1. Topics to be covered in this lesson
1.3 Dimensional analysis and applications
1.4 Limitations of Dimensional method
1.5 Accuracy and Precision
1.6 Significant figures
1.7 Errors in measurement
1.8 Measuring Instruments
Measurement of any physical quantity involves its comparison with a certain basic, reference standard called unit.
Any standard unit must have two properties
The result of measurement of a physical quantity is expressed by a number (numerical measure) accompanied by a unit.
Measurement = nu
Fundamental quantities : The units for these quantities are called fundamental or base units. Length, mass, time, electric current etc. are some of the fundamental quantities.
Derived quantities : They can be expressed in terms of fundamental quantities. The units of derived quantities are expressed in terms of fundamental units and they are called derived units. e.g., velocity, force, impulse etc. are derived quantities.
International system of units (SI) :- This system of units was introduced in 1971 by the general conference on weights and measures and was internationally accepted. It has seven fundamental units along with two supplementary units.
All the physical quantities represented by derived units can be expressed in terms of some combination of seven fundamental quantities. These seven fundamental quantities are called seven dimensions of the physical world. They are denoted with square brackets [ ].
The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity.
1. In this type of representation the magnitudes are not considered. It is the quality of the type of the physical quantity that enters.
2. The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called dimension formula of the given physical quantity.
3. An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity.
e.g. [F] = [MLT–2] …(i)
1.3 DIMENSIONAL ANALYSIS AND ITS APPLICATIONS:
Principle of Homogeneity of dimensions : It states that in a correct equation, the dimensions of each term added or subtracted must be same. Every correct equation must have same dimensions on both sides of the equation.
Conversion of units : The numerical value of a physical quantity in a system of units can be changed to another system of units using the equation n[u] = constant i.e., n1u1=n2u2, where n is the numerical value and u is the unit.
By knowing the conversion factors for the base quantities and dimensional formula of the derived quantity, one can convert the numerical value of a physical quantity from one system of units to other system of units.
n2 = n1 [M1/M2]a [L1/L2]b [T1/T2]c
To find a relation among the physical quantities. If one knows the quantities on which a particular physical quantity depends and guesses that this dependence is of product type, method of dimensions are helpful in deducing their relation.
Suppose we want to find the relation between force, mass and acceleration. Let force depends on mass and acceleration as follows.
1.4 LIMITATIONS OF DIMENSIONAL METHOD
1. Dimensional method cannot be used to derive equations involving addition and subtraction.
2. Numerical constants having no dimensions cannot be obtained by method of dimensions.
3. Dimensional method fails when number of variables is more than number of equations obtained from the exponents of fundamental quantities.
4. Equations using trigonometric, exponential and logarithmic functions can not be deduced.
Some Important Points :
1. If name of unit is the name of scientist, then for e.g., 5 Ampere is wrong, correct is 5 ampere. Also 5a is wrong, correct is 5A.
2. Pure numbers are dimensionless.
3. All trigonometrical ratios, powers, exponential and logarithmic functions are dimensionless.
4. All ratio of physical quantities having same dimensional formula are dimensionless. e.g. relative density, relative permeability, dielectric constant, angles, refractive index etc.
5. Dimensions do not depend upon magnitude.
1.5 ACCURACY AND PRECISION
The closeness of the measured value to the true value of the physical quantity is known as the accuracy of the measurement.
It is the measure of the extent to which successive measurements of a physical quantity differ from one another.
Suppose the true value of a measurement is 35.75 and two measured values are 35.73 and 35.725. Here 35.73 is closest to 35.75, so its accuracy is more than 35.725 but 35.725 is more precise than 35.73 because 35.725 is measured upto after 3 decimal places.
1.6 SIGNIFICANT FIGURES
The number of digits in the measured value about the correctness of which we are sure plus one more digit are called significant figures.
Rules for counting the significant figures
Rule I : All non-zero digits are significant.
Rule II : All zeros occurring between the non zero digits are significant. For example 230089 contains six significant figures.
Rule III : All zeros to the left of non zero digit are not significant. For example 0.0023 contains two significant figures.
Rule IV : If a number ends in zeros that are not to the right of a decimal, the zeros are not significant.
For example, number of significant figures in
1.5 × 10^3 (Two)
1.50 × 10^3 (Three)
1.500 × 10^3 (Four)
Length of an object may be represented in many ways say 5 m, 5.0 m, 500 cm, 5.00 m, 5 × 102 cm. Here 5.00 m is most precise as it contains 3 significant figures.
Rules for Arithmetic Operations with Significant Figures
Rule I : In addition or subtraction, the final result should retain as many decimal places are there are in the number with the least decimal places.
Rule II : In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures.
Rounding Off of Uncertain Digits
Rule I : The preceding digit is raised by 1 if the insignificant digit to be removed is more than 5 and is left unchanged if the later is less than 5.
Rule II : When the insignificant digit to be removed is 5 and the uncertain digit is even, 5 is simply dropped and if it is odd, then the preceding digit is raised by 1.
1.7 ERRORS IN MEASUREMENT
1. Mean Absolute Error :- It is given by
2. Relative Error or Fractional Error : It is given by
3. Percentage Error
4. Combination of Errors :
(i) In Sum : If Z = A + B, then ΔZ = ΔA + ΔB, maximum fractional error in this case
i.e. when two physical quantities are added then the maximum absolute error in the result is the sum
of the absolute errors of the individual quantities.
(ii) In Difference : If Z = A – B, then maximum absolute error is ΔZ = ΔA + ΔB and maximum fractional error in this case
(iii) In Product : If Z = AB, then the maximum fractional error,
(iv) In Division : If Z = A/B, then maximum fractional error is
(v) In Power :
1.8 MEASURING INSTRUMENTS AND LEAST COUNT
The error in the measurement by an instrument is equal to least count of the instrument.
For example, a meter scale has smallest division 1 mm. This represents the least count or the absolute error in the measurement.
Let a length measured by the meter scale = 56.0 cm
This implies that x = 56.0 cm
Absolute error Δx = 1 mm = 0.1 cm
Relative error = Δx / x = 0.1 / 56.0
It consists of two scales viz main scale and vernier scale.
Vernier scales moves on the main scale. The least count of the instrument is the smallest distance between two consecvtive divisions and it is equal to 1 MSD – 1 VSD.
In the figure shown, 1 MSD = 0.1 cm
1 VSD = 0.09 cm
Least count = 1 MSD – 1 VSD = 0.01 cm
For the commonly used instruments, least counts are given below :
Least count of Vernier callipers = 1 MSD – 1 VSD
It contains a main scale and a circular scale. The circular scale is divided into a number of divisions. In other words, the complete rotation of circular scale is divided into a number of parts.
The least count of a screw gauge is pitch/no. of circular scale divisions.
If reading on main scale is 5 cm (say) and reading on circular scale is 25 divisions, then the reading is 5 cm + 25 × least count.
Least count of spherometer and Screw Gauge = Pitch / No. of CSD
FUNDAMENTAL QUANTITIES, THEIR UNIT AND DIMENSIONAL FORMULA
1. Mass ⇒ kilogram ⇒ kg ⇒ [M]
2. Length ⇒ metre ⇒ m ⇒ [L]
3. Time ⇒ second ⇒ s ⇒ [T]
4. Temperature ⇒ Kelvin ⇒ K ⇒ [K]
5. Electric current ⇒ Ampere ⇒ A ⇒ [A]
6. Luminous Intensity ⇒ candela ⇒ cd ⇒ [cd]
7. Amount of Matter ⇒ mole ⇒ mol ⇒ [mol]
SUPPLEMENTARY QUANTITY AND UNIT
1. Plane angle ⇒ radian ⇒ rad [180º = π radian]
2. Solid angle ⇒ steradian ⇒ sr