Topics to be Learn : Part-2
- Dimensions and Dimensional analysis
- Accuracy, Precision and Uncertainty in measurements
- Errors in measurements
- Significant figures
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Dimensions and Dimensional analysis :
Dimensions of a physical quantity : The dimensions of a physical quantity are the powers to which the fundamental units must be raised in order to obtain the unit of that physical quantity.
Writing of Dimensions of a physical quantity : When we represent any derived quantity with appropriate powers of symbols of the fundamental quantities, then such an expression is called dimensional formula. This dimensional formula is expressed by square bracket and no comma is written in between any of the symbols.
- The dimensions of the SI fundamental quantities length, mass, time, electric current, thermodynamic temperature, amount of substance and luminous intensity are denoted by L, M, T, I, K, C and mol, respectively.
- The dimensions of a derived quantity are written as the product or quotient of the fundamental quantities raised to appropriate powers using the defining equation of that derived quantity.
For example, suppose a quantity n in mechanics is given in terms of length (l), mass (m) and time (t) by an equation of the form
n = ao lα mβ tλ
where the exponents a, b and l are integers which can be positive, negative or zero; the factor ao is a pure number. The dimensions of n are defined to be dim n or [n] = [Lα MβTλ]
If a derived quantity does not depend on certain fundamental quantity, then the corresponding dimensional exponent is zero. In such a case, the dimensions of that
fundamental quantity is omitted from the dimensions of the derived quantity. Also, pure numerical factors in an equation, such as ½ in ½ mv2 or p in pr2 have no dimensions; hence, the numerical factor ao does not figure in the expression for dim n above.
Example :
Example:
(i) Dimensional formula of velocity
Velocity = \frac{displacement}{time}
Dimensions of velocity = \frac{[L]}{[T]} = [L1M0T1]
ii) Dimensional formula for charge.
charge = current x time
Dimensions of charge = [I] [T] = [L0M0T1I1]
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Common Physical Quantities their, SI Units and Dimensions :
Some Common Physical Quantities their, SI Units and Dimensions :
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Dimensionless derived quantities : There are some derived quantities for which the defining equation is such that all of the dimensional exponents in the expression for its dimension are zero.
- This is true, in particular, for any quantity that is defined as the ratio of two quantities of the same kind. Such quantities are described as being dimensionless, or alternatively as being of dimension one.
- The coherent derived unit for such dimensionless quantities is always the number one, 1, since it is the ratio of two identical units for two quantities of the same kind.
Examples :
Examples :
- Refractive index, which is defined as the ratio of the speed of light in free space to that in the medium.
- Plane angle, which is defined as the ratio of the length of arc of a circle to the radius of the circle.
- Relative density, which is defined as the ratio of the density of a material to that of water.
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Requirements that must be satisfied by a formula or a physical equation involving dimensional quantities :
A physical equation involving dimensional quantities must meet certain requirements which may be summarized as follows :
- The dimensions of the quantities on both sides of an equation must be the same.
- Only quantities having the same dimensions may be added or subtracted. It follows from this that all the additive terms of an equation must have the same dimensions. This is called the principle of dimensional homogeneity: a physical equation must be dimensionally consistent.
- Any two dimensional quantities may be multiplied together or divided, one by the other. The dimensions of the resulting product or quotient are then the product or quotient of the dimensions of the individual factors.
Dimensional analysis : A physical equation must be dimensionally consistent. This is called the principle of dimensional homogeneity. Thus, dimensional analysis is a method of analysing the dimensions
Uses of Dimensional Analysis :
Uses of Dimensional Analysis:
(i) To check the correctness of physical equations:
In any equation relating different physical quantities, if the dimensions of all the terms on both the sides are the same then that equation is said to be dimensionally correct.
Example : Consider the first equation of motion.
v = u + at
Dimension of L.H.S = [v] = [LT-1]
[u] =[LT-1]
[at] = [LT-2] [T] = [LT-1]
Dimension of R.H.S= [LT-1]+ [LT-1]
[L.H.S] = [R.H.S]
As the dimensions of L.H.S and R.H.S are the same, the given equation is dimensionally correct.
(ii) To establish the relationship between related physical quantities:
Example :
The period T of oscillation of a simple pendulum depends on length l and acceleration due to gravity g. Let us derive the relation between T, l, g :
Suppose T ∝ la
and T ∝ gb
... T ∝ lagb
T = k lagb,
where k is constant of proportionality and it is a dimensionless quantity and a and b are rational numbers.
Equating dimensions on both sides,
[M0L0T1] = k [L1]a [LT-2]b
= k [La+b T-2b]
[L0T1] = k [La+bT-2b]
Comparing the dimensions of the corresponding quantities on both the sides we get
a + b = 0
∴ a = -b
and
-2b=1
∴b = -1/2
∴a = -b = -(-1/2)
∴ a = 1/2
∴ T = k l1/2g -1/2
∴T = k\sqrt{\frac{l}{g}}
The value of k is determined experimentally and is found to be 2π
∴ T = 2π\sqrt{\frac{l}{g}}
(iii) To find the conversion factor between the units of the same physical quantity in two different systems of units:
Let us use dimensional analysis to determine the conversion factor between joule (SI unit of work) and erg (CGS unit of work).
Let 1 J = x erg
Dimensional formula for work is [M1L2T-2]
Substituting in the above equation, we can write
[M11L12T1-2]= x [M21L22T2-2]
x = [M11L12T1-2]/ [M21L22T2-2]
or, x = [\frac{M_1}{M_2}]^1[\frac{L_1}{L_2}]^2[\frac{T_1}{T_2}]^{-2}
where suffix 1 indicates SI units and 2 indicates CGS units.
In SI units, L, M, T are expressed in m, kg and s and in CGS system L, M, T are represented in cm, g and s respectively.
∴ x = [\frac{kg}{g}]^1[\frac{m}{cm}]^2[\frac{s}{s}]^{-2} = [(10^3)\frac{kg}{g}]^1[(100)\frac{m}{cm}]^2[(1)\frac{s}{s}]^{-2}
∴ x = 103 x 104 = 107
∴ 1 joule = 107 erg
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Limitations of Dimensional Analysis :
Limitations of Dimensional Analysis:
Dimensional analysis has the following limitations :
(1) If the correct equation contains some more terms of the same dimension, it is not possible to know about their presence using dimensional equation.
Examples :
- (i) In the usual notation, the equation
- s = ½ ut, s = ½ ut + at and s = a2t3/u are all dimensionally correct yet obviously incorrect, i.e., they do not correspond to real situations.
- (ii) The displacement of a particle in a sinusoidal disturbance is correctly given by x = A sinωt,
- where [A] = [L] and [ω] = [T-1]. But since a trigonometrical ratio (viz., sine, etc.) is a dimensionless derived quantity, the equation x = A sin wt would also be dimensionally correct, but may not correspond to any physical situation. Moreover, since the argument of a trigonometrical ratio must be the dimensionless derived quantity — plane angle, even the equation x =A tan w2t2 would be dimensionally correct.
2) The value of dimensionless constant can be obtained with the help of experiments only.
3) Dimensional analysis can not be used to derive relations involving trigonometric, exponential, and logarithmic functions as these quantities are dimensionless.
4) This method is not useful if constant of proportionality is not a dimensionless quantity.
Example : Gravitational force between two point masses is directly proportional to product of the two masses and inversely proportional to square of the distance between the two
F= m1m2/r2
Let F = G(m1m2/r2)
The constant of proportionality 'G' is not dimensionless. Thus earlier method will not work.
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Q. Are all constants dimensionless or unitless?
Answer :
A proportionality constant in a formula need not necessarily be dimensionless. For example, the universal gravitational constant G in Newton's law of gravitation, the permittivity in Coulomb's law, thermal conductivity K in the formula for rate of heat flow through a conductor, Planck's constant h, etc. are all dimensional constants.
Error of measurement : The determination of a physical quantity may involve measurement of two or more basic quantities. These measured quantities, and the derived quantities calculated from them, can never be exact. That is to say, there is always some uncertainty depending on factors like the measuring instruments, measuring techniques and some uncontrollable factors like personal errors, random fluctuations, etc. The uncertainty in the measured value of a quantity is termed an error.
[Note : Every measurement has an error. Every calculated value which is based on measured values has an error. Exact measurements are not possible.]
The errors are broadly divided into two categories : a) Systematic errors b) Random errors
Systematic errors : A systematic error is one which is constant throughout a set of readings.
- Systematic errors often arise because the measuring instrument and the experimental arrangement are not ideal, and the correction factor is not taken into account out of ignorance or disregarded out of sheer carelessness.
- Radiation loss or gain in calorimetric experiments and most types of instrumental errors are systematic errors.
Each of these errors tends to be in one direction, either positive or negative. The sources of systematic errors are as follows:
Sources of systematic errors :
(1) Instrumental errors : All instruments and standards possess some uncertainties. Examples where an instrument introduces a systematic error are as follows :
- An instrument with a zero-error : If an ammeter does not read zero when the current in the circuit is zero, all subsequent readings with the meter will be either more or less, consistently.
- Inaccurate calibration of scale : For example, each millimetre marked on a ruler is actually 0.95 mm, or a stopwatch used for time measurement may be running slightly fast or slow.
- An instrument is used under external or environmental conditions different from those at which it was calibrated.
- An instrument having irregular scale markings introduces random errors.
- Example : If a ruler has irregular millimeter markings, then a length measured using different parts of the ruler will give slightly different values.
(2) Personal errors : Random errors which can be attributed to the experimenter are known as personal errors. They may be due to parallax, judgment, response time, fatigue or eye-strain, bias of the observer, carelessness in taking observations etc. could result in such errors.
- Example : While measuring the length of an object with a ruler, it is necessary to look at the ruler from directly above. If the observer looks at it from an angle, the measured length will be wrong due to parallax.
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Random errors : Random errors are always present in an experiment and may be detected only by repeating a measurement many times.
Simple examples of sources of random errors are as follows.
Besides a couple of instrumental and personal errors which can be random, minute changes in the experimental conditions also introduce a random error in the measurements.
Sources of random errors :
Sources of random errors :
- Variation in the surrounding temperature
- Variation in the Earth's magnetic field
- Mechanical vibrations
- Fluctuations in electric power supply (which may affect the output of a voltage source or give rise to stray electrical signals), etc.
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How the errors of measurement can be reduced :
How the errors of measurement can be reduced :
A systematic error is neither revealed nor eliminated by repeated measurements with the same apparatus. Hence, the only course is to discover the possible sources of systematic errors in the experimental method and apparatus. Any systematic error that we know about should be reduced to a level small compared with the random errors by taking into account the appropriate correction factor.
In order to reduce the errors due to random effects,
- The magnitude of the measured quantity should be as large as possible.
- The least count of the measuring instrument should be small.
- Experimental conditions such as temperature, pressure, humidity, etc. should remain constant to within tolerable limits.
- A large number of measurements of the same quantity should be made.
- The mean value, i.e., the arithmetic average, is then taken as the best estimate for the true value of the quantity measured.
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Accuracy : A measured value is said to be accurate if it is relatively free from systematic errors. Thus, the term accurate refers to how closely a measured value is to the accepted or most probable value of the quantity. The lesser the errors, the more is the accuracy.
Example :
Example of accuracy : Suppose two wooden rulers are each measured three times against a standard stainless steel ruler. Ruler A is found to have lengths 99.8 cm, 99.9 cm and 99.7 cm, giving an average length of 99.8 cm; ruler B is found to have lengths 100.1 cm, 99.9 cm and 99.7 cm. Here, ruler B, with an average length of 99.9 cm, is closer to the calibrated value of 100 cm and, hence, more accurate.
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Precision : A measured value is said to be precise if the random error is small. Thus, the term precise refers to how reproducible measurements of the same quantity are.
| Q. If ten students are asked to measure the length of a piece of cloth up to a millimeter using a metre scale, do you think their answers will be identical? Give reasons.
Ans. No, their answers will not be same. A cloth also stretches according to pressure of hold, so that its length is not accurate to a millimetre. Hence, due to random errors, not just ten students’, but in repeated measure, even a single student's measurements will not be accurate to a millimeter and identical. This lack of accuracy is also the reason a cloth seller's metre scale has a least count of 5 cm.
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Estimation of errors :
Estimation of error:
Most probable value : The mean value, i.e., the arithmetic average, of a large number of measurements (assumed to be free from systematic errors) of the same quantity is called the most probable value of the quantity.
Absolute error : For a set of measurements of the same quantity, the positive difference between each individual value and the most probable value gives the absolute error in that value.
(i) Suppose a1, a2, ..., an are the individual values of n successive measurements of the same quantity. The most probable value of the measured quantity is taken as the arithmetic mean ( \overline{a} ) of these values :
\overline{a} = (a1 + a2 +, ... + an)/n =1
Thus, the absolute error in the value a1 is | a1 −\overline{a}], that in the value a2 is [a2 −\overline{a}], and so on.
The mean absolute error in the measurement of the quantity is :
Δx = \frac{\left | a_1-\overline{a}\right |+\left | a_2-\overline{a}\right | + .....\left | a_n-\overline{a}\right |}{n}
= \frac{1}{n}\sum_{t=i}^{n}\left | a_i-\overline{a}\right |
Mean absolute error : For a set of measurements of the same quantity, the arithmetic mean of all the absolute errors is called the mean absolute error in the measurement of that quantity.
Relative error = \frac{Δ_a}{\overline{a}}
elative error : The ratio of the mean absolute error in the measurement of a physical quantity to its most probable value is called the relative error in the measurement of that quantity.
Mean percentage error = relative error x 100%
= \frac{Δ_a}{\overline{a}} x 100%
Mean percentage error : The relative error in a measurement multiplied by 100, gives the mean percentage error in the measurement of a quantity.
In general, scientists accept as the standard true value a result that has been determined reproducibly in many laboratories by different methods and has the smallest percentage error.
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The rules to determine the error in a calculated quantity from the errors in each of the quantities used in the calculation are as follows :
- The absolute error in the sum or difference of the quantities is equal to the sum of the absolute errors in the individual quantities.
- The relative error in a product or quotient is the sum of the relative errors in the individual quantities.
- The relative error in a quantity raised to the nth power is n times the relative error in the quantity itself.
The combination of errors is shown symbolically in the table below :
| Relation between y and a, b |
Relation between the errors |
| y = a + b
y = a — b
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Δy = Δa + Δb |
| y = a.b
y = a/b
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\frac{Δy}{y}=\frac{Δa}{a}+\frac{Δb}{b} |
| y= an |
\frac{Δy}{y}=n\frac{Δa}{a} |
| y= an +bm |
\frac{Δy}{y}=n\frac{Δa}{a}+m\frac{Δb}{b} |
Combination of errors :
Combination of errors:
a) Errors in sum and in difference:
Suppose two physical quantities A and B have measured values A ± ΔA and
B ± ΔB, respectively, where ΔA and ΔB are their mean absolute errors. We wish to find the absolute error ΔZ in their sum.
Z=A+B
Z ± ΔZ = (A ± ΔA)+(B ± ΔB)
= (A+B) ± ΔA ± ΔB
± ΔZ = ± ΔA ± ΔB,
For difference, i.e., if Z = A-B,
Z ± ΔZ = (A ± ΔA)-(B ± ΔB)
= (A-B) ± ΔA± ΔB
± ΔZ = ± ΔA± ΔB,
There are four possible values for ΔZ, namely (+ ΔA - Δ B), (+ΔA+ΔB), (-ΔA-ΔB),
(-ΔA+ΔB). Hence maximum value of absolute error is ΔZ = ΔA+ΔB in both the cases.
When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.
b) Errors in product and in division:
Suppose Z=AB and measured values of A and B are (A ± ΔA) and (B ± ΔB) Then
Z ± ΔZ= (A ± ΔA) (B ± ΔB)
= AB ± AΔB ± BΔA ± ΔAΔB
Dividing L.H.S by Z and R.H.S. by AB we get
(1±\frac{ΔZ}{Z}) = (1±\frac{ΔB}{B} ± \frac{ΔA}{A}±( \frac{ΔA}{A})(\frac{ΔB}{B}))
Since ΔA/A and ΔB/B are very small we shall neglect their product. Hence maximum relative error in Z is
\frac{ΔZ}{Z}=\frac{ΔA}{A}+\frac{ΔB}{B}
This formula also applies to the division of two quantities.
Thus, when two quantities are multiplied or divided, the maximum relative error in the result is the sum of relative errors in each quantity.
c) Errors due to the power (index) of measured quantity:
Suppose
Z =A3 = A.A.A
\frac{ΔZ}{Z}=\frac{ΔA}{A}+\frac{ΔA}{A}+\frac{ΔA}{A}
from the multiplication rule above.
Hence the relative error in Z =A3 is three times the relative error in A. So if Z = An
\frac{ΔZ}{Z}=n\frac{ΔA}{A}
In general, if Z = \frac{A^pB^q}{C^r}
\frac{ΔZ}{Z}=p\frac{ΔA}{A}+q\frac{ΔB}{b}+r\frac{ΔC}{C}
The quantity in the formula which has large power is responsible for maximum error.
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Significant figures : The value of a measured or calculated quantity should be reported in a way that indicates the precision with which the quantity is known.
The number of figures or digits used to write this value should include only the digits that are known reliably plus the first uncertain digit.
The reliable digits plus the first uncertain digit are known as significant figures.
Examples :
Example ; A length quoted as 6.4 cm has two significant figures. This implies that the actual length could be as small as 6.3 cm or as large as 6.5 cm. That is, the digit 6 in 6.4 is certain, but there is uncertainty in the last digit.
In general, in a measured value, the last digit to the right is taken to be inexact or uncertain but all digits farther to the left are assumed to be exact. However, it is necessary to record the last digit to indicate the precision of the measurement :
Suppose we measure the length of a metal rod using a metre scale of least count 0.1cm.
The measurement is done three times and the readings are 15.4, 15.4, and 15.5 cm. The most probable length which is the arithmetic mean as per our earlier discussion is 15.43. Out of this we are certain about the digits 1 and 5 but are not certain about the last 2 digits because of the least count limitation.
The number of digits in a measurement about which we are certain, plus one additional digit, the first one about which we are not certain is known as significant figures or significant digits.
Thus in above example, we have 3 significant digits 1, 5 and 4.
The larger the number of significant figures obtained in a measurement, the greater is the accuracy of the measurement. If one uses the instrument of smaller least count, the number of significant digits increases.
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Rules for determining the number of significant figures :
Rules for determining the number of significant figures :
1) The result of a measurement is reported as a number with only one uncertain digit. This last digit to the right, which may be zero, is considered significant as it indicates the precision of measurement.
2) All the nonzero digits are significant,
- Example : If the volume of an object is 268.43 cm3, there are five significant digits which are 2,6,8,4 and 3.
3) All the zeros between two nonzero digits are significant,
- Example : m = 365.02 g has 5 significant digits.
4) If the number is less than 1, the zero (s) between the decimal point and the first non-zero digit are not significant.
- Example : In 0.001205, the underlined zeros are not significant. Thus the above number has four significant digits.
5) The zeros on the right hand side of the last nonzero number are significant (but for this, the number must be written with a decimal point),
- Example : 1.500 or 0.01500 have both 4 significant figures each. On the contrary, if a measurement yields length L given as L = 125 m = 12500 cm = 125000 mm, it has only three significant digits.
6) If the result of a calculation with measured values (which are always inexact, i.e., they have limited number of significant figures) contains more than one uncertain digit, it should be rounded off as follows: The last (rightmost) significant digit to be retained is left unchanged if and only if the first insignificant digit to be dropped is less than 5, otherwise it is increased by1.
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Order of magnitude : To avoid the ambiguities in determining the number of significant figures, it is necessary to report every measurement in scientific notation (i.e., in powers of 10) i.e., by using the concept of order of magnitude.
The order of magnitude of a physical quantity is its magnitude expressed to the nearest integral power of ten.
To find the order of magnitude of a physical quantity, its magnitude is expressed as a number that lies between 0.5 and 5 multiplied by an appropriate integral power of 10. The power of 10 along with the unit then gives the order of magnitude of the quantity.
The magnitude of any physical quantity can be expressed as A×10n where ‘A’ is a number such that 0.5 ≤ A<5 and ‘n’ is an integer called the order of magnitude.
Example :
Example : (i) radius of Earth = 6400 km = 0.64×107m
The order of magnitude is 7 and the number of significant figures are 2.
(ii) Magnitude of the charge on electron e = 1.6×10−19 C
Here the order of magnitude is -19 and the number of significant digits are 2.
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