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The SI System of Units
The internationally accepted standard units of the fundamental physical quantities are given in Table
Dimensions of Physical Quantities
The dimensions of a physical quantity are the powers to which the fundamental units of mass (M), length (L) and time (T) must be raised to represent the unit of that quantity. The dimensional formula of a physical quantity is an expression that tells us how and which of the fundamental quantities enter into the unit of that quantity. In mechanics, the dimensional formula is written in terms of the dimensions of mass, length and time (M, L and T). In heat and thermodynamics, in addition to M, L and T, we need to mention the dimension of temperature in kelvin (K). In electricity and magnetism, in addition to M, L and T, we need to mention the dimension of current or charge per unit time (A or QT– 1).
Principle of Homogeneity of Dimensions
Consider a simple equation,
A + B = C.
If this is an equation of Physics, i.e. if A, B and C are physical quantities, then this equation says that one physical quantity A, when added to another physical quantity B, gives a third physical quantity C. This equation will have no meaning in Physics if the nature (i.e. the dimensions) of the quantities on the left-hand side of the equation is not the same as the nature of the quantity on the right-hand side. For example, if A is a length, B must also be a length and the result of addition of A and B must express a length. In other words, the dimensions of both sides of a physical equation must be identical. This is called the principle of homogeneity of dimensions.
Note : 1. Trigonometric functions (sin, cos, tan, cot, etc.) are dimensionless. The arguments of these functions are also dimensionless.
2. Exponential functions are dimensionless. Their exponents are also dimensionless.
The number significant figure in any measurement indicates the degree of precision of that measurement. The degree of precision is determined by the least count of the measuring instrument. Suppose a length measured by a metre scale (of least count = 0.1 cm) is 1.5 cm, then it has two significant figures, namely 1 and 5. Measured with a vernier callipers
(of least count = 0.01 cm) the same length is 1.53 cm and it then has three significant figures. Measured with a screw gauge (of least count = 0.001 cm) the same length may be 1.536 cm which has four significant figures. It must be clearly understood that we cannot increase the accuracy of a measurement of changing the unit. For example, suppose a measurement of mass yields a value 39.4 kg. It is understood that the measuring instrument has a least count of 0.1 kg. In this measurement, three figures 3, 9 and 4 are significant. If we change 39.4 kg to 39400 g or 39400000 mg, we cannot change the accuracy of measurement. Hence 39400 g or 39400000 mg still have three significant figures; the zeros only serve to indicate only the magnitude of measurement.
Estimation of Appropriate Significant Figures in Calculations The importance of significant figures lies in calculation to find the result of addition or multiplication of measured quantities having a different number of significant figures. The least accurate quantity determines the accuracy of the sum or product. The result must be rounded off to the appropriate digit.
Rules for Rounding off
The following rules are used for dropping figures that are not significant
1. If the digit to be dropped is less than 5, the next (preceding) digit to be retained is left unchanged. For example, if a number 5.34 is to be rounded off to two significant figures, the digit to be dropped is 4 which is less than 5. Hence the next digit, namely 3, is not changed. The result of the indicated roundingoff is therefore, 5.3.
2. If the digit to be dropped is more than 5, the preceding digit to be retained is increased by 1. For examples 7.536 is rounded off as 7.54 to three significant figures.
3. If the digit to be dropped happens to be 5, then (a) the preceding digit to be retained is increased by 1 if it is odd, or (b) the preceding digit is retained unchanged if it is even. For example, 6.75 is rounded off to 6.8 to two significant figures and 4.95 is rounded off to 5.0 but 3.45 is rounded off to 3.4.
Rule for Finding Significant Figures
1. For addition and subtraction, we use the following rule.
Find the sum or difference of the given measured quantities and then round off the final result such that it has the same number of digits after the decimal place as in the least accurate quantity (i.e., the quantity which has the least number of significant figures)
Least Counts of Some Measuring Instruments
1. Least count of metre scale = 1 mm = 0.1 cm
2. Vernier constant (or least count) of vernier callipers
= value of 1 main scale division – value of 1 vernier
scale division = 1 M.S.D. – 1 V.S.D
Let the value of 1 M.S.D = a unit
If n vernier scale divisions coincide with m main scale divisions, then value of
3. Least count of a micrometer screw is found by the formula
where pitch = lateral distance moved in one complete rotation of the screw.
Order of Accuracy: Proportionate Error
The order of accuracy of the result of measurements is determined by the least counts of the measuring instruments used to make those measurements. Suppose a length x is measured with a metre scale, then the error in x is ± △x, where △x = least count of metre scale = 0.1 cm. If the same length is measured with vernier callipers of least count 0.01 cm, then △x = 0.01 cm.
3. Error in Product and Division: Suppose we determine the value of a physical quantity u by measuring three quantities x, y and z whose true values are related to u by the equation
Let the expected small errors in the measurement of quantities x, y and z be respectively ± ðx, ± ðy and ± ðz so that the error in u by using these observed quantities is ± ðu. The numerical values of ðx, ðy and ðz are given by the least count of the instruments used to measure them. Taking logarithm of both sides we have
Partial differentiation of the above equation gives
The proportional or relative error in u is △u/u . The values of △x, △y and △z may be positive or negative and in some cases the terms on the right hand side may counteract each other. This effect cannot be relied upon and it is necessary to consider the worst case which is the case when all errors add up giving an error △u given by the equation:
Thus to find the maximum proportional error in u, multiply the proportional errors in each factor (x, y and z) by the numerical value of the power to which each factor is raised and then add all the terms so obtained. The sum thus obtained will give the maximum proportional error in the result of u. When the proportional error of a quantity is multiplied by 100, we get the percentage error of that quantity.
An Important Tip
To find the dimensional formula of the required quantity, recall any relation which relates that quantity with other quantities whose dimensions we already know. For example, to find the dimensional formula for capacitance C, we can use relation
Similarly, to find the dimensional formula for magnetic field B, we can use relation F = q v B or F = BIL. or B = μnI (here n = Number of turns per unit length).
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