Gravitational constant : According to Newton?s law of gravitation \[F=G\frac{{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\] or \[G=\frac{F{{r}^{2}}}{{{m}_{1}}{{m}_{2}}}\]
Substituting the dimensions of all physical quantities \[[G]=\frac{[ML{{T}^{-2}}][{{L}^{2}}]}{[M][M]}=[{{M}^{-1}}{{L}^{3}}{{T}^{-2}}]\]
(ii) Plank constant : According to Planck \[E=h\nu \] or \[h=\frac{E}{\nu }\]
Substituting the dimensions of all physical quantities \[[h]=\frac{[M{{L}^{2}}{{T}^{-2}}]}{[{{T}^{-1}}]}=[M{{L}^{2}}{{T}^{-1}}]\]
(iii) Coefficient of viscosity : According to Poiseuille?s formula \[\frac{dV}{dt}=\frac{\pi p{{r}^{4}}}{8\eta l}\] or \[\eta =\frac{\pi p{{r}^{4}}}{8l(dV/dt)}\]
Substituting the dimensions of all physical quantities \[[\eta ]=\frac{[M{{L}^{-1}}{{T}^{-2}}][{{L}^{4}}]}{[L][{{L}^{3}}/T]}=[M{{L}^{-1}}{{T}^{-1}}]\]
(3) To convert a physical quantity from one system to the other : The measure of a physical quantity is nu = constant
If a physical quantity X has dimensional formula [MaLbTc] and if (derived) units of that physical quantity in two systems are \[[M_{1}^{a}L_{1}^{b}T_{1}^{c}]\] and \[[M_{2}^{a}L_{2}^{b}T_{2}^{c}]\] respectively and \[{{n}_{1}}\] and \[{{n}_{2}}\] be the numerical values in the two systems respectively, then \[{{n}_{1}}[{{u}_{1}}]={{n}_{2}}[{{u}_{2}}]\]
\[\Rightarrow \] \[{{n}_{1}}[M_{1}^{a}L_{1}^{b}T_{1}^{c}]={{n}_{2}}[M_{2}^{a}L_{2}^{b}T_{2}^{c}]\]
\[\Rightarrow \] \[{{n}_{2}}={{n}_{1}}{{\left[ \frac{{{M}_{1}}}{{{M}_{2}}} \right]}^{a}}{{\left[ \frac{{{L}_{1}}}{{{L}_{2}}} \right]}^{b}}{{\left[ \frac{{{T}_{1}}}{{{T}_{2}}} \right]}^{c}}\]
where \[{{M}_{1}},\,\,{{L}_{1}}\] and \[{{T}_{1}}\] are fundamental units of mass, length and time in the first (known) system and \[{{M}_{2}},\,{{L}_{2}}\] and \[{{T}_{2}}\] are fundamental units of mass, length and time in the second (unknown) system. Thus knowing the values of fundamental units in two systems and numerical value in one system, the numerical value in other system may be evaluated.
Example : (i) conversion of Newton into Dyne.
The Newton is the S.I. unit of force and has dimensional formula \[[ML{{T}^{-2}}]\].
So 1 N = 1 kg-m/ \[{{\sec }^{2}}\]
By using \[{{n}_{2}}={{n}_{1}}{{\left[ \frac{{{M}_{1}}}{{{M}_{2}}} \right]}^{a}}{{\left[ \frac{{{L}_{1}}}{{{L}_{2}}} \right]}^{b}}{{\left[ \frac{{{T}_{1}}}{{{T}_{2}}} \right]}^{c}}\]
\[=1\,{{\left[ \frac{kg}{gm} \right]}^{1}}\,{{\left[ \frac{m}{cm} \right]}^{1}}{{\left[ \frac{sec}{sec} \right]}^{-2}}\]
\[=1\,{{\left[ \frac{{{10}^{3}}gm}{gm} \right]}^{1}}\,{{\left[ \frac{{{10}^{2}}cm}{cm} \right]}^{1}}{{\left[ \frac{sec}{sec} \right]}^{-2}}\]\[={{10}^{5}}\]
\[\therefore \,1\,N={{10}^{5}}\]Dyne
(ii) Conversion of gravitational constant (G) from C.G.S. to M.K.S. system
The value of G in C.G.S. system is \[6.67\times {{10}^{-8}}\] C.G.S. units while its dimensional formula is \[[{{M}^{-1}}{{L}^{3}}{{T}^{-2}}]\]
So \[G=6.67\times {{10}^{-8}}\,c{{m}^{3}}/g\,{{s}^{2}}\]
By using \[{{n}_{2}}={{n}_{1}}{{\left[ \frac{{{M}_{1}}}{{{M}_{2}}} \right]}^{a}}{{\left[ \frac{{{L}_{1}}}{{{L}_{2}}} \right]}^{b}}{{\left[ \frac{{{T}_{1}}}{{{T}_{2}}} \right]}^{c}}\]
\[=6.67\times {{10}^{-8}}{{\left[ \frac{gm}{kg} \right]}^{-1}}{{\left[ \frac{cm}{m} \right]}^{3}}{{\left[ \frac{sec}{sec} \right]}^{-2}}\]
\[=6.67\times {{10}^{-8}}{{\left[ \frac{gm}{{{10}^{3}}gm} \right]}^{-1}}{{\left[ \frac{cm}{{{10}^{2}}cm} \right]}^{3}}{{\left[ \frac{sec}{sec} \right]}^{-2}}\]
\[=6.67\times {{10}^{-11}}\]
\[\therefore \,\,G=6.67\times {{10}^{-11}}\,\] M.K.S. units
(4) To check the dimensional correctness of a given physical relation : This is based on the 'principle of homogeneity'. According to this principle the dimensions of each term on both sides of more...
Although dimensional analysis is very useful it cannot lead us too far as,
(1) If dimensions are given, physical quantity may not be unique as many physical quantities have same dimensions. For example if the dimensional formula of a physical quantity is \[[M{{L}^{2}}{{T}^{-2}}]\]it may be work or energy or torque.
(2) Numerical constant having no dimensions [K] such as (1/2), 1 or \[2\pi \] etc. cannot be deduced by the methods of dimensions.
(3) The method of dimensions can not be used to derive relations other than product of power functions. For example,
\[s=u\,t+\,(1/2)\,a\,{{t}^{2}}\] or \[y=a\sin \omega \,t\]
cannot be derived by using this theory (try if you can). However, the dimensional correctness of these can be checked.
(4) The method of dimensions cannot be applied to derive formula if in mechanics a physical quantity depends on more than 3 physical quantities as then there will be less number (= 3) of equations than the unknowns (>3). However still we can check correctness of the given equation dimensionally. For example \[T=2\pi \sqrt{{I}/{mgl}\;}\]can not be derived by theory of dimensions but its dimensional correctness can be checked.
(5) Even if a physical quantity depends on 3 physical quantities, out of which two have same dimensions, the formula cannot be derived by theory of dimensions, e.g., formula for the frequency of a tuning fork \[f=(d/{{L}^{2}})\,v\] cannot be derived by theory of dimensions but can be checked.
Significant figures in the measured value of a physical quantity tell the number of digits in which we have confidence. Larger the number of significant figures obtained in a measurement, greater is the accuracy of the measurement. The reverse is also true.
The following rules are observed in counting the number of significant figures in a given measured quantity.
(1) All non-zero digits are significant.
Example : 42.3 has three significant figures.
243.4 has four significant figures.
24.123 has five significant figures.
(2) A zero becomes significant figure if it appears between two non-zero digits.
Example : 5.03 has three significant figures.
5.604 has four significant figures.
4.004 has four significant figures.
(3) Leading zeros or the zeros placed to the left of the number are never significant.
Example : 0.543 has three significant figures.
0.045 has two significant figures.
0.006 has one significant figure.
(4) Trailing zeros or the zeros placed to the right of the number are significant.
Example : 4.330 has four significant figures.
433.00 has five significant figures.
343.000 has six significant figures.
(5) In exponential notation, the numerical portion gives the number of significant figures.
Example : \[1.32\times {{10}^{-2}}\] has three significant figures.
\[1.32\times {{10}^{4}}\] has three significant figures.
While rounding off measurements, we use the following rules by convention:
(1) If the digit to be dropped is less than 5, then the preceding digit is left unchanged.
Example : \[x=7.82\] is rounded off to 7.8,
again \[x=3.94\] is rounded off to 3.9.
(2) If the digit to be dropped is more than 5, then the preceding digit is raised by one.
Example : x = 6.87 is rounded off to 6.9,
again x = 12.78 is rounded off to 12.8.
(3) If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit is raised by one.
Example : x = 16.351 is rounded off to 16.4,
again x = 6.758 is rounded off to 6.8.
(4) If digit to be dropped is 5 or 5 followed by zeros, then preceding digit is left unchanged, if it is even.
Example : x = 3.250 becomes 3.2 on rounding off,
again x = 12.650 becomes 12.6 on rounding off.
(5) If digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd.
Example : x = 3.750 is rounded off to 3.8,
again x = 16.150 is rounded off to 16.2.
A quantity which can be measured and by which various physical happenings can be explained and expressed in the form of laws is called a physical quantity. For example length, mass, time, force etc.
On the other hand various happenings in life e.g., happiness, sorrow etc. are not physical quantities because these can not be measured.
Measurement is necessary to determine magnitude of a physical quantity, to compare two similar physical quantities and to prove physical laws or equations.
A physical quantity is represented completely by its magnitude and unit. For example, 10 metre means a length which is ten times the unit of length. Here 10 represents the numerical value of the given quantity and metre represents the unit of quantity under consideration. Thus in expressing a physical quantity we choose a unit and then find that how many times that unit is contained in the given physical quantity, i.e.
Physical quantity (Q) = Magnitude × Unit = n × u
Where, n represents the numerical value and u represents the unit. Thus while expressing definite amount of physical quantity, it is clear that as the unit(u) changes, the magnitude(n) will also change but product 'nu' will remain same.
i.e. n u = constant, or \[{{n}_{1}}{{u}_{1}}={{n}_{2}}{{u}_{2}}=\text{constant}\]; \ \[n\propto \frac{1}{u}\]
i.e. magnitude of a physical quantity and units are inversely proportional to each other. Larger the unit, smaller will be the magnitude.
(1) Ratio (numerical value only) : When a physical quantity is the ratio of two similar quantities, it has no unit. e.g. Relative density = Density of object/Density of water at 4oC Refractive index = Velocity of light in air/Velocity of light in medium Strain = Change in dimension/Original dimension
(2) Scalar (magnitude only) : These quantities do not have any direction e.g. Length, time, work, energy etc. Magnitude of a physical quantity can be negative. In that case negative sign indicates that the numerical value of the quantity under consideration is negative. It does not specify the direction. Scalar quantities can be added or subtracted with the help of ordinary laws of addition or subtraction.
(3) Vector (magnitude and direction) : These quantities have magnitude and direction both and can be added or subtracted with the help of laws of vector algebra e.g. displacement, velocity, acceleration, force etc.
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