UNITS AND MEASUREMNT
Fundamental quantities:
Physics relies on the measurement of physical quantities, where specific quantities are designated as fundamental or base quantities. These base quantities include length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.
Units:
When measuring a physical quantity, it is compared to a standard, known as a unit. For instance, if the length of a rod is stated as 5 m, it signifies that the length of the rod is five times the standard unit of measurement, which is the meter.
Fundamental Unit:
Fundamental or base units refer to the units associated with fundamental or base quantities. These base units are listed in a table, representing the standard measurements for length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.
Table 1
Derived Unit
The units of other physical quantities can be expressed as combination of base units. Such units are called derived units.
Example: Unit of force is kgms-2 (or Newton). Unit of velocity is ms-1.
System of Units:
A complete set of fundamental and derived units is called a system of unit.
Different system of units:
The different systems of units are CGS system FPS (or British) system, MKS system and SI system. A comparison of these systems of unit is given in the table below, (for length, mass and time)
Table 2
Note: The first three systems of units were used in earlier times. Presently we use SI system.
International System Of Unit (Si Unit):
The internationally accepted system of unit for measurement is system international d’ unites (French for International System of Units). It is abbreviated as SI.
The SI system is based on seven fundamental units and these units have well defined and internationally accepted symbols, (given in table –1)
Solid Angle and Plane Angle:
Other than the seven base units, two more units are defined.
1.Plane angle (dq):
It is defined as ratio of length of arc (ds) to the radius, r.
- The unit of plane angle is radian.
- Its symbol is rad.
2.Solid Angle (dW):
It is defined as the ratio of the intercepted area (dA) of spherical surface, to square of its radius.
$d\Omega=\frac{dA}{r^2}$
- The unit of solid angle is steradian.
- The symbol is Sr.
Measurement Of Length
Two methods are used to measure length
- direct method
- indirect method.
The metre scale, Vernier caliper, screwgauge, spherometer are used in direct method for measurement of length. The indirect method is used if range of length is beyond the above ranges.
Measurement of Large Distances:
Parallax Method:
Parallax method is used to find distance of planet or star from earth. The distance between two points of observation (observatories) is called base. The angle between two directions of observation at the two points is called parallax angle or parallactic angle (q).
The planet ‘s’ is at a distance ‘D’ from the surface of earth. To measure D, the planet is observed from two observatories A and B (on earth). The distance between A and B is b and q be the parallax angle between direction of observation from A and B.
AB can be considered as an arch A h B of length ‘b’ of a circle of radius D with its center at S. (Because q is very small, $\frac bD<<1$ Thus from arch-radius relation.
$$\theta=\frac bD$$
$$D=\frac b\theta$$
Estimation Of Very Small Distances:
Size Of Molecule
Electron microscope can measure distance of the order of 0.6A0 (wavelength of electron).
Range Of Lengths:
The size of the objects in the universe varies over a very wide range. The table (given below) gives the range and order of lengths and sizes of some objects in the universe.
Units for short and large lengths
1 fermi = 1f = 10-15m
1 Angstrom = 1A° = 10-10m
1 astronomical unit = 1AU = 1.496 × 1011m
1 light year = 1/y = 9.46 × 1015m
(Distance that light travels with velocity of 3 × 108 m/s in 1 year)
1 par sec = 3.08 × 1016m = 3.3 light year
(Par sec is the distance at which average radius of earth’s orbit subtends an angle of 1 arc second).
Measurement Of Mass
Mass is basic property of matter. The S.l. unit of mass is kg. While dealing with atoms and molecules, the kilogram •is an inconvenient unit. In this case there is an important standard unit called the unified atomic mass unit( u).
1 unified atomic mass unit = lu = (1/12)th of the mass of carbon-12
Range Of Masses:
The masses of the objects in the universe vary over a very wide range which is given in the table.
Measurement Of Time
To measure any time interval we need a clock. We now use an atomic standard of time, which is based on the periodic vibrations produced in a cesium atom. This is the basis of the cesium clock sometimes called atomic clock.
Definition of second:
One second was defined as the duration of 9, 192, 631, 770 internal oscillations between two hyperfine levels of Cesium-133 atom in the ground state.
Range and Order of time intervals
Error:
The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error.
Systematic errors:
Systematic errors are those errors that tend to be in one direction, either positive or negative.
Sources of systematic errors
- Instrumental errors
- Imperfection in experimental technique or procedure
- personal errors
1.Instrumental errors:
Instrumental error arise from the errors due to imperfect design or calibration of the measuring instrument.
eg: In Vernier Callipers, the zero mark of vernier scale may not coincide with the zero mark of the main scale.
2.Imperfection in experimental technique or procedure:
To determine the temperature of a human body, a thermometer placed under the armpit will always give a temperature lower than the actual value of the body temperature. Other external conditions (such as changes in temperature, humidity, velocity……..etc) during the experiment may affect the measurement.
3.Personal Errors:
Personal error arise due to an individual’s bias, lack of proper setting of the apparatus or individual carelessness etc.
Random errors
The random errors are those errors, which occur irregularly and hence are random with respect to sign and size. These can arise due to random and unpredictable fluctuations in experimental conditions (eg. unpredictable fluctuations in temperature, voltage supply, etc.)
Least Count Error
The smallest value that can be measured by the measuring instrument is called its least count. The least count error is the error associated with the resolution of the instrument. By using instruments of higher precision, improving experimental technique etc, we can reduce least count error.
Absolute Error, Relative Error And Percentage Error:
The magnitude of the difference between the true value of the quantity and the measured value is called absolute error in the measurement. Since the true value of the quantity is not known, the arithmetic mean of the measured values may be taken as the true value.
Explanation:
Suppose the values obtained in several measurements are a1, a2, a3,………,an. Then arithmetic mean can be written as
$$a_{mean}=\frac{a_1+a_2+a_3\dots\dots.+a_n}n$$
a.The absolute error,
$\Delta\;a_1=a_{mean}–a_1$
$\trianglea_2=a_{mean}–a_2$
$\trianglea_n=a_{mean}–a_n$
b.Mean absolute error:
The arithmetic mean of all the absolute errors is known as mean absolute error. The mean absolute error in the above case,
$a_{mean}=\frac{\vert\triangle a_1\vert+\vert\triangle a_2\vert+\vert\triangle a_3\vert\dots\dots.+\vert\triangle a_n\vert}n$
c.Relative error:
The relative error is the ratio of the mean absolute error () to the mean value ().
$$Relative\;error=\frac{\triangle a_{mean}}{a_{mean}}$$
d.Percentage error:
The relative error expressed in percent is called the percentage error (δa).
$$Percentage\;error,\delta a=\frac{\triangle a_{mean}}{a_{mean}}\times100\%$$
Combination Of Errors:
When a quantity is determined by combining several measurements, the errors in the different measurements will combine in some way or other.
1.Error of a sum or a difference:
Rule: 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.
Explanation:
Let two quantities A and B have measured values A ± ∆A and B ± ∆B respectively. ∆A and ∆B are the absolute errors in their measurements. To find the error Dz that may occur in the sum
z = A + B,
Consider
z + ∆z = (A ± ∆A) + B ± ∆B = (A + B) ± ∆A ± ∆B
The maximum possible error in the value of z is given by,
∆z = ∆A + ∆B
Similarly, it can be shown that, the maximum error in the difference.
Z = A – B is also given by
∆z = ∆A + ∆B
2.Error of product ora quotient:
Rule: When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.
Explanation:
Suppose Z=AB and the measured values of A and B are A + ∆A and B + ∆B. They
Z + ∆Z = (A + ∆A) (B + ∆B)
= AB ± B∆A ± A∆B ± ∆A∆B
Dividing LHS by Z and RHS by AB, we get
$$1\;\pm\;\frac{\triangle Z}Z=1\;\pm\;\frac{\triangle A}A\pm\;\frac{\triangle B}B\pm\left(\frac{\triangle A}A\right)\frac{\triangle B}B$$
3.Errors in case of a measured quantity raised to a power:
Suppose Z = A2
$\frac{\triangle Z}Z=\;\frac{\triangle A}A+\frac{\triangle A}A=2\frac{\triangle A}A$
Hence, the relative error in A2 is two time the error in A.
In general, if
$Z=\frac{A^pB^q}{C^r}$
Then,
$$\frac{\triangle Z}Z=p\left(\frac{\triangle A}A\right)+q\left(\frac{\triangle B}B\right)+r\left(\frac{\triangle C}C\right)$$
Hence the rule: The relative error in a physical quantity raised to the power K is the K times the relative error in the individual quantity.
Significant Figures
Every measurement involves errors. Hence the result of measurement should be reported in a way that indicates the precision of measurement.
Normally, the reported result of measurement is a number that includes all digits in the number that are known reliable plus the first digit that is uncertain. The reliable digits plus the first uncertain digit are known as significant digits or significant figures.
Example:
- The length of the rod measured is 3.52cm. Here there are 3 significant figures. The digits 3 and 5 are reliable and the last digit 2 is uncertain.
- The mass of a body measured as 3.407g. Here are four significant figures.
When the measurement becomes more accurate, the number of significant figure is increased.
Rules to find significant figures:
- All the non zero digits are significant.
Example:
Question 1.
Find significant figure of
2500
263.25
Answer:
In this case, there are two nonzero numbers. Hence significant figure is 2.
In this, there are 5 nonzero numbers. Hence significant figure is 5.
- All the zeros between two nonzero digits are significant, no matter where the decimal point
is,
Example:
Question 2.
Find the significant figure
2.05
302.005
2000145
Answer:
Significant figure is 3
Significant figure is 6
Significant figure is 7
- If the number is less than 1, the zeros on the right of decimal point but to the left to the first nonzero digits are not significant.
Example:
Question 1.
Find the significant figure of
0.002308
0.000135
Answer:
4 significant figures
3 significant figures
- The terminal zeros in a number without a decimal point are not significant.
Example:
Question 1.
Find the significant figure of
12300
60700
Answer:
3
3
Note: But if the number obtained is on the basis of actual measurement, all zeros to the right of last non zero digit are significant.
Example: If distance is measured by a scale as 2010m. This contain 4 significant figures.
- The terminal zeros in a number with a decimal point are significant.
Example:
Question 1.
Find the significant figure of
3.500
0.06900
4.7000
Answer:
4
4
5
Method to find significant figures through scientific notation:
In this notation, every number is expressed as , where a is a number between 1 and 10 and b is any positive or negative power. In this method, we write the decimal after the first digit.
Example:
4700m =4.700 × 103m
The power of 10 is irrelevant to the determination of significant figures. But all zeros appearing in the base number in the scientific notation are significant. Hence each number in this case has 4 significant figures.
Significant figures in numbers:-
Rules for Arithmetic operations with significant figures:
1.Rules for multiplication or division:
In multiplication or division, the computed result should not contain greater number of significant digits than in the observation which has the fewest significant digits.
Examples:
- 53 × 2.021 =107.113
The answer is 1.1 × 102 since the number 53 has only 2 significant digits.
(ii) 3700 10.5 = 352.38
The answer is 3.5 × 102 since the minimum number of significant figure is 2 (in the number 3700)
2.Rules for Addition and Subtraction:
In addition or subtraction of given numbers, the same number of decimal places is retained in the result as are present in the number with minimum number of decimal places.
Examples:
(i) 76.436 +
12.5
88.936
The answer is 88.9, since only one decimal place is found in the number 12.5.
(ii) 43.6495 +
4.31
47.9595
The answer is 47.96 since only two decimal places are to be retained.
(iii) 8.624 –
3.1726
5.4514
The answer is 5.451
(iv) 6.5 × 10-5 – 2.3 × 10-6 = 6.5 × 10-5 – 0.23 × 10-5
= 6.27 × 10-5
The answer is = 6.3 × 10-5
Dimensions And Dimensional Analysis
All physical quantities can be expressed in terms of seven fundamental quantities. (Mass, length, time, temperature, electric current, luminous intensity and amount of substance). These seven quantities are called the seven dimensions of the physical world.
- The dimensions of the three mechanical quantities mass, length and time are denoted by M, L and T. Other dimensions are denoted by K (for temperature), I (for electric current), cd (for luminous intensity) and mol (for the amount of substance).
- The letters [L], [M], [T] etc. specify only the nature of the unit and not its magnitude. Since area may be regarded as the product of two lengths, the dimensions of area are represented as $\left[L\right]\times\left[L\right]=\left[L\right]^2$.
- Similarly, volume being the product of three lengths, its dimensions are represented by$\left[L\right]^3$.Density being mass per unit volume, its dimensions are $\frac M{\left[L\right]^3}$ or $M^1L^{-3}$
- Thus, the dimensions of a physical quantity are the powers to which the fundamental units of length, mass, time must be raised to represent it.
Note: The dimensions of a physical quantity and the dimensions of its unit are the same.