Unit and Measurements
Category : Railways
Unit and Measurements
PHYSICAL QUANTITIES
Those quantities which can describe the laws of physics and possible to measure are called physical quantities.
A physical quantity is that which can be measured. Physical quantity is completely specified;
If it has
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Only numerical value Ex. Refractive index, dielectric constant etc. |
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Only magnitude ex, Mass, charge etc. |
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Magnitude and direction Displacemnt, torque etc. |
Types of physical Quantities
The physical quantities which do not depend upon other Physical quantities are called fundamental quantities. In M.K.S System the fundamental quantities are mass, length and time
In standard International system the fundamental quantities are mass, length, time, temperature, illuminatig power current and amount of substance, The physical quantities which depend on fundamental quantities are called derived quantities e, g. speed, acceleration, force, etc
UNITS
The unit of a physical quantity is the reference standard used to measure it.
For the measurement of a physical quantity a definite magnitude of quantity is taken as standard and the name given to this standard is called unit.
Properties of Unit
(a) The unit should be well-defined.
(b) The unit should be of some suitable size.
(c) The unit should be easily reproducible.
(d) The unit should not change with time.
(e) The unit should not change with physical conditions like pressure, temperature etc.
(f) The unit should be universally acceptable.
Types of Units
The units defined for the fundamental quantities are called fundamental or base units.
Base quantities and their SI unit
One kilogram is defined as the mass of a platinum iridium cylinder kept in National Bureau of Weights and
Measurements, Paris.
The distance travelled by light in vacuum in 1/299,792,458 second or it is equal to 1650763.73 wavelength emitting
From \[K{{r}^{86}}\]
The time interval in which Cesium-133 atom vibrates
9, 192, 631, 770times.
It is defined as the (1/273.16) fraction of thermo dynamic temperature of triple point of water.
Triple Point of Water is the temperature at which ice, water and water vapours co-exist.
The amount of current which produces a force of \[2\times {{10}^{7}}\]N per unit length acts between two parallel wires of infinite length and negligible cross section area placed at 1m distance in vacuum.
The amount of intensity on \[1/60000\,\,{{m}^{2}}\] area of black body in the direction perpendicular to its surface at freezing point of platinum 2042 K at pressure of 101325\[N/{{m}^{2}}\].
It is the amount of a substance which has same number of elementary entities as in 12gm of carbon-12.
The units defined for the derived quantities are called derived units
Unit of speed
\[(Speed/velocity)=\frac{\text{distance}(displacement)}{time}\]
\[\Rightarrow (Unit\,\,of\,\,speed/velocity)=\frac{metre}{\sec .}=m{{s}^{-1}}\]
Unit of acceleration
Acceleration; a=\[\frac{v}{t}\]
\[\Rightarrow Unit\,\,of\,\,'a'=\frac{m/s}{s}=m/{{s}^{2}}\]
Unit of force
Force F=ma \[\Rightarrow Unit\,\,of\,\,force=kg\,\,m/{{s}^{2}}=newton\]
System of Units
A complete set of fundamental and derived units is known as the system of units.
S.I. Prefixes of power 10
|
Perfix |
Symbol |
Power of 10 |
|
Exa |
E |
\[{{10}^{18}}\] |
|
Peta |
P |
\[{{10}^{15}}\] |
|
tera |
T |
\[{{10}^{12}}\] |
|
Giga |
G |
\[{{10}^{9}}\] |
|
mega |
M |
\[{{10}^{6}}\] |
|
kilo |
K |
\[{{10}^{3}}\] |
|
hecto |
h |
\[{{10}^{2}}\] |
|
deca |
Da |
\[{{10}^{1}}\] |
|
metre |
m |
\[10{}^\circ =1\] |
|
deci |
D |
\[{{10}^{-1}}\] |
|
centi |
c |
\[{{10}^{-2}}\] |
|
milli |
m |
\[{{10}^{-3}}\] |
|
micro |
u |
\[{{10}^{-6}}\] |
|
nano |
n |
\[{{10}^{-9}}\] |
|
pico |
p |
\[{{10}^{-12}}\] |
|
femto |
f |
\[{{10}^{-15}}\] |
|
atto |
a |
\[{{10}^{-18}}\] |
Practical units of length
MEASUREMENT OF LARGE DISTANCE
Parallax method: It is used to measure large distances such as the distance of a planet or a star.
If, S is the position of a planet, A and B are the positions of eyes observing planet S from two different positions on the earth called basis.
Angle, \[\theta \]is called the Parallax angle or Parallactic angle.
Distance between A and B = b Therefore, the distance of the planet' S' from the earth
ACCURACY, PRECISION AND ERRORS IN MEASUREMENT
Accuracy of measurement: It depends upon the number of significant figures in it. The higher the accuracy, the higher the number of significant figures.
Precision of measurement: It depends upon the least count of the measuring instrument. The smaller the least count, the more precise the measurement.
Errors in measurement: It is the difference in the true value and the measured value of the quantity.
Least count error: It is the smallest value that can be measured by the measuring instrument.
Absolute error: It is the difference in the true value or mean value and measured value of a quantity. Suppose, are the measured value then,
Suppose, \[{{a}_{1}},\,\,{{a}_{2}},\,\,{{a}_{3}}............{{a}_{n}}\] are the measured value then,
True value or mean value\[{{a}_{m}}=\frac{{{a}_{1}}+{{a}_{2}}+{{a}_{3}}+.......+{{a}_{n}}}{n}\]
where n is the number of observations.
Absolute error =
\[\begin{matrix}
\Delta {{a}_{1}} & = & {{a}_{m}}-{{a}_{1}} \\
\Delta {{a}_{2}} & = & {{a}_{m}}-{{a}_{2}} \\
\Delta {{a}_{n}} & = & {{a}_{m}}-{{a}_{n}} \\
\end{matrix}\]
Relative error: It is the ratio of mean absolute error to the mean value of quantity measured. Mean absolute error (Ad)
i.e.. Relative error = \[\frac{Mean\,\,absolute\,\,error\,\,(\Delta \bar{a})}{Mean\,\,value\,\,({{a}_{mean}})}\]
Percentage error (e a): When the relative error is expressed in per cent it is called percentage error.
Mean absolute error (Aa)
Percentage error \[=\]\[\frac{Mean\,\,absolute\,\,error\,\,(\Delta \bar{a})}{Mean\,\,value\,\,({{a}_{mean}})}\times 100%\]
Combination of Errors
(a) Error of a sum or a difference: In sum or difference of quantities the absolute error in final result is the sum of the absolute errors in the individual quantities. i.e., \[\Delta Z=\Delta A+\Delta B\]
(b) Error of a product or a quotient: In multiplication or division, the relative error in the result is equal to the sum of the relative errors in the measured quantities. i.e. \[\frac{\Delta Z}{Z}=\frac{\Delta A}{A}+\frac{\Delta \Beta }{B}\]
Error in quantity raised to some power: The relative error in a physical quantity raised to the power p is the p times the relative error in the measured quantity.
i.e; \[\frac{\Delta Z}{Z}=p\frac{\Delta A}{A}\]
SIGNIFICANT FIGURES
Significant figures are the number of digits up to which we are sure about their accuracy. In the measured quantity significant figures are those digits which are known reliably plus the first digit that is uncertain.
Rules in counting the number of significant figures in a given measured quantity:
(i) All non-zero digits are significant.
(ii) All zeroes occuring between two non-zero digits are sigificant.
(iii) All zeroes to the right of a decimal point and to the left of a non-zero digit are not significant.
(iv) All zeroes to the right of the last non-zero digit are not significant. On the other hand, all zeroesto the right of the last non-zero digit are significant provided they come from a measurement.
(v) Power of ten are not counted in significant figures. e. g. \[2.5\times {{10}^{-29}}\]has only 2 significant figures.
ROUNDING OFF
Rules of Rounding off Uncertain Digits
(a) The preceding digit is raised by 1 if the uncertain digit to be dropped is more than 5 and is left unchanged if the latter is less than 5.
Example: x = 5.686 is rounded off to 5.69 (as 6 > 5)
x = 3.462 is rounded off to 3.46 (as 2 < 5)
(b) If the uncertain digit to be dropped is 5, the preceeding digit raised by 1 if it is odd and is left unchanged if it is even digit.
Example: 7.735 is rounded off to three significant figures becomes 7.74 as preceeding digit is odd.
7.745 is rounded off to 7.74 as preceeding digit is even.
DIMENSIONS OF PHYSICAL QUANTITIES
The limit of a derived quantity in terms of necessary basic units is called dimensional formula and the raised powers on the basic units are dimensions.
Force is the product of mass and acceleration
Force = mass \[\times \] acceleration
= mass \[\times \] (length)/\[{{(time)}^{2}}\]
Dimensions of force are
[M][L]/\[{{[T]}^{2}}\]=\[[ML{{T}^{-2}}]\]
i.e. 1 in mass, 1 in length and 2 in time.
DIMENSIONAL FORMULAE AND
DIMENSIONAL EQUATIONS
The expression which shows how and which fundamental quantities represent the dimensions of a physical quantity is known as dimensional formula.
Ex. The dimensional formula off force is \[{{[MLT]}^{-2}}\]
When a dimensional formula is equated to its physical quantity then the equation is called dimensional equation.
Ex. Dimensional equation of force:
By F=ma
\[\Rightarrow \]Dimension equation of force, F =\[[{{M}^{1}}][{{L}^{1}}{{T}^{-2}}]\]
\[=[ML{{T}^{-2}}]\]
Ex. Dimensional equation of energy:
By E = W = force \[\times \] displacement
Dimensional equation of energy, E \[=[{{M}^{1}}{{L}^{1}}{{T}^{-2}}][{{L}^{1}}]\]
\[=[M{{L}^{2}}{{T}^{-2}}]\]
Dimensions of physical quantities in mechanics
|
Physical quantity |
dimensions |
Physical quantity |
dimensions |
|
Distance displacement, length/depth/thickness, wavelength |
\[[{{M}^{0}}L{{T}^{0}}]\] |
Force, weight tension centripetal force |
\[[ML{{T}^{-2}}]\] |
|
Mass, inertia, intertial mass, gravitational mass |
\[[M{{L}^{0}}{{T}^{0}}]\] |
Work energy torque moment of couple heat |
\[[M{{L}^{2}}{{T}^{-2}}]\] |
|
Speed, velocity |
\[[{{M}^{0}}L{{T}^{-1}}]\] |
Linear momentum impulse |
\[[ML{{T}^{-1}}]\] |
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Acc.(a) acc. due to gravity(g) |
\[[{{M}^{0}}L{{T}^{-2}}]\] |
Surface tension |
\[[M{{L}^{0}}{{T}^{-2}}]\] |
|
Angular velocity, velocity gradient, decay constant (\[\lambda \]) linear frequency |
\[[{{M}^{0}}{{L}^{0}}{{T}^{-1}}]\] |
Pressure, coefficient of elasticity young?s modulus bulk modulus stress |
\[[M{{L}^{-1}}{{T}^{-2}}]\] |
|
Wave number propagation constant (k) ridburg constant |
\[[{{M}^{0}}{{L}^{-1}}{{T}^{0}}]\] |
Plank?s constant, angular momentum |
\[[M{{L}^{2}}{{T}^{-1}}]\] |
|
Gravitational constant(G) |
\[[{{M}^{-1}}{{L}^{3}}{{T}^{-2}}]\] |
Coefficient of viscosity |
\[[M{{L}^{-1}}{{T}^{-1}}]\] |
Dimensions in heat
|
Physical quantity |
Dimensions |
|
Temperature |
\[[{{M}^{0}}{{L}^{0}}{{T}^{0}}\theta ]\] |
|
Latent heat |
\[[{{M}^{0}}{{L}^{2}}\,{{T}^{-2}}{{\theta }^{0}}]\] |
|
Specific heat |
\[[{{M}^{0}}{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}]\] |
|
Coefficient of thermal expansion |
\[[{{M}^{0}}{{L}^{0}}{{T}^{0}}{{\theta }^{-1}}]\] |
|
Coeff. of thermal conductivity |
\[[ML{{T}^{-3}}{{\theta }^{-1}}]\] |
|
Mechanical equivalent of heat(J) |
\[[{{M}^{0}}{{L}^{0}}{{T}^{0}}]\] |
|
Stephan constant |
\[[M{{L}^{0}}{{T}^{-3}}{{K}^{-4}}]\] |
|
Wien's constant |
\[[{{M}^{0}}L{{T}^{0}}\theta ]\] |
|
Boltzmann constant |
\[[M{{L}^{2}}{{T}^{-2}}{{\theta }^{-1}}]\] |
Dimensions in electricity
|
Physical quantity Dimension |
Physical quantity Dimension |
|
Charge \[[{{M}^{0}}{{L}^{0}}AT]\] |
Electric \[[{{M}^{-1}}{{L}^{-3}}{{T}^{4}}{{A}^{2}}]\] permittivity |
|
Current \[[{{M}^{0}}{{L}^{0}}{{T}^{0}}A]\] |
Resistance Reactance\[[M{{L}^{2}}{{T}^{-3}}{{A}^{-2}}]\] Impedance |
|
Potential gradient Electric field \[[M\,\,L\,\,{{T}^{-3}}{{A}^{-1}}]\] Intensity of electric field |
Electrical conductance Admittance Susceptance |
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Potential difference Potential \[[M{{L}^{2}}{{T}^{-3}}{{A}^{-1}}]\] Potential energy Electromotive force |
Electric flux \[[M{{L}^{3}}{{T}^{-3}}{{A}^{-1}}]\] |
|
Electrical\[[{{M}^{-1}}{{L}^{-2}}{{T}^{4}}{{A}^{2}}]\] capacitance
|
Specific \[[M{{L}^{3}}{{T}^{-3}}{{A}^{-2}}]\] resistance |
Dimensions in electricity
|
Physical quantity |
Dimensions |
|
Magnetic induction |
\[[M{{L}^{0}}{{T}^{-2}}{{A}^{-1}}]\] |
|
Permeability of magnet (u) |
\[[ML{{T}^{-2}}{{A}^{-2}}]\] |
|
Self inductance or mutual inductance |
\[[M{{L}^{2}}{{T}^{-2}}{{A}^{-2}}]\] |
|
Bohr magneton\[({{u}_{B}})\] |
\[[{{M}^{0}}{{L}^{2}}{{T}^{0}}A]\] |
Dimensionless quantity
Principle of Homogeneity
The dimensions of both sides i.e. dimensions of left side of physical quantity and right side of physical quantity in an equation are same.
Ex. \[s=ut+\frac{1}{2}g{{t}^{2}}\]
\[L.H.S\,\,\,\,\,\,\,\,\,R.H.S\]
\[[L]=[L{{T}^{-1}}.T]+[L{{T}^{-2}}.{{T}^{2}}]\]
\[[L]=[L]+[L]\]
APPLICATIONS OF DIMENSIONAL ANALYSIS
Limitations of Dimensional Analysis
(i) While deriving a formula the proportionality constant cannot be found.
(ii) The formula for a physical quantity depending on more than three other physical quantities cannot be derived. It can be checked only.
The equations of the type v = u ± at cannot be derived.
They can be checked only.
(m) The equations containing trigonometrical functions
\[(\sin \theta ,\cos \theta ,\,\,etc)\], logarithmic functions (log x, log\[{{x}^{3}}\], etc) and exponential functions \[({{e}^{x}},{{e}^{{{x}^{2}}}},etc)\] cannot be derived. They can be checked only.
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