Chapter 2: Units and Measurement

Introduction From ancient times to the present there has been a need for measuring things accurately. In India a carpenter buys wood by the cubic foot, petrol is sold by the litre gold by the gram land by the acre, vegetables by the kg and cable by the metre. In the United States, a carpenter pays for wood by the board-foot, petrol is sold by the gallon, and a jeweler sells gold by the ounce, land is sold by the acre, fruits and vegetables are sold by the pound, and electric cable is sold by the yard. Buyers and sellers have always tried to cheat each other by wrongly representing the quantity of the product exchanged.

When the ancient Egyptians built the pyramids, they measured the stones they cut using body dimensions every worker could easily understand. Small distances were measured in digits (the width of a finger) and longer distances in cubits (the length from the tip of the elbow to the tip of the middle finger; 1 cubit = 28 digits). The Romans were famous road builders and measured distances in paces (1 pace = two steps). Archaeologists have uncovered ancient Roman roads and found milestones marking each 1000 paces (mil is Latin for 1000). The Danes were a seafaring people and interested in knowing the depth of water in shipping channels. They measured soundings in fathoms (the distance from the tip of the middle finger on one hand to the tip of the middle finger on the other) so navigators could easily visualize how much clearance their boats would have.In England distances were defined with reference to body features of the king. A yard was the circumference of his waist; an inch was the width of his thumb, and a foot the length of his foot. Scientists today prefer to use the metric system. The metric system did not evolve from a variety of ancient measurement systems, but was a logical, simplified system developed in Europe during the seventeenth and eighteenth centuries. The metric system is now the mandatory system of measurement in every country of the world except the United States, Liberia and Burma (Myanmar). In 1960, an international conference was called to standardize the metric system. The international System of Units (SI) was established in which all units of measurement are based upon seven base units: meter (distance), kilogram (mass), second (time), ampere (electrical current), Kelvin (temperature), mole (quantity), and candela (luminous intensity). The metric system simplifies measurement by using a single base unit for each quantity and by establishing decimal relationships among the various units of that same quantity. For example, the meter is the base unit of length and other necessary units are simple multiples or sub-multiples:

The table shows the SI prefixes and symbols. Throughout this book we use the metric system of measurement.

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Table 1: SI Prefixes and SymbolsFactor Decimal Representation Prefix Symbol

1018 1,000,000,000,000,000,000 exa E

1015 1,000,000,000,000,000 peta P

1012 1,000,000,000,000 tera T

109 1,000,000,000 giga G

106 1,000,000 mega M

103 1,000 kilo k

102 100 hecto h

101 10 deka da

100 1

10-1 0.1 deci d

10-2 0.01 centi c

10-3 0.001 milli m

10-6 0.000 001 micro m

10-9 0.000 000 001 nano n

10-12 0.000 000 000 001 pico p

10-15 0.000 000 000 000 001 femto f

10-18 0.000 000 000 000 000 001 atto a

Unit: A unit of measurement is a definite amount of a physical quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same physical quantity. Any other value of the physical quantity can be expressed as a simple multiple of the unit of measurement.

Fundamental or Base Units and Derived Units: A fundamental unit of measurement is a defined unit that cannot be described as a function of other units. Not all quantities require a unit of their own. Using physical laws, units of quantities can be expressed as combinations of units of other quantities. Thus only a small set of base units is required, the other units are derived units.

International System of UnitsThe International System of Units (SI) defines seven fundamental units of measurement.

SI Fundamental Units

The International System of Units (SI) defined seven basic units of measure from which all other SI units are derived. These SI base units or commonly called metric units are:

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Measure Unit Symbol Area of Science

Time Second s All

Length or distance Meter or Metre m All

Mass Kilogram kg Physics

Electric Current Ampere A Physics

Temperature Kelvin K Physics

Luminous Intensity Candela cd Optics

Amount of Substance Mole mol Chemistry

Plane angle: Radian describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. More generally, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, θ = s /r, where θ is the subtended angle in radians, s is arc length, and r is radius. As the ratio of two lengths, the radian is a pure number and the symbol "rad" is usually omitted. In the absence of any symbol radians are assumed, and when degrees are meant the symbol ° is used. A complete revolution is 2π radians (shown here with a circle of radius one and thus circumference 2π).It follows that the magnitude in radians of one complete revolution (360 degrees) is the length of the entire circumference divided by the radius, or 2πr /r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees.Solid Angle: In geometry, a solid angle (symbol: Ω) is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large the object appears to an observer looking from that point. In the SI System of Units, a solid angle is a dimensionless unit of measurement called a steradian (symbol: sr).

Measurement of Length

Measurement of Large Distances: Parallax MethodIn order to calculate how far away a star is, astronomers use a method called parallax. Because of the Earth's revolution about the sun, near stars seem to shift their position against the farther stars. This is called parallax shift. By observing the distance of the shift and knowing the diameter of the Earth's orbit, astronomers are able to calculate the parallax angle across the sky. The smaller the parallax shift, the farther away from earth the star is. This method is only accurate for stars within a few hundred light-years of Earth. When the stars are very far away, the parallax shift is too small to measure.

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Measurement of Very Small Distances: **See power point

Range of Lengths

Measurement of Mass

Unified Atomic Mass Unit:The standard – throughout chemistry, physics, biology, etc – is the unified atomic mass unit (symbol u). It is defined as 1/12 (one-twelfth) of the mass of an isolated carbon-12 atom. The symbol amu, which stands for atomic mass unit is defined in terms of oxygen As it’s a unit of mass, the atomic mass unit (u) has a value, in kilograms = 1.660 538 782(83) x 10-27 kg. The kilogram is defined in terms of a bar of platinum-iridium alloy, sitting in a vault in Paris. It is important to recognize that the unified atomic mass unit is not an SI unit, but one that is accepted for use with the SI. The kilogram and unified atomic mass unit are related via a primary SI unit, the mole, which is defined as the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12. The number of atoms there are in a mole of an element is given by Avogadro’s number. The Dalton (symbol D, or Da) is the same as the unified atomic mass unit. In microbiology and biochemistry, many molecules have hundreds, or thousands, of constituent atoms, so it’s convenient to state their masses in terms of ‘thousands of unified atomic mass units’, so the convention is to use kDa (kilodaltons).

Range of Masses

Measurement of TimeHumans have been measuring time for a relatively short period in our long history. The desire to synchronize our activities came about 5,000 or 6,000 years ago as our nomadic ancestors began to settle and build civilizations. Before that, tome was divided only into daylight and night, with days for hunting and working and nights for sleeping. But as people began to feel the need to coordinate their actions, they needed a unified system of keeping time.The second (symbol: s) is the base unit of time in the International System of Units and is also a unit of time in other systems of measurement; it is the second division of the

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hour by sixty, the first division by 60 being the minute. Between AD 1000 and 1960 the second was defined as 1/86,400 of a mean solar day. Between 1960 and 1967, it was defined in terms of the period of the Earth's orbit around the Sun. But it is now defined more precisely in atomic terms. Astronomical observations of the 19th and 20th centuries revealed that the mean solar day is slowly lengthening, thus the sun–earth motion is no longer considered a suitable basis for definition of time. With the advent of atomic clocks, it became feasible to define the second based on fundamental properties of nature. Since 1967, the second has been defined to be the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.

When exposed to certain frequencies of radiation, such as radio waves, the subatomic particles called electrons that orbit an atom's nucleus will "jump" back and forth between energy states. Clocks based on this jumping within atoms can therefore provide an extremely precise way to count seconds.

Time is one of the seven fundamental physical quantities in the International System of Units. Time is used to define other quantities, such as velocity and acceleration.

Accuracy, Precision of Instruments and Errors in Measurement

Error: Error is the actual difference between a measured value and the known standard value. In other words, the errors in experimental data are those that are always unknown to some extent and carry some amount of uncertainty. A reasonable definition of experimental uncertainty may be taken as the possible value the error may have. The uncertainty may vary a great deal depending upon the circumstances of the experiment. Accuracy: The accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's actual value.Precision: The precision of a measurement system, is the degree to which repeated measurements under unchanged conditions show the same results.Systematic ErrorsThere may be certain fixed errors which will cause repeated readings to be in error by roughly some amount for some unknown reasons. These are called systematic errors and are inherent errors of instruments or methods. These errors always give a constant deviation. On the basis of the sources of errors, systematic errors may be divided into following sub-categories:Instrumental ErrorsNo apparatus can be constructed to satisfy all specifications completely. This is the reason of giving guarantees within a limit. Therefore, a manufacturer always mentions the minimum possible errors in the construction of the instruments.Imperfections in Experimental Technique or ProcedureFollowing are some of the reasons of errors in results of the indicating instruments:(a) Construction of the Scale: There is a possibility of error due to the division of the scale not being uniform and clear.(b) Fitness and Straightness of the Pointer: If the pointer is not fine and straight, then it always gives the error in the reading.(c) Parallax: Without a mirror under the pointer there may be parallax error in reading.(d) Efficiency or skill of the observer : Error in the reading is largely dependent upon the skill of the observer in reading the measurement accurately.

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Personal ErrorsIt is due to the variation in final adjustment of the measuring apparatus. The error varies from person to person.Random ErrorsAfter corrections have been applied for all the parameters whose influences are known, there is some deviation left. These are random errors and their magnitudes are not constant. Persons performing the experiment have no control over the origin of these errors. These errors may be either positive or negative. In the case of random errors, the law of probability may be applied. Generally, these errors may be minimized by taking the average of a large number of readings.Least Count ErrorThe "Least Count" of any measuring equipment is the smallest quantity that can be measured accurately using that instrument. Thus Least Count indicates the degree of accuracy of measurement that can be achieved by the measuring instrument.All measuring instruments used in physics have a least count. A meter ruler's least count is 0.1 centimeter; an electronic scale may have a least count of 0.001g, a vernier caliper a least count of 0.02 millimeters.

Absolute Error, Relative Error and Percentage ErrorAbsolute Error: Absolute error is defined as the magnitude of difference between the actual and the individual values of any quantity in question. Say we measure any given quantity for n number of times and a1, a2 , a3 …..an are the individual values then:

Arithmetic mean am = [a1+a2+a3+ .....an]/n

am=

Definition of absolute error: The magnitude of the difference between the individual measurements and the true value of the quantity is called the absolute error of the measurement. By definition:

Δa1= am – a1

Δa2= am – a2

…am – an

Mean Absolute Error = Δamean=

Note: While calculating absolute mean value, we do not consider the +- sign in its value.

Relative Error or Fractional Error: This is defined as the ration of mean absolute error to the mean value of the measured quantity:

δa =mean absolute value/mean value = Δamean/am

Percentage Error: It is the relative error measured in percentage.

Percentage Error =mean absolute value/mean value x 100= Δamean/amx100

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Significant Figures

The number of significant figures in a result is simply the number of figures that are known with some degree of reliability. The number 13.2 is said to have 3 significant figures. The number 13.20 is said to have 4 significant figures.

Rules for deciding the number of significant figures in a measured quantity:

(1) All nonzero digits are significant:

1.234 g has 4 significant figures, 1.2 g has 2 significant figures.

(2) Zeroes between nonzero digits are significant:

1002 kg has 4 significant figures, 3.07 mL has 3 significant figures.

(3) Leading zeros to the left of the first nonzero digits are not significant; such zeroes merely indicate the position of the decimal point:

0.001o C has only 1 significant figure, 0.012 g has 2 significant figures.

(4) Trailing zeroes that are also to the right of a decimal point in a number are significant:

0.0230 ml has 3 significant figures, 0.20 g has 2 significant figures.

(5) When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant:

190 miles may be 2 or 3 significant figures, 50,600 calories may be 3, 4, or 5 significant figures.

The potential ambiguity in the last rule can be avoided by the use of standard exponential, or "scientific," notation. For example, depending on whether the number of significant figures is 3, 4, or 5, we would write 50,600 calories as:

5.06 × 104 calories (3 significant figures) 5.060 × 104 calories (4 significant figures), or 5.0600 × 104 calories (5 significant figures)

By writing a number in scientific notation, the number of significant figures is clearly indicated by the number of numerical figures in the 'digit' term as shown by these examples.

Exact Number: An exact number is exact if it is known with complete certainty. Most exact numbers are integers: exactly 12 inches are in a foot, there might be exactly 23 students in a class. Exact numbers are often found as conversion factors or as counts of objects. Exact numbers can be considered to have an infinite number of significant

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figures. Thus, the number of apparent significant figures in any exact number can be ignored as a limiting factor in determining the number of significant figures in the result of a calculation.

Rules for mathematical operations: In carrying out calculations, the general rule is that the accuracy of a calculated result is limited by the least accurate measurement involved in the calculation.

(1) In addition and subtraction, the result is rounded off to the last common digit occurring furthest to the right in all components. Another way to state this rule is as follows: in addition and subtraction, the result is rounded off so that it has the same number of digits as the measurement having the fewest decimal places (counting from left to right). For example: 100 (assume 3 significant figures) + 23.643 (5 significant figures) = 123.643, which should be rounded to 124 (3 significant figures). Note, however, that it is possible two numbers have no common digits (significant figures in the same digit column).(2) In multiplication and division, the result should be rounded off so as to have the same number of significant figures as in the component with the least number of significant figures. For example, 3.0 (2 significant figures) × 12.60 (4 significant figures) = 37.8000which should be rounded to 38 (2 significant figures).

Rules for rounding off numbers

(1) If the digit to be dropped is greater than 5, the last retained digit is increased by one. For example, 12.6 is rounded to 13. (2) If the digit to be dropped is less than 5, the last remaining digit is left as it is. For example, 12.4 is rounded to 12. (3) If the digit to be dropped is 5, and if any digit following it is not zero, the last remaining digit is increased by one. For example, 12.51 is rounded to 13. (4) If the digit to be dropped is 5 and is followed only by zeroes, the last remaining digit is increased by one if it is odd, but left as it is if even. For example, 11.5 is rounded to 12, 12.5 is rounded to 12. This rule means that if the digit to be dropped is 5 followed only by zeroes, the result is always rounded to the even digit. The rationale for this rule is to avoid bias in rounding: half of the time we round up, half the time we round down.

Sample problems on significant figures

1. 37.76 + 3.907 + 226.4 = ?

2. 319.15 - 32.614 = ?

3. 104.630 + 27.08362 + 0.61 = ?

4. 125 - 0.23 + 4.109 = ?

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5. 2.02 × 2.5 = ?

6. 600.0 / 5.2302 = ?

7. 0.0032 × 273 = ?

8. (5.5)3 = ?

9. 0.556 × (40 - 32.5) = ?

10. 45 × 3.00 = ?

11. What is the average of 0.1707, 0.1713, 0.1720, 0.1704, and 0.1715?

12. What is the standard deviation of the numbers in question 11?

13. 3.00 x 105 - 1.5 x 102 = ? (Give the exact numerical result, and then express that result to the correct number of significant figures).

Answer key to sample problems on significant figures

1. 37.76 + 3.907 + 226.4 = 268.1

2. 319.15 - 32.614 = 286.54

3. 104.630 + 27.08362 + 0.61 = 132.32

4. 125 - 0.23 + 4.109 = 129 (assuming that 125 has 3 significant figures).

5. 2.02 × 2.5 = 5.0

6. 600.0 / 5.2302 = 114.7

7. 0.0032 × 273 = 0.87

8. (5.5)3 = 1.7 x 102

9. 0.556 × (40 - 32.5) = 4

10. 45 × 3.00 = 1.4 x 102This answer assumes that 45 has two significant figures; however, that is not unambiguous, because there is no decimal point, and because it is not expressed in scientific notation. If 45 is an exact number (e.g., a count), then the result should be 1.35 x 102.

11. What is the average of 0.1707, 0.1713, 0.1720, 0.1704, and 0.1715?The average of these numbers is calculated to be 0.17118, which rounds to 0.1712 .

12. What is the standard deviation of the numbers in question 11?

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The result that you get in calculating the standard deviation of these numbers depends on the number of digits retained in the intermediate digits of the calculation. For example, if you used 0.1712 instead of the more accurate 0.17118 as the mean in the standard deviation calculation that would be wrong (don't round intermediate results or you will introduce propagated error into your calculations).The Mathworks MATLAB std function gives 6.379655163094639e-004Microsoft Excel gives 0.000637965516309464000000000000A Hewlett-Packard 50G calculator gives 6.37965516309e-04These results should be rounded to 0.0006380, which is 6.380 x 10-4 (expressed in scientific notation).13. 3.00 x 105 - 1.5 x 102 = ? (Give the exact numerical result, and then express that result to the correct number of significant figures).Doing the math right is the first step. Then, click here to send your answer and to receive an answer key and explanation by email.

Dimensions of Physical Quantities

Dimensions: Physical dimension is a generic description of the kind of quantity being measured. Physical dimension is an inherent and unvarying property for a given quantity. A given quantity can only ever have one specific physical dimension. The converse, however, is not true: different physical quantities can have the same physical dimension.They are the powers (or exponents) to which the units of base quantities are raised for representing a derived unit of that quantity.

Examples: Dimensional formula of volume [M0L3T0]

Dimensional formula of velocity [M0LT−1]

Dimensional formula of acceleration [M0LT−2]

Examples:• The height of a building, the diameter of a proton, the distance from the Earth to the Moon, and the thickness of a piece of paper all involve a measurement of a length, L, of some kind. They are all measured in different ways, and might have their values assigned using different units, but they all share the common feature of being a kind of length.• Time intervals, timestamps, or any other measurement characterizing duration, all share the common feature of having a physical dimension of time, T. You will never be able to assign a value to a time coordinate using units of length—length and time are fundamentally distinct and inequivalent physical dimensions.When we talk about a quantity’s physical dimension, using symbols such as L for length, or T for time, we are not using those symbols to represent an actual variable—we will not assign any particular values or use any specific units for those symbols. Instead, when we write, “this quantity has physical dimension L”, the symbol L really just means “…having a length unit”. To make this distinction explicit, we will henceforth set aside symbols in square brackets whenever making a statement about physical dimensions. So: all distances, lengths, heights, and widths have dimension of length, [L], all timestamps and time intervals have dimension of time, [T], and so forth. The physical dimension of a quantity is determined by how we measure that quantity—and to do that, we need to define an appropriate unit for the measurement. We can make different

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choices for the units we use (e.g. the SI system versus the British Engineering system, BE), but whatever those alternative unit choices are, they must all have the same physical dimension. For example: meters, inches, furlongs, and nautical miles are all very different units, but the all share the common feature of being lengths, [L].

Dimensional Formulae and Dimensional Equations

Dimensional Formula: Dimensional formula of a derived physical quantity is the “expression showing powers to which different fundamental units are raised”.

Ex: Dimensional formula of Force F = [M^1L^1T^{-2}]

Dimensional equation: When the dimensional formula of a physical quantity is expressed in the form of an equation by writing the physical quantity on the left hand side and the dimensional formula on the right hand side, then the resultant equation is called Dimensional equation.

Ex: Dimensional equation of Energy is E = [M^1L^2T^{-2}] .

Question: How can you derive Dimensional formula of a derived physical quantity.

Ans: We can derive dimensional formula of any derived physical quantity in two ways

i) Using the formula of the physical quantity: Ex: let us derive dimensional formula of Force.

Force F= ma; substitute the dimensional formula of mass m →[M] ; acceleration →[LT^{-2}]

We get F → [M][L T^{-2}]; F →[M^1L^1T^{-2}] .

ii) Using the units of the derived physical quantity. Ex: let us derive the dimensional formula of momentum.

Unit of Momentum ( p ) → [kg-m sec^{-1}] ;

kg is unit of mass → [M] ; is unit of length → [L] ; sec is the unit of time →[T]

Substitute these dimensional formulas in above equation we get p →[M^1L^1T^{-1}].

• Quantities having no units, can not possess dimensions: Trigonometric ratios, logarithmic functions, exponential functions, coefficient of friction, strain, Poisson’s ratio, specific gravity, refractive index, Relative permittivity, Relative permeability. All these quantities neither possess units nor dimensional formulas.

• Quantities having units, but no dimensions: Plane angle, angular displacement, solid angle. These physical quantities possess units but they does not possess dimensional formulas.

• Quantities having both units & dimensions: The following quantities are examples of such quantities.

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Area, Volume,Density, Speed, Velocity, Acceleration, Force, Energy etc.

Dimensional Analysis and its ApplicationsDimensional AnalysisOnly quantities with like dimensions may be added(+), subtracted(-) or compared (=,<,>). This rule provides a powerful tool for checking whether or not equations are dimensionally consistent. It is also possible to use dimensional analysis to suggest plausible equations when we know which quantities are involved.Example of checking for dimensional consistencyConsider one of the equations of constant acceleration,s = ut + 1/2 at2. (1)

The equation contains three terms: s, ut and 1/2at2. All three terms must have the same dimensions.

s: displacement = a unit of length, L ut: velocity x time = LT-1 x T = L 1/2at2 = acceleration x time = LT-2 x T2 = L

All three terms have units of length and hence this equation is dimensionally valid. Of course this does not tell us if the equation is physically correct, nor does it tell us whether the constant 1/2 is correct or not.

Checking the Dimensional Consistency of EquationsBased on the principle of homogeneity of dimensions only that formula is correct in which the dimensions of the various terms on one side of the relation are equal to the respective dimensions of these terms on the other side of the relation.Example:

Check the correctness of the relation where l is length and t is time period of a

simple pendulum; g is acceleration due to gravity.Solution:

Dimension of L.H.S = t = T

Dimension of R.H.S = (2π is a constant);

Dimensionally, L.H.S = R.H.S; therefore, the given relation is correct.

Equations in physics must be dimensionally consistent. It is extremely useful to perform a dimensional analysis on any doubtful equation according to the following rules:Two quantities can only be added or subtracted if they are of the same dimension.Two quantities can only be equal if they are of the same dimension.

Note that only the dimension needs to be the same, not the units. It is perfectly valid to write 12 inches = 1 foot because both of them are lengths, L = L, even though their units

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are different. However, it is not valid to write x inches = t seconds because they have different dimensions, L and T.

Deducing Relationships among Physical QuantitiesExample:The centripetal force, F acting on a particle moving uniformly in a circle may depend upon the mass (m), velocity (v) and radius (r) of the circle. Derive the formula for F using the method of dimensions.Solution:Let F = kmavbrc … (i)Where, k is the dimensionless constant of proportionality, and a, b, c are the powers of m, v, r respectively.On writing the dimensions of various quantities in (i), we get[M1L1T−2] = Ma [LT−1]b Lc

= MaLbT−bLc

or M1L1T−2 = MaLb + cT−b

On applying the principle of homogeneity of dimensions, we geta = 1,b= 2,b + c = 1 …(ii)From (ii), c = 1 − b = 1 − 2 = −1On putting these values in (i), we getF = km1v2r−1

Which is the required relation for centripetal force.

http://cbse-notes.blogspot.in/2012/05/cbse-class-xi-physics-ch2-dimension.html

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