Measurement in Physics : Measurement in Physics : Fundamental Units :
Measurement of a physical quantity involves:
The standard or unit in which the quantity is being measured
The numerical value representing the number of times the quantity contains that unit.
The physical quantities which do not depend upon other quantities are called fundamental quantities. In M.K.S. system the fundamental quantities are mass, length and time, while in more general Standard International (S.I.) system the Fundamental quantities are mass, length, time, temperature, luminous intensity, current and amount of substance.
The units of fundamental quantities are called fundamental units and are discussed below. Slide 3: Derived Units :
The units of physical quantities which may be derived from fundamental units are called derived units, for example:
Unit of area: area = length × breadth
unit of area = unit of length × unit of breadth
= m × m = m2
Unit of Velocity: velocity =
unit of velocity = = = ms-1
Hence m2 and ms-2 are derived units. Slide 4: Systems of Units :
There are following principal system of units:
1. C.G.S System : length ? centimetre (cm), mass ? gram (g) time ? second (s).
2. F.P.S System : length ? foot (ft), mass ? pound (lb), time ? second (s).
3. M.K.S. System: length ? metre (m), mass ? kilogram (kg), time ? second (s).
4. S.I. System : It has SEVEN fundamental units. length ? metre (m), mass ? kilogram (kg), Slide 5: time ? second (s).temperature ? kelvin (K),luminous intensity ? candela (cd),electric current ? ampere (A),amount of substance ? mole (mol).
In S.I. system there are two supplementary units.
Radian (rad) : Unit of plane angle
Steradian (st) : Unit of solid angle MASS : Mass of a body is defined as the quantity of matter in the body, which can never be zero.
Usually, we define two types of masses of a body: inertial mass and gravitational mass.
The inertial mass of a body is a measure of inertia of the body and the gravitational mass of a body is a measure of gravitational pull of earth on the body.
The values of the two types of masses of a body are identical.
The unit of mass is kilogram. MASS LENGTH : Length of an object may be defined as the distance of separation between any two points at the extreme ends of the object.
The unit of length is metre. LENGTH TIME : The idea of passage of time occurred first from the motion of moon across the sky, then from the formation of day and night as a result of rotation of earth around its axis, and so on.
Today, even a child is familiar with what time is. But it is very difficult to define time precisely.
According to Einstein, ‘time is what a clock reads’.
Any phenomenon that repeats itself regularly can serve as a measure of time.
Human heart which beats regularly is an inbuilt clock in every human being.
The unit of time is second. TIME ORDER OF MAGNITUDE : The order of magnitude of a number is the power of ten closest to the number.
Following table gives us some of the commonly used prefixes for power of ten.
Positive Powers of 10 ORDER OF MAGNITUDE Slide 10: Negative Powers of 10 Precision of measuring instruments : Measurements are done with the help of measuring for example, one can use a metre scale to measure lengths.
The accuracy of a measuring instrument depends upon several factors, like the limit or the resolution of the measuring instrument.
The precision of a number is often indicated by following it with the symbol and a second number indicating the maximum likely error. As an illustration, suppose the diameter of a steel rod is given as 56.47 0.02 m. So, the true value is unlikely to be less than 56.45 m or greater than 56.49 mm.
Precision can also be expressed in terms of the maximum likely fractional error or percentage error. A resistor marked “47 10%” probably has a true resistance differing from 47 by no more than 10% of 47 . So, the value is between 42 and 52 Precision of measuring instruments Slide 12: SIGNIFICANT FIGURES :
Significant figures in the measured value of a physical quantity tells us 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.
COMMON RULES FOR COUNTING SIGNIFICANT FIGURES :
Following are some of the common rules for counting significant figures in a given expression :
All zeros occurring between two non zero digits are significant. For example: x = 5008 has four significant figures. Again x = 7.0102 has five significant figures.
All non zero digits are significant.
For example: x = 7284 has four significant figures. Again x = 457 has only three significant digits. ROUNDING OFF : While rounding off measurements, we use the following rules by convention:
If the digit to be dropped is more than 5, then the preceding digit is raised by one.
For example, x = 6.87 is rounded off to 6.9. Again x = 12.78 is rounded off to 12.8 .
If the digit to be dropped is less than 5, then the preceding digit is left unchanged.
For example, x = 7.82 is rounded off to 7.8 . Again x = 3.94 is rounded off to 3.9 . ROUNDING OFF Slide 14: If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit is raised by one.
For example, x = 16.351 is rounded off to 16.4. Again x = 6.758 is rounded off to 6.8 .
If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd.
For example, x = 3.750 is rounded off to 3.8. Again x = 16.150 is rounded off to 16.2 .
If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is left unchanged, if it is even.
For example, x = 3.250 becomes 3.2 on rounding off, Again x = 12.650 becomes 12.6 on rounding off. ARITHMENTICAL OPERATIONS WITH SIGNIFICANT FIGURES : (i) Addition and subtraction :
In addition or subtraction, the number of decimal places in the result should equal the smallest number of decimal places of terms in the operation.
For example, the sum of three measurements of length; 2.1 m, 1.78 m and 2.046 m is 5.926m, which is rounded off to 5.9 m (upto smallest number of decimal places).
In the subtraction of quantities of nearly equal magnitudes, accuracy is almost destroyed. For example, if x = 42.87m and y = 12.86m, then
x – y = 12.87 - 12.86 = 0.01 m. The difference has only one significant figure, whereas x and y have four significant digits each. ARITHMENTICAL OPERATIONS WITH SIGNIFICANT FIGURES Slide 16: (ii) Multiplication and Division :
In multiplication and division, the number of significant figures in the product or in the quotient is the same as the smallest number of significant figures in any of the factors.
For example, suppose x = 3.8 and y = 0.125. Therefore, xy = (3.8) (0.125) = 0.475.
As least number of significant figures is 2 (in x = 3.8). Therefore, xy = 0.475 = 0.48 is rounded off to two significant figures.