Chapter 2: Units and Measurement

Chapter 2: Units and MeasurementIntroduction

An accurate and consistent system of measurement is the foundation of a healthy economy. In the United States, a carpenter pays for wood by the board-foot, while a motorist buys petrol 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. Without a consistent, honest system of measurement, world trade would be thrown into chaos. Buyers and sellers have always tried to defraud each other by wrongly representing the quantity of the product exchanged. From ancient times to the present there has been a need for measuring things accurately.When the ancient Egyptians built the pyramids, they measured the stones they cut using body dimensions every worker could 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 particularly 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. English farmers, however, estimated lengths in something they could more easily relate to: furlongs, the length of an average plowed furrow.

Many English units are specific to certain professions or trades. A sea captain reports distances in nautical miles and depths in fathoms, while a horse trainer measures height in hands and distance in furlongs. Unfortunately, most people have no idea what nautical miles, fathoms, hands, or furlongs are because they only use the more common measures of miles, yards, inches.

The early English settlers brought the Customary system of measurement with them to the American colonies. Although the Customary system is still widely used in America, scientists prefer to use the metric system. Unlike the English (Customary) 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:

1 meter = 0.001 kilometer = 1,000 millimeters =1,000,000 micrometers = 1,000,000,000 nanometers

The table 1 shows the SI prefixes and symbols. Throughout this book we use the metric system of measurement.

Table 1: SI Prefixes and Symbols

FactorDecimal RepresentationPrefixSymbol

10181,000,000,000,000,000,000exaE

10151,000,000,000,000,000petaP

10121,000,000,000,000teraT

1091,000,000,000gigaG

1061,000,000megaM

1031,000kilok

102100hectoh

10110dekada

1001

10-10.1decid

10-20.01centic

10-30.001millim

10-60.000 001microm

10-90.000 000 001nanon

10-120.000 000 000 001picop

10-150.000 000 000 000 001femtof

10-180.000 000 000 000 000 001attoa

Unit: You measure things by defining a standard unit and then stating the measurement in terms of multiples of that unit. A unit of measurement is a definite magnitude 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 units is required. These units are taken as the base units. Other units are derived units. Derived units are a matter of convenience, as they can be expressed in terms of basic units. Which units are considered base units is a matter of choice.International System of Units

The International System of Units (SI) defines seven fundamental units of measurement. SI base 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:

MeasureUnitSymbolArea of Science

TimeSecondsAll

Length or distanceMeter or MetremAll

MassKilogramkgPhysics

Electric CurrentAmpereAPhysics

Temperature KelvinKPhysics

Luminous IntensityCandelacdOptics

Amount of SubstanceMolemolChemistry

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, = s /r, where is the subtended angle in radians, s is arc length, and r is radius. Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, s = r. As the ratio of two lengths, the radian is a "pure number" that needs no unit symbol, and in mathematical writing the symbol "rad" is almost always omitted. In the absence of any symbol radians are assumed, and when degrees are meant the symbol is used. A complete revolution is 2 radians (shown here with a circle of radius one and thus circumference 2).

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 2r /r, or 2. Thus 2 radians is equal to 360 degrees, meaning that one radian is equal to 180/ degrees.

Solid Angle: In geometry, a solid angle (symbol: ) 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).

Problem areas

There are some variations or problem areas in these fundamental units:

Length: Although multiples or fractions of a meter are useful in most sciences, the unit is impractical in Astronomy. Instead, the fundamental unit of length in Astronomy is the light-year, which is the distance that light travels in kilometers in one year.

Mass: It would seem more intuitive to define the fundamental of mass as a gram. However, the SI decision was to say that 1000 grams or a kilogram was fundamental.

Electric current: Since electric current is man-made and depends on a number of factors, the ampere does not seem appropriate as a fundamental unit. The ampere is defined as:

"The constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in a vacuum, would produce between these conductors a force equal to 2 107 Newton per meter of length."

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.

Measurement of Very Small Distances

See power pointRange of LengthsMeasurement of MassUnified Atomic Mass UnitThe 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 its 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 Avogadros 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 its convenient to state their masses in terms of thousands of unified atomic mass units, so the convention is to use kDa (kilodaltons).

Range of MassesMeasurement 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 [source: Beagle]. Before that, we divided time only into daylight and night, with bright days for hunting and working and dark nights for sleeping. But as people began to feel the need to coordinate their actions, to be prompt for public gatherings and such, they needed a unified system of keeping time.But even the best mechanical pendulums and quartz crystal-based clocks develop discrepancies. Far better for timekeeping is the natural and exact "vibration" in an energized 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.

It is no surprise then that the international standard for the length of one second is based on atoms. Since 1967, the official definition of a second is 9,192,631,770 cycles of the radiation that gets an atom of the element called cesium to vibrate between two energy states.

Inside a cesium atomic clock, cesium atoms are funneled down a tube where they pass through radio waves . If this frequency is just right 9,192,631,770 cycles per second then the cesium atoms "resonate" and change their energy state.

A detector at the end of the tube keeps track of the number of cesium atoms reaching it that have changed their energy states. The more finely tuned the radio wave frequency is to 9,192,631,770 cycles per second, the more cesium atoms reach the detector.

The detector feeds information back into the radio wave generator. It synchronizes the frequency of the radio waves with the peak number of cesium atoms striking it. Other electronics in the atomic clock count this frequency. As with a single swing of the pendulum, a second is ticked off when the frequency count is met.

The first quality atomic clocks made in the 1950s were based on cesium, and such clocks honed to greater precisions over the decades remain the basis used to keep official time throughout the world.

In the United States, the top clocks are maintained by the National Institutes of Standards and Technology (NIST) in Boulder, Colo., and the United States Naval Observatory (USNO) in Washington, D.C.

The NIST-F1 cesium atomic clock can produce a frequency so precise that its time error per day is about 0.03 nanoseconds, which means that the clock would lose one second in 100 million years.

Super-accurate timekeeping is integral to many elements of modern life, such as high-speed electronic communications, electrical grids and the Global Positioning System (GPS) and of course knowing when your favorite television show comes on.

The second (symbol: s) is the base unit of time in the International System of Units (SI)[1] and is also a unit of time in other systems of measurement (abbreviated s or sec[2]); it is the second division of the hour by sixty, the first division by 60 being the minute.[3] Between AD 1000 (when al-Biruni used seconds) and 1960 the second was defined as 1/86,400 of a mean solar day (that definition still applies in some astronomical and legal contexts).[4][5] Between 1960 and 1967, it was defined in terms of the period of the Earth's orbit around the Sun in 1900,[6] but it is now defined more precisely in atomic terms. Seconds may be measured using mechanical, electric or atomic clocks.

Astronomical observations of the 19th and 20th centuries revealed that the mean solar day is slowly but measurably lengthening and the length of a tropical year is not entirely predictable either; thus the sunearth motion is no longer considered a suitable basis for definition. 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.[1]

In 1997, the CIPM affirmed that the preceding definition "refers to a caesium atom at rest at a temperature of 0 K."[1]

SI prefixes are frequently combined with the word second to denote subdivisions of the second, e.g., the millisecond (one thousandth of a second), the microsecond (one millionth of a second), and the nanosecond (one billionth of a second). Though SI prefixes may also be used to form multiples of the second such as kilosecond (one thousand seconds), such units are rarely used in practice. The more common larger non-SI units of time are not formed by powers of ten; instead, the second is multiplied by 60 to form a minute, which is multiplied by 60 to form an hour, which is multiplied by 24 to form a day.

The second is also the base unit of time in the centimetre-gram-second, metre-kilogram-second, metre-tonne-second, and foot-pound-second systems of units.

Unit of time (second) Abbreviations: CGPM, CIPM, BIPM

The unit of time, the second, was defined originally as the fraction 1/86 400 of the mean solar day. The exact definition of "mean solar day" was left to astronomical theories. However, measurement showed that irregularities in the rotation of the Earth could not be taken into account by the theory and have the effect that this definition does not allow the required accuracy to be achieved. In order to define the unit of time more precisely, the 11th CGPM (1960) adopted a definition given by the International Astronomical Union which was based on the tropical year. Experimental work had, however, already shown that an atomic standard of time-interval, based on a transition between two energy levels of an atom or a molecule, could be realized and reproduced much more precisely. Considering that a very precise definition of the unit of time is indispensable for the International System, the 13th CGPM (1967) decided to replace the definition of the second by the following (affirmed by the CIPM in 1997 that this definition refers to a cesium atom in its ground state at a temperature of 0 K):

The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom.

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 so defining time in terms of such quantities would result in circularity of definition.[24] An operational definition of time, wherein one says that observing a certain number of repetitions of one or another standard cyclical event (such as the passage of a free-swinging pendulum) constitutes one standard unit such as the second, is highly useful in the conduct of both advanced experiments and everyday affairs of life. The operational definition leaves aside the question whether there is something called time, apart from the counting activity just mentioned, that flows and that can be measured. Investigations of a single continuum called spacetime bring questions about space into questions about time, questions that have their roots in the works of early students of natural philosophy.

Accuracy, Precision of Instruments and Errors in MeasurementError: The actual difference between a measurement value and the known standard value.Error in the measurement of a physical quantity is its deviation from actual value. If an experimenter knew the error, he or she would correct it and it would no longer be an error. In other words, the real errors in experimental data are those factors that are always vague 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. Perhaps it is better to speak of experimental uncertainty instead of experimental error because the magnitude of an error is uncertain.

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, related to reproducibility and repeatability, is the degree to which repeated measurements under unchanged conditions show the same results

Systematic ErrorsAt this point, we may mention some of the types of errors that cause uncertainty is an experimental in measurement. First, there can always be those gross blunders in apparatus or instrument construction which may invalidate the data. Second, there may be certain fixed errors which will cause repeated readings to be in error by roughly some amount but for some unknown reasons. These are sometimes called systematic errors. Third, there are the random errors, which may be caused by personal fluctuation, random electronic fluctuation in apparatus or instruments, various influences of friction, etc.

These are inherent errors of apparatus or method. 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 Errors

None of the apparatus can be constructed to satisfy all specifications completely. This is the reason of giving guarantee within a limit. Therefore, a manufacturers always mention the minimum possible errors in the construction of the instruments.

Imperfections in Experimental Technique or Procedure

Following 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 Skillness of the Observer : Error in the reading is largely dependent upon the skillness of the observer by which reading is noted accurately.

Personal Errors

It is due to the indefiniteness in final adjustment of measuring apparatus. For example, Maxwell Bridge method of measuring inductances, it is difficult to find the differences in sound of head phones for small change in resistance at the time of final adjustment. The error varies from person to person.

Random Errors

After corrections have been applied for all the parameters whose influences are known, there is left a residue of deviation. These are random error and their magnitudes are not constant. Persons performing the experiment have no control over the origin of these errors. These errors are due to so many reasons such as noise and fatigue in the working persons. These errors may be either positive or negative. To these errors the law of probability may be applied. Generally, these errors may be minimized by taking 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 has a least count of 0.001g, although this may vary; a vernier caliper has a least count of 0.02 millimeters, although this too may vary; and micrometer screwgauge's least count is 0.01 millimeter and of course a conventional ruler has .01m.

The Least Count is the discrimination of a vernier instrument. All measuring instruments used in the subject of physics can be used to measure various types of objects, but all do so without considering the detail of accuracy.

No measuring instrument used in physics is accurate and always has an error when readings are taken. Even the latest technology used in measuring objects also have an error where reading are concerned. Various names can be given to this error. The Least Count, uncertainty or maximum possible error are the terms normally used in a physics course, although this may vary with different syllabuses.

The error made in an instrument can be compared with another by calculating the percentage uncertainty of each of the readings obtained. The one with the least uncertainty is always taken to measure objects, as all measurements are required with accuracy in mind. The percentage uncertainty is calculated with the following formula: (Maximum Possible error/Measurement of the Object in question) *100

The smaller the measurement, the larger the percentage uncertainty. The least count of an instrument is indirectly proportional to the accuracy of the instrument.

Absolute Error, Relative Error and Percentage Error

Absolute 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:a1= am a1a2= am a2am anMean Absolute Error = amean= 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:a =mean absolute value/mean value = amean/amPercentage Error: It is the relative error measured in percentage.

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

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:

0.001 oC 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. This approach is a reasonable convention to follow.What is an "exact number"?

Some numbers are exact because they are 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 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) = 37.8000

which 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

Instructions: print a copy of this page and work the problems. When you are ready to check your answers, go to the next page.

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 = ?

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 102

This 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?

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-004

Microsoft Excel gives 0.000637965516309464000000000000

A Hewlett-Packard 50G calculator gives 6.37965516309e-04

These 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

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 [M0LT1]

Dimensional formula of acceleration [M0LT2]

Dimensions of Physical Quantities

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.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 lengthlength and time are fundamentally distinct and inequivalent physical dimensions.

When we talk about a quantitys physical dimension, using symbols such as L for length, or T for time, we are not using those symbols to represent an actual variablewe 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 quantityand to do that, we need to define an appropriate unit for the measurement. We can make different 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].http://cbse-notes.blogspot.in/2012/05/cbse-class-xi-physics-ch2-dimension.htmlDimensional 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 Fma ; 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, poissons ratio, specific gravity, refractive index, Relative permittivity, Relative permeability. All these quantities nighter 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.

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

Dimensional Analysis and its Applications

Dimensional Analysis

Only 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 consistency

Consider 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

According to this principle, 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 =

(2 is a constant)

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

As stated above, in physics we derive the physical quantities of interest from the set of fundamental quantities of length, mass, and time. When a quantity is broken down to its makeup in terms of the fundamental quantities we call this breakdown its dimension. The dimension, therefore, represents the fundamental type of the quantity. When indicating the dimension of a quantity only, we use capital letters and enclose them in brackets. Thus, the dimension of length is represented by [L], mass by [M], and time [T].

We use a lot of equations in physics; these equations must be dimensionally consistent. It is extremely useful to perform a dimensional analysis on any equation about which you are unsure. If the dimensional consistency is not there, it cannot be a correct equation. The rules are simple:

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.

Notice 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 are different. However, it is not valid to write x inches = t seconds because they have different dimensions, [L] [T].Deducing Relationships among Physical Quantities

Based on the principle of homogeneity of dimensions

Example:

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

[M1L1T2] = Ma [LT1]b Lc

= MaLbTbLc

M1L1T2 = MaLb + cTb

On applying the principle of homogeneity of dimensions, we get

a = 1,

b= 2,

b + c = 1 (ii)

From (ii), c = 1 b = 1 2 = 1

On putting these values in (i), we get

F = km1v2r1

OR

This is the required relation for centripetal force._1455566247.unknown

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