## 1. Units

Introduction:

Measurement of any physical quantity involves comparison with a certain basic, arbitrary chosen, Internationally accepted reference standard called units. Any measured quantity is expressed by a number and a unit. The units for the fundamental or base quantities are called fundamental or base units. The units of all other physical quantities can be expressed as combinations of base units are called derived units. A complete set of these units, both the base units and derived units, is known as the system of units.

Need for measurement

There is a need for measurement in physical quantity. Without it, how can you define things? By measuring a physical quantity we define the properties of a material which the physical quantity is associated with. For example, distance, speed, mass, pressure, force, momentum, and energy.

Think this yourself, what is going to be the meaning of the length of an object if we cannot measure it. We will then inform its length subjectively as ‘long’, ‘short', etc. That makes no sense anymore now.

A physical quantity is a property that can be quantified by the measurement of a material or system. A physical quantity may be represented as the combination of a unit and a numerical value. For example, by specifying methods for calculating them, we outline distance and time, while we describe average speed by stating that it is measured as traveled distance divided by travel time. Measures of physical quantities are represented in unit units, which are numerical meanings

Physical quantities haven't just existed before the dawn of mankind. We made them up to serve our own inquisitive purposes. We have discovered that there are certain properties in the world around us, which can be altered in proportions but not in nature.

Heavier. More. Higher. Stronger. These are all comparative terms. And to compare, we need a basis for measurements to take place. Because, without one, there is no logical way to distinguish between different levels or amounts of the same property.

Units of measurement

There are two types of units of measurement.

• Fundamental/ Base units
• Derived units

Fundamental units: The fundamental units are the base units defined by the International System of Units. These units are not derived from any other unit, therefore they are called fundamental units. There are seven fundamental units.

Derived units. The quantities that are derived using the fundamental quantities are called derived quantities. The units that are used to measure these derived quantities are called derived units. Examples: Force, Velocity, Density, Heat, Power, Energy, Momentum, Acceleration. Some derived units are given in the table below.

Difference between fundamental and derived units

System of units:

In earlier times scientists of different countries were using different systems of units for measurement.  We have here four types of system of units, one of which is S.I units which is accepted globally.

System of units are classified mainly into four types:

1. C.G.S. system: It stands for Centimetre-Gram-Second system. In this system, length, mass and time are measured in centimeters, grams and seconds respectively.

2. M.K.S. system: It stands for Metre-Kilogram-Second system. In this system, length, mass and time are measured in meters, kilograms and seconds respectively.

3. F.P.S. system: It stands for Foot-Pound-Second system. In this system, length, mass and time are measured in the foot, pound and second respectively.

4. S.I. system: It stands for System International. This system has replaced all other systems mentioned above. It has been internationally accepted and is being used all over the world. As the SI units use a decimal system, conversion within the system is very simple and convenient.

## 2. measurements

Measurement :

In everyday experience, it is necessary to make measurements. Anytime one interacts with the environment around, he/she is making measurements of physical quantities. For this reason, it is important that measurements are available as friendly as possible for specialized and non-specialized people. Measurement is available if the measurement system is accessible and easy to use

Any mechanical quantity can be expressed in terms of three fundamental quantities, mass, length and time.

For example, speed is a length divided by time. Force is mass times acceleration and is, therefore, a mass times a distance divided by the square of a time.

Measurement of Mass

The mass of an object is defined by Newton’s Laws. It is the resistance offered by an object to acceleration. In the SI system, we use kilograms to measure mass. But large quantities of matter like the mass of a mountain or the Earth or stars or the entire Universe is measured indirectly by using Newton’s Law of gravitation or other such equations

What are 3 ways to measure mass?

Mass is the amount of matter in an object. A number of tools exist for measuring mass in different environments. These include balances and scales, measurement transducers, vibrating tube sensors, Newtonian mass measurement devices and the use of gravitational interaction between objects.

Measurement of Time

All our activities depend on time. For example, to know the duration of a journey, to meet a schedule at work, to know whether it's day or night, to know the heartbeat, to know the amount of time taken by the computer to perform an operation, etc. Hence it is very important to measure time.

Two natural periodic events which were used in ancient times to measure time were the occurrence of the full moon and sunrise

• The time from one full moon to the next full moon was a month.
• The time from one sunrise to the next sunrise was called a day.

To measure time is to measure the length of time. We know the following units of measuring time: second, minute, hour, day, week, fortnight, month, year, and century. There are various methods of measuring time in different parts of the world.

These instruments can be anything that exhibits two basic components:

(1) a regular, constant, or repetitive action to mark off equal increments of time.

(2) a means of keeping track of the increments of time and of displaying the result

Standard time

As a standard, the atomic standard of time is now used, which is measured by Cesium or Atomic clock. In a Cesium clock, a second is equal to 9,192,631,770 vibrations of radiation from the transition between two hyperfine levels of cesium-133 atoms.

Measurement of Length

Length is the term used for identifying the size of an object or distance from one point Length is a measure of how long an object is or the distance between two points. It is used for identifying the size of an object or distance from one point to another.

What is used to measure length?

Measuring length means measurement of the length of any object with the help of measuring tools like a ruler, measuring tape, etc.

Measurement of length is defined as the act of measuring the length of objects in some specified units which can be standard or non-standard.

Tools that can be used to measure length include rulers, vernier calipers, micrometer screw gauges, measuring tape and odometers. The most precise tool used to measure length are vernier calipers.

Parallax method: for measuring large distances.

To measure large distances between objects methods such as the echo method, the laser method, sonar method, radar method, triangulation method and Parallax method are used.

Parallax is the effect whereby the position or direction of an object appears to differ when viewed from different positions. When I say parallax it means that we are viewing the same object but from two different positions.

Example:  Let us suppose you have some object. Let's say you have a candle that is quite far from you. You first close your right eye and view the candle with your left eye.  Then you open your right eye and view the candle by closing your left eye and trying to view the same candle which is located at the same position.

You will find that there is a change in position. This is because you observed the same object from two different positions. So, the distance between these two different observation points is known as the basis and this phenomenon is known as parallax.

Measuring the distance of distant stars or planets with the parallax method.

### Distance of Planet From Earth

We will see another example here now. Let us suppose we have to find the distance of a distant planet from Earth. We observe the planet from Earth from two different observation points. Let us say we take an observation point as A (taken during June), and the other observation point as B(taken during December). We observe the planet from these two points.

Since the object is very far off, this object can be treated as a small point and the distance between points A and B are the basis. The basis, in this case, is nothing but the diameter of the earth. If we know the angle, that is the angle subtended at this planet from both the observation points then we can find out the value of D.

tan P= R/d    here R=mean distance of the earth from sun= 1 A.U.

P is parallax angle  ; Distance d=1 A.U / tan P

for small P    , tan PP  so  d= R/P = 1 A.U. /P

Accuracy and Precision of measuring instruments.

Accuracy is the degree of closeness to the true value. Precision is the degree to which an instrument or process will repeat the same value. In other words, accuracy is the degree of veracity while precision is the degree of reproducibility.

Precision is defined as the closeness between two or more measured values to each other. Suppose you weigh the same box five times and get close results like 3.1, 3.2, 3.22, 3.4, and 3.0 then your measurements are precise. Remember: Accuracy and Precision are two independent terms.

Accuracy refers to how closely the measured value of a quantity corresponds to its “true” value. Precision expresses the degree of reproducibility or agreement between repeated measurements. The more measurements you make and the better the precision, the smaller the error will be.

The top left image shows the target hit at high precision and accuracy. The top right image shows the target hit at a high accuracy but low precision. The bottom left image shows the target hit at a high precision but low accuracy. The bottom right image shows the target hit at low accuracy and low precision.

### More Examples

• If the weather temperature reads 28 °C outside and it is 28 °C outside, then the measurement is said to be accurate. If the thermometer continuously registers the same temperature for several days, the measurement is also precise.
• If you take the measurement of the mass of a body of 20 kg and you get 17.4,17,17.3 and 17.1, your weighing scale is precise but not very accurate. If your scale gives you values of 19.8, 20.5, 21.0, and 19.6, it is more accurate than the first balance but not very precise.

Some frequently asked questions related to Accuracy and precisions

1. Why are accuracy and precision important in measurement?

In order to get the most reliable results in a scientific inquiry, it is important to minimize bias and error, as well as to be precise and accurate in the collection of data. Both accuracy and precision have to do with how close a measurement is to its actual or true value.

1. How do you measure instrument accuracy?

The accuracy formula provides accuracy as a difference of error rate from 100%. To find accuracy we first need to calculate the error rate. And the error rate is the percentage value of the difference between the observed and the actual value, divided by the actual value

1. What is better accuracy or precision?

Precision is how close measure values are to each other, basically how many decimal places are at the end of a given measurement. Precision does matter. Accuracy is how close a measured value is to the true value. Accuracy matters too, but it's best when measurements are both precise and accurate.

1. Which of the following is the most accurate instrument for measuring length?

Screw gauge of least counts 0. 001 cm is the most precise instrument for measuring length as a device with a minimum least count is more suitable for measuring length.

1. What is the importance of accuracy?

Accuracy is to ensure that the information is correct and without any mistakes. Information accuracy is important because the lives of people depend on it like the medical information at the hospitals, so the information must be accurate.

1. What is the importance of precision?

Precision in scientific investigations is important in order to ensure we are getting the correct results. Since we typically use models or samples to represent something much bigger, small errors may be magnified into large errors during the experiment.

1. How can you improve accuracy?

The accuracy can be improved through the experimental method if every single measurement is made more accurate, e.g. through the choice of equipment. Implementing a method that reduces systematic errors will improve accuracy.

1. How do accuracy and precision relate to significant figures?

Accuracy refers to how closely a measured value agrees with the correct value. Precision refers to how closely individual measurements agree with each other. In any measurement, the number of significant figures is critical.

1. What are the three ways to ensure that you are making an accurate and precise measurement?

What three steps can you take to ensure that your measurements are both accurate and precise? First, use a high-quality measurement tool. Next, measure carefully. Finally, repeat the measurement a few times.

Error in measurements

Sometimes we fail to know the exact measurement and the values vary giving rise to errors. In this article, let us learn about measurement, errors in measurement, types of errors and how to avoid the errors.

An error may be defined as the difference between the measured value and the actual value. For example, if the two operators use the same device or instrument for measurement. It is not necessary that both operators get similar results. The difference that occurs between both the measurements is referred to as an ERROR.

There are three types of errors that are classified on the basis of the source they arise from; They are:

• Gross Errors
• Random Errors
• Systematic Errors

### Gross Error

This category basically takes into account human oversight and other mistakes while reading, recording, and readings. For example, the person taking the reading from the meter of the instrument he may read 23 as 28.

### Random Error

Random errors are those errors, which occur irregularly and hence are random. These can arise due to random and unpredictable fluctuations in experimental conditions (Example: unpredictable fluctuations in temperature, voltage supply, mechanical vibrations of experimental set-ups, etc

### Systematic Error

Systematic errors can be better understood if we divide them into subgroups; They are:

• Environmental Errors

This type of error arises in the measurement due to the effect of external conditions on the measurement. The external condition includes temperature, pressure, and humidity

• Observational Errors

These are the errors that arise due to an individual’s bias, lack of proper setting of the apparatus, or an individual’s carelessness in taking observations. The measurement errors also include wrong readings due to Parallax errors.

• Instrumental Errors

Instrumental Errors: These errors arise due to faulty construction and calibration of the measuring instruments. Such errors arise due to the hysteresis of the equipment or due to friction.

## Errors  in Calculation

Different measures of errors include:

### Absolute Error

The difference between the measured value of a quantity and its actual value gives the absolute error. It is the variation between the actual values and measured values. It is given by

Absolute error = |VA-VE|

### Percent Error

It is another way of expressing the error in measurement. This calculation allows us to gauge how accurate a measured value is with respect to the true value. Percent error is given by the formula

Percentage error (%) = (VA-VE) / VE) x 100

### Relative Error

The ratio of the absolute error to the accepted measurement gives the relative error. The relative error is given by the formula:

Relative Error = Absolute error / Actual value

Combination of Error

a) Error of a sum or a difference

When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.

Z = A + B

We have by addition, Z ± ΔZ = (A ± ΔA) + (B ± ΔB).

The maximum possible error in Z

ΔZ = ΔA + ΔB

For the difference Z = AB, we have

Z ± Δ Z = (A ± ΔA) – (B ± ΔB) = (AB) ± ΔA ± ΔB

or, ± ΔZ = ± ΔA ± ΔB

The maximum value of the error ΔZ is again ΔA + ΔB.

(b) Error of a product or a quotient

When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.

Suppose Z = AB and the measured values of A and B are A ± ΔA and B ± ΔB. Then

Z ± ΔZ = (A ± ΔA) (B ± ΔB) = AB ± B ΔA ± A ΔB ± ΔA ΔB.

Dividing LHS by Z and RHS by AB we have,

1 ± (ΔZ/Z) = 1 ± (ΔA/A) ± (ΔB/B) ± (ΔA/A)(ΔB/B).

Since ΔA and ΔB are small, we shall ignore their product.

Hence the maximum relative error

ΔZ/ Z = (ΔA/A) + (ΔB/B).

(c) Error in case of a measured quantity raised to a power

The relative error in a physical quantity raised to the power k is the k times the relative error in the individual quantity.

Suppose Z = A2,

Then,

ΔZ/Z = (ΔA/A) + (ΔA/A) = 2 (ΔA/A).

Hence, the relative error in A2 is two times the error in A.

In general, if Z = (Ap Bq)/Cr

Then,

ΔZ/Z = pA/A) + qB/B) + r C/C).

## How To Reduce Errors In Measurement

Keeping an eye on the procedure and following the below-listed points can help to reduce the error.

• Make sure the formulas used for measurement are correct.
• Cross-check the measured value of a quantity for improved accuracy.
• Use the instrument that has the highest precision.
• It is suggested to pilot test measuring instruments for better accuracy.
• Use multiple measures for the same construct.
• Note the measurements under controlled conditions.

Significant figures

Significant figures in the measured value of a physical quantity tell the number of digits in which we have confidence. Larger the number of significant figures obtained in a measurement, the greater the accuracy of the measurement. The reverse is also true.

Important Rules for counting significant figures

• All the non-zero digits are significant.

• All the zeros between two non-zero digits are significant, no matter where the decimal point is, if at all.

• If the number is less than 1, the zero(s) on the right of the decimal point but to the left of the first non-zero digit are not significant.

[In 0.00 2308, the underlined zeroes are not significant].

• The terminal or trailing zero(s) in a number without a decimal point are not significant.

[Thus 123 m = 12300 cm = 123000 mm has three significant figures, the trailing zero(s) being not significant.] However, you can also see the next observation.

• The trailing zero(s) in a number with a decimal point are significant.

[The numbers 3.500 or 0.06900 have four significant figures each.]

• For a number greater than 1, without any decimal, the trailing zero(s) are not significant.

• For a number with a decimal, the trailing zero(s) are significant.

Rules for Arithmetic Operations with Significant Figures

(1) In multiplication or division, the final result should retain as many significant figures as there are in the original number with the least significant figures.

(2) In addition or subtraction, the final result should retain as many decimal places as there are in the number with the least decimal places.

Rounding off the Uncertain Digits

(i) If the digit dropped is less than 5, then the preceding digit is left unchanged.

(ii) If the digit to be dropped is more than 5, then the preceding digit is raised by one.

(iii) If the digit to be dropped is 5 followed by digits other than zero, then the preceding digit is raised by one.

(iv) If the digit to be dropped is 5 or 5 followed by zeroes, then the preceding digit is left unchanged if it is even.

(v) If the digit to be dropped is 5 or 5 followed by zeroes then the preceding digit is raised by one if it is odd.

Selected prefixes used in metric system

## 3. Dimensions

Dimensions

Each derived quantity requires proper power for fundamental quantities so as to represent it. The powers of fundamental quantities, through which they are to be raised to represent unit-derived quantity, are called dimensions. In other words, the dimensions of a physical quantity are the powers to which the base quantities (fundamental quantities) are raised to represent that quantity.

Thus, the dimensions of a physical quantity are the powers(or exponents) to which the fundamental units of length, mass, time, etc. must be raised to represent it or the dimension of the units of a derived physical quantity is defined as the number of times the fundamental units of length, mass, time, etc appear in the physical quantity.

A few examples are :

Some Physical quantities and their dimensions

Dimensional Analysis

Dimensional analysis is the practice of checking relations amongst physical quantities by identifying their dimensions and units of measurement. However, dimensional analysis is possible only if the dimensions of various terms on either side of the equation are the same. This rule is known as the principle of homogeneity of dimensions. The principle is based on the fact that two quantities of the same dimension only can be added, subtracted, or compared.

Also, dimensional analysis is an amazing tool for checking whether or not equations are dimensionally correct. It is also possible to use dimensional analysis to generate plausible equations if we know the quantities involved. Quantification of the size and shape of things can be done using dimensional analysis. The mathematical study of the nature of objects is possible today, thanks to dimensional analysis.

We have already learned to express most physical quantities in terms of basic dimensions. We will now learn about dimensional analysis and its applications with the help of fundamental quantities like mass, length, time, etc.

## Principle of Homogeneity of Dimensions

The equations depicting physical situations must have the same dimensions. This principle is based on the fact that only two quantities of the same dimension can be added, subtracted, or compared.

The principle of Homogeneity states that “dimensions of each of the terms of a dimensional equation on both sides should be the same.”

As in the above equation dimensions of both sides are not the same; this formula is not correct dimensionally, so it can never be physically accurate.

An equation of the form x= a+b+c+....  is dimensionally correct if and only if the variables x, a, b, and c all have the same dimension. This principle may be applied to differential equations and integral equations, as well as to algebraic equations.

For example :

1.  F=mv^2/r^2

By substituting dimensions of the physical quantities in the above relations

[MLT^(-2) ]=[M] [LT^(-1) ]^2  / [L]^2   ;  we have [MLT^(-2)]=[MT^(-2)]

As in the above equation, dimensions of both sides are not  the same , the formula is not correct dimensionally , so it can never be physically accurate.

Note

•  If  [M]^a [L]^b [T]^c   = [M]^x [L]^y [T]^z  then from the principle of homogeneity we have a=x, b=y and c=z.
• A dimensionally correct equation may not be physically correct, but a physically correct equation must be dimensionally correct.

## Dimensional Analysis as Factor Label Method

Units can be converted from one system to the other. The method used for this is called the factor label method, unit factor method, or dimensional analysis.

In this method, a unit can be converted from one system to another by using a conversion factor that describes the relationship between units. It is based on the fact that the ratio of each fundamental quantity in one unit with their equivalent quantity in another unit is equal to one.

For example:

How many minutes are there in 5 hours?

Solution : 1 hour= 60 minutes , so 3 hours = 3×60=180 minutes

(Here the conversion factor from hours to minutes is 60)

We use conversion factors accordingly so that the answer comes in the desired unit and biased results are avoided. In this way, each fundamental quantity like mass, length, and time is converted into another desired unit system using the conversion factor.

## Applications of Dimensional Analysis

Dimensional analysis is used to solve problems in real-life physics. We make use of dimensional analysis for five prominent reasons:

1. Finding unit of a physical quantity in a given system of units
2. Finding dimensions of physical constant or coefficients
3. Converting a physical quantity from one system to the other
4. Checking the dimensional correctness of a given physical relation
5. Tool for research to derive new relations

1. Finding Unit of a Physical Quantity in a Given System of Units

James Clerk Maxwell and Jenkin invented the dimensional formula in the early 1860’s for unit conversion, and the modern concept of dimension started in 1863 with Maxwell. He synthesized earlier formulations by Fourier, Weber, and Gauss.
To write the formula of a physical quantity, we find its dimensions using dimensional analysis. Now in the dimensional formula replacing M, L and T with the fundamental units of the required system, we get the unit of the physical quantity.

However, sometimes to this unit, we further assign a specific name, e.g.,

Work = Force×Displacement

So , [W]=[MLT^(-2)]×[L]=[ML^2 T^(-2)]

So its units in the C.G.S. system will be gcm^2/s^2  which is called erg while in the M.K.S. system will be  kgm^2/s^(2 )  which is called joule.

2. Finding Dimensions of Physical Constant or Coefficients

Dimensions of a physical quantity are unique and special. We write an equation by putting a proportional constant calculated by substituting the dimensional formulae of all other physical quantities.

(i) Gravitational constant: According to Newton’s law of gravitation

F=G (m1 m2)/r^2    or   G=(F r^2)/(m1 m2)

Substituting the dimensions of all physical quantities

[G]= ([MLT^(-2)] [ L^2])/([M][M])= [M^(-1) L^3 T^(-2)]

(ii) Planck constant: According to Planck   E= hυ or  h=E/υ

Substituting the dimensions of all physical quantities

[h]= ([ML^2 T^(-2)])/([T^(-1)])= [ML^2 T^(-1)]

(iii) Coefficient of viscosity: According to Poiseuille’s formula

dV/dt=(Πpr^4)/8ηl  or  n=(Πpr^4)/(8 (dV/dt))

Substituting the dimensions of all physical quantities

[η]=([ML^(-1) T^(-2)][L^4])/([L][L^3 T^(-1)])=[ML^(-1) T^(-1)]

3. Converting a Physical Quantity from One System to the Other

The measure of a physical quantity is  nu= constant

If a physical quantity X has the dimensional formula [M^a L^b T^c] and if its derived units of that physical quantity in two systems are [M_1^a^ L_1^b T_1^c  ] and [M_2^a L_2^b  T_2^c  ]  respectively and n1 and n2  are the numerical values in the two systems respectively, then the system

Respectively.

n1[M_1^a^ L_1^b T_1^c  ]  = n2 [M_2^a L_2^b  T_2^c  ]

n2= n1[M_1^a^ L_1^b T_1^c  ] / [M_2^a L_2^b  T_2^c  ]

Where, M1, L1 and T1= basic units of mass, length, and time in the first (known) system

M2,  L2 and T2= basic units of mass, length, and time in the second (unknown) system

Example: conversion of Newton into dynes.

### 4. Checking the Dimensional Correctness of a Given Physical Relation

This application is based on the principle of homogeneity of dimensions. According to this rule, only those terms can be added or subtracted with the same dimensions.

If  X=A ±(BC)^DEF

Then according to the principle of homogeneity

[X]=[A]=[(BC)^2]=[((DEF)])

If the dimensions of each term on both sides of an equation are the same, then the equation is dimensionally correct.

Example:     s= ut+(1/2)at^2

By substituting respective dimensions of the physical quantities in the above equation –

[L]=[LT^(-1)][T]-[LT^(-2)][T^2]   so [L]=[L]-[L]

As in the above equation, the dimensions of each term on both sides of an equation are the same; the equation is dimensionally correct. However, from equations of motion, we know that s= ut-(1/2)at^2

### Suppose one knows the dependency of a physical quantity on other quantities which is of the product type. In that case, using the dimensional analysis method, a relation between the quantities can be derived.

Derivation of stoke’s law using dimensional Analysis

The force of viscosity on an object in a fluid depends on the radius of the object, coefficient of viscosity and its velocity in the fluid.

From the dimensional analysis we have F= krηv, the value of  K=6Π

So viscous force F=6Πηrv

Derivation of centripetal force using dimensional analysis.

Centripetal force required to move an object in a circular path depends on its mass, velocity and radius of the circular path.