FSc ICS Physics Part 1 XI 11th Chapter 1 Measurements Notes Long Questions

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Introduction of Physics

Ever since man has started to observe, think and reason he has been wondering about the world around him. He tried to find ways to organize the disorder prevailing in the observed - facts about the natural phenomena and material things in an orderly manner. His attempts » resulted in the birth of a single discipline of science, called natural philosophy.

There was a huge increase in the volume of scientific knowledge up till the beginning of nineteenth century and it was found necessary to classify the study of nature into two branches.

i. Biological Sciences

ii. Physical Sciences

Biological sciences which deal with living things.

Physical sciences which concern with non-living things.

Physics is an important and basic part of physical sciences besides its other disciplines such as chemistry, astronomy, geology etc.

Physics is an experimental science and the scientific method emphasizes the need of accurate measurement of various measurable features of different phenomena or of man made objects. This chapter emphasizes the need of thorough understanding and practice of measuring techniques and recording skills; ' '

1.1 Introduction to Physics

At the present time, there are three main frontiers of fundamental science.

First, the world of the extremely large, the universe itself, Radio telescopes now gather information from the far side of the universe and have recently detected, as radio waves, the “firelight” of the big bang which probably started off the expanding universe nearly 20 billion ‘years ago‘.

Second, the world of the extremely small, that of the particles such as, electrons, protons, neutrons, mesons and others.

The third frontiers the world of complex matter; lt is also the World of "middle-sized" things, from molecules at one extreme to‘ the Earth at the other. This is all fundamental physics, which is the heart of science. " 

What is Physics?

According to one definition, physics deals with the study of matter .and energy and the relationship between them.

The study of physics involves investigating such things as the laws of motion, the structure of space and time, the nature and type of forces that hold different materials together, the interaction between different particles, the interaction of electromagnetic radiation with matter and soon.

Branches of Physics

By the end of 19th century many physicists started believing that every thing about physics has been discovered. However, about the beginning of the twentieth century many new experimental facts revealed that the laws formulated by the previous investigators need modifications. Further researches gave birth to many new disciplines in physics.

Nuclear physics which deals with atomic nuclei,

Particle physics which is concerned with the ultimate particles of which the matter is composed.

Relativistic mechanics which deals with velocities approaching that of light.

Solid state physics which is concerned with the structure and properties of solids,

but this list is by no means exhaustive".

Other Branches of Science

Physics is most fundamental of all sciences and provides other branches of science, basic principles and fundamental laws. This overlapping of physics and other fields gave birth to new branches such as physical chemistry, biophysics, astrophysics, health physics etc.

Role of Physics in Technology

Physics also plays an important role in the development of technology and engineering. ‘

Science and technology are a potent force for change in the outlook of mankind. Then information media and fast means of communications have brought all parts of the world in close contact with one another. Events in one part of the world immediately reverberate round the globe.

We are living in the age of information technology. The computer networks are products of chips developed from the basic ideas of physics. The chips are made of silicon. Silicon can be obtained from sand. It is up-to us whether we make a sand castle or a computer out of it.

1.2 Physical Quantities

The foundation of physics rests upon physical quantities in terms of which the laws of physics are expressed. Therefore, these quantities have to be measured accurately. Among these are mass, length, time, velocity, force, density, temperature, electric current, and numerous others. '

Physical quantities are often divided into two categories:

i. Base quantities

ii. Derived quantities.

Derived quantities are those whose definitions are based on other physical quantities. Velocity, acceleration and force etc. are usually viewed as derived quantities.

Base quantities are not defined in terms of other physical quantities. The base quantities are the minimum number of those physical quantities in terms of which other physical quantities can be defined. Typical examples of base quantities are length, mass and time. -

Measurement of Base Quantities

The measurement of a base quantity involves two steps.

first, the choice of a standard, and 

second, the establishment of a ~ the standard so that a number and a unit are determined as the measure of that quantity.

Characteristics of an Ideal Standard:

An ideal standard has two principal characteristics-:'it“ is , accessible and it is invariable. These two requirements are often incompatible and a compromise has to -be made ‘ between them.

1.3 International System of Units

In 1960, an international committee agreed on a set of definitions and [standard to describe the . physical quantities. The system that was established is called the - System International (SI).

Due to the simplicity and convenience with which the units  in this system‘ are amenable to arithmetical manipulation, it is in universal use by the world's scientific community and ' by most nations. The system international (SI) is built up from‘ three kinds of units: 

i. Base units

ii. Supplementary units

iii. Derived units.-

Base Units:

There are seven base units for various physical‘ quantities. namely: "length, mass, time, temperature, electric current, thermodynamic temperature, intensity and amount of a substance (with special reference to the number of particles).

Supplementary Units:

The General Conference on Weights and Measures has not yet classified certain units of the SI under either base units or derived units. These SI units are called supplementary units. For the time being this class contains-only two units of purely geometrical quantities, which are plane angle and the solid angle

Radian:

The radian is the plane angle between two radii of a circle, which cut off on the circumference an arc, equal in length to the radius,.

Steradian:

The steradian is the solid angle (three-dimensional angle) subtended at the center of a sphere by, an area of its surface equal-to the square of radius of the sphere.

Derived Units

SI units for measuring all other physical quantities are derived from the base and supplementary units.

Scientific Notation:

Numbers are expressed in standard form called scientific notation, which employs powers of ten. The internationally accepted practice is that there should be only one non-zero digit left of decimal. Thus, the number 134.7 should be written as 1.347 x 10(2).

Conventions for Indication Units

Use of SI units requires special care more particularly in writing prefixes.

Following points should be kept in mind while using units.

(i) Full name of" the unit does not begin with a capital . letter even if named after a scientist e.g. newton.

(ii) The symbol of unit named after a scientist has initial capital letter such as N for newton.

(iii) The prefix should be written before the unit without any space, such as 1 x 10(-3) m, is written as 1 mm.

(iv) A combination of base units is written each with one space apart. For example, newton meter is (written as N m).

(v) Compound prefixes are not allowed. For example, 1μμF may be written as 1pF. . '

(vi) A number such as 5.0 x 104 cm may be expressed in scientific notation as 5.0 x 10(2) m.

(vii) When a multiple of a base unit is raised to a power, the power applies to the whole multiple and not the base unit alone. Thus, 1 km(2) = 1 (km)2 = 1 x 10(6) m2.

(viii) Measurement in practical work should be recorded immediately in the most convenient unit, e.g., micrometer screw gauge measurement in mm, and the mass of calorimeter in grams (g). But before calculation for the result, all measurements must be converted to the appropriate SI base units.

1.4 Errors and Uncertainties

All physical measurements are uncertain or imprecise to some extent. It is very difficult to eliminate all possible errors or uncertainties in a measurement.

The error may occur due to» 

(1) negligence or inexperience of a person

(2) the faulty apparatus

(3) inappropriate method or technique.

The uncertainty may occur due to inadequacy or limitation of an instrument, natural variations of the object being measured or natural imperfections of a person's senses. However, the uncertainty is also usually described as an error in a measurement. There are two major types of errors.

(i) Random error

(ii) Systemic error

Random error is said to occur when repeated measurements of the quantity, give different values under the same conditions. It is due to some unknown causes.

Repeating the measurement several times and taking an average can reduce then effect of random errors.

Systematic error refers to an effect that influences all measurements of a particular quantity equally. It produces a consistent difference in readings. It occurs to some definite rule.

It may occur due to:

i. zero error of instruments.

ii. poor calibration of instruments or incorrect markings etc.

Systematic error can be reduced by comparing the instruments with another which is known to be more accurate. Thus for systematic error, a correction factor can be applied.

1.5 Significant Figures

As stated earlier physics is based on measurements. But unfortunately whenever a physical quantity is measured, there is inevitably some uncertainty about its determined value. This uncertainty may be. due to a number of reasons. One reason is the type of instrument, being used. We know that every measuring instrument is calibrated to a certain smallest division and this fact put a limit to the degree of accuracy which may be achieved’ while measuring with it.

Suppose that we want to measure the length of 'a straight line with the help of a meter rod calibrated in millimeters. Let the end point of the line lies between 10.3 and 10.4 cm marks. By convention, if the end of the line does not touch or cross the midpoint of the smallest division, the reading is confined to the previous division. In case the end of the line seems to be touching or have crossed the midpoint, the reading is extended to the next division.

By applying the above rule the position of the edge of a line recorded as 12.7 cm with the help of a meter rod calibrated in millimeters may lie between 12.65 cm and 12.75 cm. Thus in this example the maximum uncertainty is 1 0.05 cm. it is, in fact, equivalent to an uncertainty of 0.1 cm equal to the least count of the instrument divided into two parts, half above and half below the recorded reading.

The uncertainty or accuracy in the value of a "measured quantity can be indicated conveniently by using significant 1-figures. The recorded value of the length of the straight line i.e. 12.7 cm contains three digits (1, 2, 7) out of which two digits (1 and 2) are accurately. known while the third digit i.e. 7 is a doubtful one. As a rule:

In any measurement, the accurately known digit and first doubtful digit are called significant figures.

In other words, a significant figure is the one which is known to be reasonably reliable.- if. the above mentioned measurement is taken by a better measuring instrument which is exact up-to a hundredth of a centimeter, it would have been recorded as 12.70 cm. in this case, the number of -significant figures is four.

Thus, we can say that as we improve the quality of our measuring instrument and techniques, we extend the measured result to more and more significant figures and correspondingly improve the experimental accuracy of the result.

While calculating a result from the measurements, it is important to give due attention to significant figures and we must know the following rules in deciding how many “significant figures are to be retained in the final result.

General Rules

1. All digits 1,2,3,4,5,6,7,8,9 are significant. However, zeros may or may not be significant. ln case of  zeros, the following rules may be adopted.

i) A zero between two significant figures is itself ‘significant.

ii) Zeros to the left of significant figures are not significant. For example, none of the zeros in 0.00467 or 02.59 is significant. ‘

iii) Zeros to the right of a significant figure may or  may not be significant.

              a. In decimal fraction, zeros to the right of a significant figure are significant. For example, all the zeros in 3.570 or 7.4000 are significant. 

               b. In integers such as 8,000 kg, the. number of significant zeros is determined by the accuracy' of the measuring instrument. if the measuring scale 1 has a least count of 1 kg then there are four 1 significant figures written in scientific notation as 8.000 x 10(3) kg. if the least count of the scale is 10 kg, then the number of significant figures will be 3 written in scientific notation as 8.00 x 10(3) kg and so on. "

d) When a measurement is recorded in scientific notation or standard form, the figures other than the powers of ten are significant figures. For example, a measurement recorded as . 8.70 x 104 kg has three significant figures.

2. In multiplying or dividing numbers, keep a number of significant figures in the product or quotient not more than that contained in the least accurate factor i.e., the factor containing the least number of significant figures. For example, the computation of the following using a calculator, gives

Round off rules:

i. If the first digit dropped is less than 5, the last digit retained should remain unchanged.

ii. If the first digit dropped is more than 5, the digit to be retained is increased by one.

iii. If the digit to be dropped is 5, the previous digit which ‘ is to be retained, is increased by one if it is odd and retained as such if it is even. For example, The following numbers are rounded off to three significant figures as follows. The digits are deleted one by one. 

43.75 is rounded off as. 43.8

56.8546 is rounded off as 56.8 l

73.650 . is rounded-off as 73.6

64.350 is rounded off as 64.4

1.6 Precision and Accuracy

In measurements made in physics, the terms precision and accuracy are  frequently used. They should be distinguished clearly. The precision of a measurement is determined by the instrument or device being used and the accuracy of a measurement depends on the fractional or percentage uncertainty in that measurement.

For example, when the length of an object is recorded as 25.5 cm by using a meter rod having smallest division in millimeter, it is, the difference of two readings of the initial and final positions. The uncertainty in the single reading as discussed before is taken as  ±0.05 cm which is now doubled and is called absolute uncertainty equal to ±O.1cm.

Absolute uncertainty, in fact, is equal to the least count of the measuring instrument. 

Precision or absolute uncertainty (least count) = ± 0.1 cm

Fractional uncertainty = 0.1cm/25.5cm = 0.004

Percentage uncertainty = 0.1/25.5 x 100/100 = 0.4/100 = 0.4%

Another measurement taken by Vernier calipers with least count as 0.01 cm is recorded as 0.45 cm.

It has Precision or absolute uncertainty (least count) = ± 0.01 cm

Fractional uncertainty = 0.01cm/0.45cm =0.02

Percentage uncertainty = 0.01/0.45 x 100/100 = 0.2%

Conclusion:

Thus the reading 25.5 cm taken by meter rule is although less precise but is more accurate having less percentage uncertainty or error.

Whereas the reading 0.45 cm taken by Vernier calipers is more precise but is less accurate. In fact, it is the relative measurement which is important. The smaller a physical quantity, the more precise instrument should be used. Here the measurement 0.45 cm demands that at more precise instrument, such as micrometer screw gauge, with least count 0.001 cm, should have been used. Hence, we can conclude that:

A precise measurement is the one which has less absolute uncertainty and an accurate measurement is the one which has less fractional or percentage uncertainty or error.

1.7 Assessment of Total uncertainty in the final result

To assess the total uncertainty or error, it is necessary to evaluate the likely uncertainties in all the factors involved in that calculation. The maximum possible uncertainty or error in the final result can be found as follows.

1. For addition and subtraction

Absolute uncertainties are added:

For example, the distance x determined by the difference between two separate position measurements -

x1=10.5 ± 0.1 cm

x2 = 26.8 1 0.1 cm is recorded as

x = x2-x1= 26.8 ±0.1 - 10.5 ±0.1

2. For multiplication and division

Percentage uncertainties are added.

For example the maximum possible uncertainty in the value of resistance R of a conductor determined from the measurements of potential difference V and resulting current flow I by using 

R = V/I is found as follows

V= 5.2 ± 0.1 V

I = 0.84 ± 0.05A '

The %age uncertainty for V is = 0.1V/5.2V x 100/100 = 2%.

The %age uncertainty for I is = 0.05A/0/84 x 100/100 = 6%.

Hence total uncertainty in the value of resistance R when V is divided by I is 8%. The result is thus quoted as 

R = 5.2V/0.84A = 6.19 V/A= 6.19 ohms with a % age uncertainty of 8% that is:

R = 6.2 * 8/100 = 0.5

Thus, R = 6.2 ± 0.5ohms

The result is rounded off to two significant digits because both V and R have two significant figures and uncertainty, being an estimate only, is recorded by one significant figure. ' '

3. For power factor:

Multiply the percentage uncertainty by that power.

For example, in the calculation of the volume of a sphere using V = 4πr(3)/4

%age uncertainty in V= 3 x % age uncertainty in radius r.

As uncertainty is multiplied by power factor, it increases the precision demand of measurement. If the radius of a small sphere is measured as 2.25 cm by a Vernier calipers with least count 0.01 cm, then the radius r is recorded as r

r = 2.25 ± 0.01 cm

Absolute uncertainty = Least count = ± 0.01 cm 

%age uncertainty in r = 0.01cm/2.25cm x 100% = 0.4% ‘

Total percentage uncertainty in V = 3 x 0.4 = 1.2%

Thus volume V= 4/3 X 3.14 X ( 2.25 cm)3 = 47.689 c(3) a with 1.2% uncertainty

Thus the result should be recorded as V = 47.71 ± 0.6cm(3).

4. For uncertainty in the average value of many measurements

I) Find the average value of measured values.

(ll) Find deviation of each measured value from the average value.

(Ill) The mean deviation is the uncertainty in the average value. A

For example, the six readings of the micrometer screw gauge to measure the diameter of a wire in mm

1.20,1.22,1.23,1.19,1.22,1.21

Average = (l.20+1.22+1.23+1.19+1.22+1.21)/6

Then ' Average = 1.21 mm

The deviation of the readings, which are the difference without regards to the sign, between each reading and average value are 0.01, 0.01,-0.02, 0.02, 0.01, 0,

Mean of deviations = (0.01 +0.01 +0.026+0.02 +0.01+ 0)/6 = 0.01 mm

Thus, likely uncertainty in the mean diameter 1.21 mm is 0.01 mm recorded as 1.21 ± 0.01 mm.

5 For the uncertainty in a timing experiment

The uncertainty in the time period of a vibrating body is found by dividing the least count of timing device by the number of vibrations.

For example, the time of 30, vibrations of a simple pendulum recorded by a stopwatch, accurate up-to one tenth of a second is 54.6s.

Time period T = 54.6s/30 = 1.82s

Uncertainty = Least count/No. of vibrations = 0.1s/30 = 0.003s

Thus, period T is quoted as T 1.82 ± 0.003s

Hence, it is advisable to count large number of swings to reduce timing uncertainty.

1.8 Dimensions of Physical Quantities

Each base quantity is considered a dimension denoted by a specific symbol written within square brackets. it stands for the qualitative nature of the physical quantity. For example, different -quantities such as length, breadth, diameter, light year which are measured in meter denote the same dimension and has the dimension of length [ L].

Similarly the mass and time dimensions are denoted by by [M] and [T ], respectively. Other quantifies that we measure have dimension which are combinations of these dimensions.

For example: speed is measured in meters per second. This will obviously have the dimensions of length divided by time. Hence we can write.

Dimension of speed = Dimension of length / Dimension of time

 = [L] / [T]

 = [LT(-1)]

Similarly Dimension of acceleration = Dimension of speed / Dimention of time

 = [LT(-1)] / [T]

 = [LT(-2)]

Similarly Dimension of Force

Since, F = ma

[F] = [m][a]

 = [M][LT(-2)]

 = [MLT(-2)]

Using‘ the method of dimensions called the dimensional analysis we can check the correctness of a given formula or an equation and can also derive it.

Dimensional analysis makes use of the fact that expression of the dimensions can be manipulated as algebraic quantities."

(i) Checking the homogeneity of physical equation

In order to check the correctness of an equation, we are to show that the dimensions of the quantities on both sides of the equation are the same, irrespective of the form of the formula. This is called the principle of homogeneity of dimensions. "

(ii) Deriving a possible formula

The success of this method for deriving a relation for a physical quantity depends on the correct guessing of various factors on which the physical quantity depends.