Physics - Chapter One

Chapter One 

Physical Quantities and Their Measurements

Figure: An Atomic clock made for measuring time very accurately 

When you think about science the image of different instruments, inventions, research laboratories etc appear in your mind. But the instrument, research or laboratory are not the subject matter of science rather the actual subject matter of science is its scientific view . The greatest contribution of this civilization has come from science and the scientific view of the human beings of the world. To explore the mysteries of science sometimes they explained logically, sometimes examined in laboratories and some other times they are scrutinized intensively in nature. From ancient times to modem times the scientists are relentlessly advancing the science. A continuous description of the gradual development of physics has been described in this chapter.

If we read the history of physics we will see that it is developed by the combined effort of theoretical and experimental scientists. In order to do research in the laboratory, we need to measure different quantities accurately. How the units are built up, how they can be measured and what type of instruments will be used to measure them will also be discussed in this chapter.

     By the end of this chapter we will be able to- 

• Explain the scope and gradual development of physics.

• Describe the objectives of physics.

• Explain the physical quantities [ with units and magnitude] as the origin of physics.

• Explain the measurement and necessity of units.

• Explain the difference between fundamental and derived quantities.

• Explain the international system of units.

• Calculate the dimensions of quantities.

• Calculate the transformation of prefixes of multiple and sub-multiple units.

• Express the concepts of physics and its theories by using scientific names, symbols and notations.

• Measure the physical quantities by using different apparatuses.

• Explain the mechanism of exactness and accuracy of measurement.

• Determine the area and volume of uniform objects by using simple instruments.

• Determine the length, mass, area and volume of uniform objects used in our daily lives.

Physical Quantities and Their Measurements                                                                                            3 

1.1 Physics:  

Physics is the most ancient branch of science. It is because scientists started studying and practicing astronomy, the most important branch of physics, before the flourishment of other branches of science. Besides being the oldest branch, physics can be said to be the most fundamental branch. Chemistry flourished on the basis of physics .Biology flourished on the basis of Chemistry, and again many other subjects flourished on the basis of Biology.

Generally we can say that the branch of science which tries to understand matter and energy and the interaction between them is called physics. You must have realized that here physics is not only the visible things around us but also the things such as molecules-atoms, electrons, protons, neutrons, quarks or string etc. Again energy may be strong and weak nuclear energy apart from our known potential energy, kinetic energy, gravitational energy or electromagnetic energy.

1.2 Scope of physics  

Since physics is the oldest and the most fundamental branch and other branches somehow flourished on the basis of physics, it is very natural that the area of physics is very wide. Not only that different technologies have flourished using different laws of physics and we are using this in our daily lives (there are examples of some instruments used in medical science in the last chapter). At present the greatest contribution behind civilization is of electronics and physics has the greatest contribution behind this technology also. Besides daily activities, from destruction of war to space exploration the contribution of physics is present. Not only that by the combination of other branches of science and physics newer branches have developed, for example astrophysics consists of astronomy and physics. In order to explain organic processes, biophysics has been built up by the combination of biology and physics, chemical physics was born by the combination of chemistry and physics. Use of physics in Geology has led to Geophysics. Medical physics has flourished using physics. Therefore the area of physics is very large and deep as well.

For the advantage of teaching-learning we divide physics into two principal parts. These are:  

Classical Physics: This contains mechanics, sound, heat and thermodynamics, electricity and magnetism and optics. 

Modern Physics: Modern Physics that built up using Quantum Mechanics and Theory of Relativity. These are molecular and atomic physics, nuclear physics, solid state physics and particle physics. 

We have mentioned earlier that many kinds of technologies have been developed in the world using physics or other branches of science. Using these technologies we have made our lives simple and meaningful. The invention of some terrible technology has not only endangered our lives but also the existence of our earth. Sometimes unjustified and unnecessary technology has spoiled the resources of our earth along with creating pollution. So remember that technology is not always good, as there are good technologis in the world, also there are bad technologis. Using your good sense/rationality, you have to understand which technology is good or bad.

Physics was not built in a single day, it has taken hundreds of years to develop. The physicists tried to explain, with the help of laws the mysterious world around them. Sometimes performing experiments those laws were accepted, changed and sometimes give up. Thus we are able to explain microscopic particle of matter to shape of largest universe and we are learning continuously. Perhaps this learning is not complete still now- scientists are trying to make it complete, it will be possible to explain everything of apparently different subject with the help of a few laws one day.

1.3 Development of physics

Modem civilization is the contribution of science. This advancement of science has not been done in a day, the modem science reached its present state gradually due to tireless labour of innumerable scientists and researchers. It has to be remembered that the exchange of information was not so easy, it was very difficult to convey the results of research to one another, books were prepared in hand written form and the number of these books were very few. Courage was necessary to express views against conventional faiths. There are examples of imprisonment or even scientists being burnt to death. But the search of knowledge was not stopped and exploring the mysteries of nature, the scientists have given us this modern science as a gift.

We can describe the history of physics by dividing it into different stages. 

1.3.1 Initial Stage (Greek, Indian subcontinent, China and the Contribution of Muslim Civilization)  

Now-a- days, what we understand by physics was started in the ancient times, by the combination of astronomy, optics, dynamics geometry, and an important branch of mathematics. The name of the Greek scientist Thales ( 624-586 BC) can be mentioned specially, because he was the first who deny explanation whice is onle based on religion, myth and extrasensory perception but without logic. Thales predicted about the solar eclipse and knew the magnetic property of load.stone. Pythagoras ( 527 BC) was remarkable among the mathematicians and scientists at that time. He had fundamental works on vibrating wire along with geometry. Greek philosopher Democritus ( 460 BC) presented the first idea that matter has indivisible unit, called atom (This name is used by modem physics). It was not acceptable to all since there was no chance to prove his idea by the scientific process. The theory that everything was made of soil, water, air and fire by the greatest philosopher and scientist Aristotle was much more oO � acceptable. Aristarchus (310 BC) first gave the idea of solar centered solar system. His follower Seleucus proved that with arguments, although these arguments have been lost with time. Greek science and mathematics had reached its highest peak at the time of Archimedes (287 BC), the greatest scientist of all times. The upward thrust of liquids is still in the content of science books. He assisted, during the war, by setting enemy ships on fire by converging the solar light using a spherical mirror. There was an another scientist of Greek era named Eratosthenes (276 BC), he found out the radius of the earth accurately at that time.

Figure 1.01: Archimedes and Al Khorgimi

After that the development of science remained stopped for almost one thousand and five hundred years. Only the civilizations of the Indians, Muslims and Chinese survived. In the Indian subcontinent Aryabhatta ( 476), Brahmagupta and Vaskar have done a lot of valuable work on mathematics and astronomy. Zero was effectively used in the Indian subcontinent (Aryabhatta). The name of Al Khorgimi (783 BC) has to be specially mentioned among the muslim mathematicians and scientists. The present name algebra has come from the book Al Zaber written by him. lbne Al Hayum (965 BC) is considered as the architect of optics. Al Masudi wrote an encyclopedia of thirty parts regarding the history of nature. Everyone knows the name of Omar K.hayam as a poet. But he was a highly famous mathematician, astronomer and philosopher. Chinese mathematicians and scientists have done a lot of work in physics. Among them the name of Shen Kuor can be mentioned (1031), he has done studies with magnets and mentioned the issue of fixing direction by using a compass during travel.

1.3.2 Developing Stage of Science

In the sixteenth and seventeenth century, a surprising revolution started in the realm of physics, the time was the era of European renaissance. In 1543 Copernicus gave the explanation of the solar centric solar system in a book. (the publisher of the book became afraid of the priest and wrote that it is not a real explanation, only a mathematical solution!) The theory was not made available to the people for a long time, Galileo (1564- 1642) brought it to the fore front. He introduced the mathematical formulation. After that he introduced the scientific process to prove the law doing experiments. Sometimes Galileo is called the father of modem science. He was in curse of church due to be the founder of solar centric solar system. He had to pass his last days in captivity. In 1687 Newton published the three laws of mechanics and the law of gravitational force, which forms the basis of force and dynamics. Scientist Newton along with Leibniz invented a new branch of mathematics called calculus with optics and other works.

 Figure 1.02: Copernicus, Galileo and Newton

Before the eighteenth century heat was considered a mass less liquid. In 1768 Count Rumford showed that heat is a kind of energy and mechanical energy can be converted into heat energy. Lord Kelvin in 1850 introduced two important laws of thermodynamics on the basis of the research work of many other scientists.

An extensive research was started on electricity and magnets at this time. In 1778 Coulomb invented the law of force between the charges. Many kinds of 00 research were started after the invention of the electric battery by Volta in 1800. � In 1820 Oersted showed that magnets can be made by the flow of electricity. In 1831 Faraday invented the exact opposite process give by Oersted. They showed that electricity can be produced by varying a magnetic field. In 1864 Maxwell (Figure 1.03) expressed the varying electric and magnetic fields with a single law, the famous Maxwell equation. Here, he also showed that actually light is an electromagnetic wave. Electricity and magnets are not different. In fact they are the two forms of the same energy. This invention was well-timed because in 1801 Young proved the wave nature of light experimentally.  

1.3.3. Introduction to Modern Physics

From the very beginning of 19th century scientists discovered that many things could not be explained with the help of existing physics. Dalton introduced atomic theory in 1803, Thomson invented an electron within the atom, In 1911 Rutherford showed that at the center of the atom there is the smallest nucleus and there are positive charges in it. But it was found that the revolving electron model around the nucleus could not be explained because according to electromagnetic theory an electron should radiate its energy and fall into the nucleus, but practically it never happens. In 1900 Max Plank invented the quantum theory and the explanation of the stability of an atom became possible. Professor Satyendranath Basu gave an appropriate mathematical explanation of the Quantum - theory of radiation. In recognition of his contributions to physics one kind of elementary particle was named the Boson. During the period of 1900-1930 many famous scientists together established the quantum theory.

Figure 1.03: Maxwell, Rutherford and Mari Curie  

An imaginary medium called ether was considered as the carrier of the electromagnetic wave. In 1887 Michelson and Morley, while trying to prove the existence of ether, showed that actually there is no ether and the velocity of ao light is equal in both stationary and moving mediums. We get this explanation from Einstein's theory of relativity in 1905. (Fig-1.04) The most wonderful equation E = mc2 is derived from the theory of relativity in which it is shown that the mass of object can be converted into energy.

In 1931 Dirac predicted the existence of an anti particle by the combination of quantum theory and the theory of relativity and it was discovered the following year.

Figure 1.04: Albert Einstein and Satyendranath Basu  

 

In 1895 Roentgen invented X-rays. In 1896 Becqurel showed that radioactive radiation is emitted from the centre of an atom. In 1899 Pierre Curie and Marie Curie (Fig:1.03) invented radium and the scientists understood that actually the atoms are not imperishable, on disintegrating these atoms may emit radioactive radiation .

1.3.4 Contemporary Physics  

Due to the invention of electronics and modem technology it is possible to make powerful particle accelerators. Accelerating the particle with high energy, new particles are being discovered. It is possible to arrange these particles systematically using the Theoretical Standard Model. It is possible to explain the structure of all the particles using a few elementary particles (and their antiparticles) though apparently we think there are an infinite numbers of new particles. It is not possible to explain the mass of the particles using the Standard Model, so to explain the mass, the existence of a new particle called the Higgs Boson was predicted. The identification of the Higgs Boson particle in the laboratory in 2013 is treated as a great success of theoretical physics.

In 1924 Hubble showed that all galaxies of the universe are moving away from one another, which indicates that the universe is expanding slowly. This means that once upon a time the whole universe was concentrated at a point. Scientists showed that fourteen billion years ago through a massive explosion called the Big Bang the universe was created and it has continued to expand. Very recently scientists showed that this expansion will never stop and every part of the universe will keep moving away from one another. Besides, the physicist have shown that they can explain only 4% of visible planets, stars, and galaxies of the universe, the concept of the mysterious dark matter and dark energy has to be accepted to explain the rest of the universe. The scientists are continuing their research in this field.

Semiconductors are created from the research of solid state physics resulting in present electronics which is the foundation of modem civilization.

1.4 Objective of Physics  

Already you know that physics is that branch of science which explains the change of position of an object with time in the presence of energy and force. Like any other science the main objective of physics is to learn, the arena of knowing in physics is very large. The objectives of physics is to unfold the mysteries of both small atoms and the vast universe. For understanding easily we can divide the objective of physics into three parts: 

1.4.1 Unfold the Mystery of Nature 

In ancient times in China, a piece of lodestone was seen to attract another by an invisible force. This special property of this special type of matter was called magnetism. Similarly in ancient Greece when a substance called amber was rubbed with wool, they attracted each other with an invisible force. This special type of property is called electricity. Many researches were conducted on it in the eighteenth century and the scientists discovered that, this is actually two different forms of the same force and this force is called an electromagnetic force. After the invention of radioactivity while explaining a special radiation called beta ray, a new type of force called weak nuclear force was discovered. Later on the physicists showed that electrometric force and weak nuclear force are different forms of the same force. Their combination is called the electro weak force. The physicists think that two other forces in nature called gravitational force and nuclear force will be brought under the same law in the future.

Physics is unveiling the mystery of nature one after another. Similarly we can say that an object consists of molecules, later on we see that the molecules consist of atoms of elements. Despite the atoms being charge neutral, there is a positive charged nucleus at the center and surrounding it electrons are revolving. Though an electron is an elementary particle, it is seen that the nucleus consists of protons and neutrons. It is there seen that neutrons and protons also consist of elementary particles called quarks. At present the research topic is whether electrons and quarks are made of strings.

1.4.2 To Know the Laws of Nature  

From time immemorial we know that if something is released from above, it falls down and seeing this we can guess that the earth attracts everything towards its center. If physics stops after pronouncing only the existence of gravitation, this is not enough at all. The knowledge will not be appropriate until accurately we know the force by which an object of a definite mass attracts another object of another definite mass and how the force varies with the distance between them. Newton explained this law of nature properly with the help of the law of gravitation. The laws of nature can be used in many other places. So, the motion of a falling body is explained with the help of gravitation, similarly the rotation of the earth about the sun can also be explained. To know the laws of nature properly the scientists analyzed them with argumentative discussions, while also conducting experiments in the laboratory. Behind the wonderful success of physics, both theoretical and experimental researches have been conducted. The main objectives of physics is to find out the laws of nature by doing research in these two different fields.

1.4.3 Development of Technology Using the Laws of Nature 

From the theory of relativity Einstein deduced the law E = mc2 and showed that mass can be converted into energy. In 1938, by breaking a nucleus Otto Hann and Fritz Strassmann showed that the amount of mass reduced, is converted into 00 energy.

Using this formula the nuclear bomb was made and dropped on Hiroshima and Nagasaki in the Second World War and millions of people were killed within a second. Not only deadly weapons can be made but also it is possible to use for the good of mankind. Using this formula, nuclear power plants are made and also in Rooppur of our country a nuclear power is being made.

Solid state physics is a branch of physics where semiconductors are studied. Mixing some special elements with semiconductors, transistors are made. With the help of this technology great development of electronics has occurred and this has made a major contribution to the present civilization. 

In this way we can show that physics has a small or large contribution in every field of technology. Contributions of physics in medical science are discussed in the last chapter of this book. 

 1.5 Physical Quantities and Their Measurements

Every one of us knows that water becomes ice when it is cooled; it becomes vapor when it is heated. Peoples have been observing it from the very ancient times. This knowledge cannot be science completely, unless we can say at what condition and at what temperature water becomes ice after freezing or raising the temperature at what condition and at what temperature does it start to boil and become vapors. This means to be science, everything has to be measured. The most important point of science is to be able to explain everything accurately after measuring them.

Table 1.01: Seven physical quantities in SI unit

In this universe anything we can measure is called a quantity. In this physical world there are innumerable quantities which can be measured. For example, we can say, length, breadth, volume, weight, temperature, colour, rigidity, position, velocity, internal material, electrical conductivity, non-conductivity, elasticity, heat conductivity, non-conductivity, density, specific heat, pressure, melting point, etc, we cannot finish this list. In brief, in the physical world the quantities are endless. You may think that to measure these infinite number of quantities, we have to prepare the definitions and units of these infinite number of quantities! Actually that is not true, you will be surprised hearing that (and definitely be happy) ifwe can fix seven units of seven quantities, then we will be able to find everything using these seven units. These seven quantities are called � fundamental quantities. When we express other an quantity using these fundamental quantities that is called a derived quantity. The fundamental quantities are length, mass, time, electric current, temperature, amount of substance and luminous intensity. Internationally recognized these seven units of seven fundamental quantities are called SI units, (SI came from the France language term 'Systeme International d'Units) and these are shown in Table 1.01.

Table 1.02: Very large to very small distance

Table 1.03: Very large to very small mass

1.5.1 Units of Measurement 
The measurement of these units is stated very clearly. For example: the distance travelled by light in vacuum in 1/299792458 second is called lm. 1 kilogram unit is still considered as the definite amount of mass, which is the mass of a cylinder made of platinum-iridium of height and diameter 3.9 cm kept in a certain building in France. (The scientists will explain this mass otherwise within few days so that nobody has to depend on a definite mass of a definite country.) The time required to complete 9,192,631,770 vibrations by a Cesium-133 (Cs133) atom is called one second. The temperature which we get when the triple point temperature of water is divided by 273.16 is called one Kelvin. More or less the definition of ampere is complex- When electric current ±lows through two wires in the same direction they attract one another. The amount of current that flows through two parallel wires separated by 1 m and if they attract one another with 2x 10-1 N force per meter, that amount of current is called one ampere. Here it is considered that the wires are of infinite length, circular cross section, and the cross section is so small that it is negligible! (You will be happy to know that plans are being made to explain this more clearly!) One mole is defined as the amount of substance that is contained in a definite number of fundamental particles (molecule, atom or ion) which is equal to the number of atoms contained in 0.012 kilogram of carbon-12 atom.

Table 1.04: Very large to very small time  

One candela unit is perhaps the most complex to understand: one candela is defined as the amount of luminous intensity of a source of light which is emitted in one steredian solid angle of luminous intensity of 1/683 part of 1 watt. But any source of light cannot be used for this, that source must be a source of light with vibrations of 540x 1012 cycle per second. The quantities distance, mass and time can be very small to very large. To give an idea about these quantities, some examples are given for distance, mass and time (table 1.02, 1.03, 1.04). Observe the tables minutely and try to feel them!

The seven units have been introduced to you formally; nobody can hope to remember them! You need not memorize them if sometimes you want to know, you can find it look up in a book or browsing the internet. You should have a practical idea about what is meant by one meter distance, what amount of mass is one kg, what time is one second, how hot is one Kelvin temperature, what 00 amount of electric current is one ampere, what is meant by one mole substance � or what amount oflight one candela is! Now let us try to give you that realistic idea. Not only do you have to know it, but also you have to feel it. In general it can be said that:

• The distance from the feet to the stomach of a person of normal height is nearly one meter.

• The water contained in one liter bottle or water of four glasses have a mass of nearly one kg.

• The time required to say three words 'one thousand one' is approximately one second.

• If three mobile phones are charging at the same time, one ampere electric current is used. (a mobile is charged at nearly 5 V. So, current consumption will be 5watt. If lights, fans, refrigerators of a residence run at 220 V and one ampere current is used then power consumption will be 220 watt!)

• If we can feel the fever of anyone by touching with hand, then it can be said that his temperature has increased by one Kelvin.

• It is difficult to realize a mole, we can say a water filled large spoon contains one mole of water molecules. In one cup water, there are ten moles.

• Light from a single candle can be said one candela.

You see that none of them are perfect measurements, but easy to realize. If you are habituated with this measurement, when in future you will calculate anything, then you will have a sense of proportion about it. 

1.5.2 Prefix 

To study science or physics we have to measure different things. Sometimes we need to measure the length of a galaxy (6xl024 m), or sometimes we have to measure the radius of the nucleus (lxI0-15 m). To measure this huge difference in the distance it is not wise to use the same type of numbers, so internationally some SI prefixes have been made. Due to this multiplier we will be able to express a large or small number by a small prefix. These prefixes are shown in Table 1.05. In our daily lives we always use these prefixes. To express distance we say 1 kilometer instead of 1000 meter. We say 1 megabyte instead of ten lac bytes to express the size of photographs of a camera.

 Table 1.05: The multiples and sub-multiples used in the SI unit

1.5.3 Dimension 

We already know that though there are an infinite number of quantities around us, we can measure them with the help of seven units only. We have to know, in which units a quantity can be expressed. Often we need to know also how this quantity is formed with which fundamental quantities (length L, time T, mass M etc.). The power of different fundamental quantities in a quantity is called its dimension. For example, we will see next that force is the product of mass and acceleration. Again, acceleration is the rate of change of velocity with time. And velocity is the rate of change of position with time.

Therefore,  

  

In this book, when we will speak about a new quantity, we will try to tell the dimension of it at the same time. You will see that, this always will help you to understand the quantity, in a different way. To indicate the dimensions of a quantity in this book, it will be shown within third bracket. For example, force F is, [F]= MLT^2.

1.5.4 Scientific symbols and notations 

The following methods are followed to write the symbol of units:

1. To express the value of a quantity, first we write a number and keep a space after it and then write the symbol of the unit. For example, 2.21 kg, 7.3 x 102 m2 or 22 K. The percentage(%) sign also follows the same rule. However no space is kept after a number to write degrees ( 0 ) minutes (') and seconds (' ').

2. Derived unit produced by multiplication is written using a space between two units, e.g. 2.35 N m

3. Derived unit produced by division is expressed as negative power or '/'. (e.g. ms-1 or m/s)

4. No punctuation mark or full stop is used with the symbols, as they are mathematical expressions but not the abbreviated form of anything.

5. The symbol of unit is written in straight font, for example, m for metre, s for second etc. But the symbol of quantities are written in italic or curved font, for example, m for mass, v for velocity etc.

6. The symbols of units are written in small letters, for example, cm, s, mol etc. But capital letters are used for those which are taken from the name of scientists (N for Newton). If there are many letters in the unit then the first letter will be capital only (the unit from the name of Pascal is Pa).

7. The prefix (k, G, M) of unit will be attached with the unit (m, W, Hz) with no space. For example, km, GW, MHz. 8. Prefixes more than kilo (103 ) will be in capital letters (M, G, T).

9. The symbols of unit will never be plural (e.g. not 25 kgs, always 25 kg).

10. We have to try to write any number or compound unit in a single line. A line break can be given between a number and a unit if it is very necessary.

1.6 Measuring Instruments

Once it was very difficult to measure different quantities of physics very accurately. The work has become much easier due to instruments based on electronics. For the physics that we will try to learn in this book, knowing how to measure distance, mass, time, temperature, electric current and voltage will be enough. Let us discuss what type of instruments we will use to measure these:

1.6.1 Scale 

A Meter scale is used to measure small lengths and definitely you may have seen it. Since this is 100 cm or lm long, it is called a meter scale. Since in many places still now inch-foot is familiar (USA is an example!), so inch is marked very often on the other side of a meter scale. One inch equals 2.54 cm.

We can measure up to the smallest division on a scale. Meter scale is generally divided up to millimeter, so using a meter scale we can measure the length of anything up to millimeter. Therefore, if we say the length of anything is 0.364 m, this means the length of this is 36 centimeter and 4 millimeter. Using a meter scale it is not possible to measure lengths smaller than this - that is, generally we can never say the length of a body is 0.3643 m. But from time to time, for microscopic purposes we have to measure the smallest length of objects of this type, this can be performed by using an interesting scale called a vernier scale. 

Vernier scale  

Let us consider that the length of an object comes in between 4 and 5 millimeter marking i.e. the length of the object is greater than 4 mm and less than 5 mm. We can use the veriner scale to find how much fraction is greater than 4 mm. This scale is attached to the main scale and can be moved forwards and backwards (Figure 1.05). In the example shown in the figure the length of 9 mm of main scale is divided into 10 divisions on the vernier scale. Therefore every division of the vernier scale equals : 9/10 mm, i.e. less than a millimeter by 1/10 millimeter. If the initial mark of the vernier scale coincides with any mark of the millimeter scale, then the next mark of the vernier scale will keep a separation of 1/10  mm from the actual millimeter mark, and the next will keep a separation of 2/10 mm, third one will keep a separation of 3/10 mm and so on.

Therefore no mark of the vernier scale will coincide with the millimeter mark of the main scale, finally the 10th mark again will coincide with the ninth millimeter mark of the main scale

Figure 1.05: Main scale and movable vernier scale 

If we keep the vernier scale in such a way that its starting is not from a millimeter mark rather it starts with a slight (e.g.t0 mm) displacement (Figure 1.06), then the number of 1/10 mm displacements by which it has moved will be the number mark 10 of the vernier scale coinciding with the millimeter mark of the main scale! Therefore, it is very easy to measure a length using a vernier scale. First of all, we have to know the difference between one division of vernier scale and one division of the main scale- this is called the vernier constant-- in brief VC. This can be calculated if we divide the length of the smallest division (1 mm) of the main scale by the total number of divisions of the vernier scale (in the figure 1.05 and 1.06 it is 10). In our example, the value of this: 

VC = 1mm/10 = 0.1 mm= 0.0001 m

To measure a length, we have to look at the vernier scale after measuring up to the last millimeter mark. Which mark of the vernier scale coincides exactly with any millimeter mark of the main scale is then found, and then that number is multiplied by the vernier constant. We will get the actual length by adding this amount with the length measured by the main scale. According to our procedure the length that is shown on the last scale of Figure 1.06 is 1.03 cm or co......� 0.013m. 

Figure 1.06: Vernier scale displaced by one, two, three divisions 

Instead of the vernier scale we can use a special type of scale called a screw gauge to measure lengths. In a screw gauge when the screw (Figure 1.07) rotates the scale moves forwards or backwards, the threads of the screw are kept very fme. After a complete rotation of the screw, the screw with the attached scale perhaps advances by an amount of I mm. This displacement of the screw is called the pitch of the screw. The circular part of the instrument, by the rotation of which the screw moves forwards or backwards is divided into 100 equal parts. For the rotation of only one division of the circular scale, the screw advances by an amount of 1/100 of pitch.

Therefore, with this scale up to 1/100 mm = 0.01 mm length measurement is possible. This is called the least count of the screw gauge 

Figure 1.07: A slide calipers with vernier scale and a screw gauge are shown 

Now-a-days instead of the vernier scale, attached dials or digital slide calipers are available by which lengths can be measured accurately.

1.6.2 Balance 

Mass cannot be measured directly, so measuring the weight the mass is generally determined. When we say the weight of an object is 1 gm or 1 kg, then actually we mean that the mass of the object is 1 gm or 1 kg. In earlier times to measure the mass of an object, a balance was used, there the mass of the object was compared with the definite mass of a known weight. Now-a-days the use of electronic balance (figure 1.08) has increased a lot. If we keep the body on the balance then the sensors of the balance can determine the weight very precisely.

Figure 1.08: Digital weigh machine

1.6.3 Stop Watch

Stop watches are used to measure time interval (figure 1.09). Once accurate stop watches were very precious things, now-a-days very accurate stop watches are available in the mobile phone at low prices due to the advancement of electronics. In a stop watch, time measurement is started at any instant of time, and by stopping the measurement of time after a definite interval, the elapsed time can be determined. An interesting matter is that the stop watch can measure the time very accurately, but we can never start or stop it with our hands with the same accuracy.

Figure 1.09: Stop watch

Investigation 1.01

1. 7 Error and Accuracy 

Error is a negative word and if the word "error in measurement" is said we think that the person who is doing the measurement is not performing his duties properly and an error has occurred! But this is not the case. Sometimes error may occur due to the person who is doing the measurement for his negligence, but we have to know that the instrument with which we measure is not error free. Therefore, there is a limit of how much accurately we can measure, i.e. it is very natural to have errors in measurement. But how much accurate the measurement is, also needs a measurement. The reliability of a result will be increased if we can inform how much accurate the result is when we publish the result of an experiment after doing it. You can give a measurement of the accuracy of your experiment if you know some conventional rules for finding the accuracy of an experiment.

Suppose, you are measuring the length of an object by a scale. How much accurately you will be able to measure the length of the object depends on how fine your scale is graduated. If the scale is graduated in 1 cm intervals then you will express your result for length in some definite number of cm. But the real length of the object is not exactly equal to this mentioned number of cm but perhaps very close to it, so there is a possibility of uncertainty in your measured length. That is why, we add that amount uncertainty with the measured result. Therefore, if we see the length is near to 4 cm, then we will say the length of the object is: 

4.0 ± 0.5 cm

Therefore the length of the object may have any value within 3.5 cm to 4.5 cm.  

Now we can discuss how accuracy can be measured. The measurement of accuracy is the absolute error. Observing the name we understand that this is the difference of the measured value with respect to the actual value. You can definitely understand that when we measure anything we don't know the actual value. So, instead of absolute error we use the most probable error. Therefore in our previous example the absolute error is

 

1±0.5 mml = 0.5 mm

We can see consider the relative error after the absolute error. Let us consider that to measure any length an error of 0.5 mm has occurred. If the length of the object is 1 mm then this error is very serious, but if the length is 1 m then the measurement is accurate enough. The concept of relative error has been introduced to give a better understanding.

Therefore, 

Relative error = Absolute error /  Measured value

So in our previous example:

Relative error: 0.5 mm/ 7 mm = 0.071

        In percentage this is: 0.071 x 100 = 7 .1 % 

Question: Suppose you have got 10 cm by measuring the length of a square shaped book. Suppose the relative error in the measurement is 10%. What is the relative error in its area?

Answer: 

Measured area of the object =10 cmxl0 cm=l00 cm2 Since the relative error of the object is 10%, hence if its length is measured, the minimum length will be 9 cm and maximum length will be 11 cm.

:. The minimum area = 9 cm x 9 cm = 81 cm^2 

and the maximum area= 11 cm x 11 cm= 121 cm^2

Therefore the absolute error: 

| 100 cm^2 - 81 cm2 | = 19 cm^2m^2

or, | 121 cm^2 - 100 cm^2 | = 21 c

Since the values are not equal, we consider the larger one i.e. the absolute error is 21 cm^2

Therefore, the relative error = 21 cm2 /100 cm2 = 0.21

In percentage, 0.21 x 100 = 21 % 

If the error in the measurement of the length is 10%, then in the case of area it will be approximately doubled. Similarly you can show that in the case of volume measurements the error will be three times!

 Question: You have measured a box by a ruler which is graduated only in cm. You have got the length, breadth and height of the box as 10 cm, 5 cm and 4 cm respectively. What is the percentage of error in your measurements?

Answer: Since your ruler is graduated only in cm, so your error is ± 0.5 cm. Therefore the error in your measurements: 

Length: 10 ± 0.5 cm

Breadth: 5 ± 0.5 cm

Height: 4 ± 0.5 cm

Your measured volume: 10 cm x 5 cm x 4 cm= 200 cm^3

Probable smallest volume:

        (10 - 0.5) cm x (5 - 0.5) cm x ( 4 - 0.5) cm= 149.625 cm^3

Probable largest volume:

        (10 + 0.5) cm x (5 + 0.5) cm x ( 4 + 0.5) cm = 259.875 cm^3

Therefore the volume, 

        149.625 cm3 < V < 259.875 cm^3

Absolute error:

From 149.625 cm^3 to 200 cm^3 is 200 cm3 - 149.625 cm^3 = 50.375 cm^3

From 200 cm^3 to 259.875 cm^3 is 259.875 cm^3 - 200 cm^3 = 59.875 cm^3

Considering the largest value we have the absolute error 59.875 cm^3

Relative error: 59.875 cm^3 /200 cm^3 x 100 = 29.9375% = 30% 

END