# Units and Measurement Class 11 notes Physics Chapter 2

1.Physical Quantities, 2.Unit, 3.Measurement, 4.The International System of Units, 5.Properties of Fundamental unit, 6.Accuracy, Precision

## ▶Physical Quantities

The quantities by mean of which laws of physics are described are known as physical quantities such as mass, velocity, force, acceleration etc.

### ▶(1) Fundamental or Base Quantities

These are treated as independent of other physical quantities. The units for these quantities are called fundamental or base units. There are seven base quantities. These are length, mass, time, electric current, temperature, luminous intensity and amount of substance.

### ▶(2) Derived Quantities

All physical quantities whose units can be expressed as combination of base units are called derived physical quantities. Thus all quantities other than seven base quantities are derived quantities e.g., velocity, acceleration, momentum, etc. Units of these quantities are called derived units.

## ▶Unit

Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit.

## ▶Measurement

It is a process of determining how large or small a physical quantity is as compared to a basic reference standard. This reference standard is called the unit of the particular physical quantity. A unit can be chosen arbitrarily but then it should be accepted internationally and should not vary from place to place.

## ▶The International System of Units

A complete set of both the base units and derived units, is called the system of units. Historically many systems of units were in use in different parts of the world. A few of them which were quite popular till recently are given below

1. CGS System : In this system centimetre, gram and second are used as the base units for length, mass and time respectively.

2. FPS System : It uses foot, pound and second as base units for length, mass and time respectively.

3. MKS System : It uses metre, kilogram and second as the base units for length, mass and time respectively.

The international system of units is now accepted internationally and is called SI system. SI is an abbreviation for System Internationale d' unites (French name for International System of Units). The SI system is a decimal system, also known as metric system, a modernised and extended form of metric systems like CGS and MKS. There are seven base units and two supplementary units in SI system. These units with their names and symbols are given below.

 Quantity Unit Symbol Mass Kilogram kg Time Second s Temperature Kelvin K Electric Current Ampere A Luminous Intensity Candela cd Length Meter m Amount of Substance Mole mol

### ▶The base units used in SI system are defined as follows

1. Metre (m) : The metre is the length of the path travelled by light in vacuum during a time interval of 1/(299, 792, 458) of a second. (1983)

2. Kilogram (kg) : 1 kg is the mass of a cylinder made of platinum-iridium alloy kept at International Bureau of Weights and Measures, at Severs near Paris, France. (1889)

3. Second (s) : One second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom. (1967)

4. Ampere (A) : One ampere is that constant current which when maintained in two straight, thin and parallel conductors of infinite length, and placed 1 metre apart in vacuum, would produce between these conductors a force of 2 × 10–7 newton per metre of length. (1948)

5. Kelvin (K) : One kelvin is the fraction 1/(273.16) of the thermodynamic temperature of the triple point of water. (1967) The triple point of water is the temperature at which ice, water and water vapour co-exist.

6. Mole (mol) : One mole is that amount of a substance which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12. (1971)

7. Candela (Cd) : One candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 1012 Hz and that has a radiant intensity in that direction of 1/683 watt per steradian. (1979)

### ▶Properties of Fundamental unit

• Base unit should be easily defined.
• It should be available everywhere.
• It should be not changeable with time and place.
• It should be copyable easily.

## Accuracy, Precision and Errors in Measurement

### ▶(i) Error

We use different kinds of instruments for measuring various quantities. But the measurement obtained by any instrument always has a degree of uncertainty related to it. We call this uncertainty, the error in the measurement. An error can arise due to the limitation of the measuring instrument or due to the skill of the person making the measurement.

For example, let us consider the measurement of the thickness of a book with the help of a metre scale. The thickness of the book is closer to 1.7 cm. We say it lies between 1.8 cm and 1.6 cm but has an uncertainty of ± 0.1 cm. This error arises due to the limitation that a metre scale cannot measure a length smaller than 0.1 cm.

### ▶(ii) Accuracy

It is a measure of how close the measured value is to the true value of the quantity. As we reduce the errors, the measurement becomes more accurate. Let the true value of a quantity is 3.9 and its measurements taken by two boys are 3.6 and 3.8. Here 3.8 is more accurate as it is closer to the true value.

### ▶(iii) Precision

It tells us as to what resolution or limit, the quantity is measured. If we measure a certain thickness by two different devices having resolutions 0.1 cm (a metre scale) and 0.01 cm (a vernier callipers), the latter will give a measurement having more precision. Thus a value 1.56mc is more precise than 1.5cm.

Note: Accuracy of a measurement is a value which tells us how close the measured value of a quantity is to its true value whereas precision tells to what resolution or extent that quantity is measured.

### ▶Types of Error

In general, the errors in measurement can be broadly classified as (a) Systematic errors, (b) Random errors.

#### ▶Systematic Errors

The errors which occur in one direction only, i.e., either positive or negative are called systematic errors. If the measured value is greater than the true value, the error is said to be positive. And if the measured value is lesser than the true value, the error is said to be negative. Some of the sources of systematic errors are as follows:

(1). Instrumental Errors : These errors arise when the measuring instrument itself has some defect in it, such as

• (i) Improper Designing or Calibration : It means the instrument is not graduated properly. For example, if an ammeter reads a current of 1.5 A, when a 2 A current is actually flowing through the circuit, it has an imperfect calibration.

• (ii) Zero Error : If the zero mark of vernier scale does not coincide with the zero mark of the main scale, the instrument is said to have zero error. A metre scale having worn off zero mark also has zero error.

(2). Imperfection in Experimental Technique or procedure : The measurement may be systematically affected by external conditions such as changes in temperature, humidity, wind velocity etc. For example, the temperature of a human body measured by a thermometer placed under the armpit will always be less than the actual temperature.

(3). Personal Errors : An experimentalist can introduce error in his measurements due to his carelessness or casual behaviour, though such a behaviour is not expected from any serious experimentalist. If a person while reading the volume of water in a beaker, habitually keeps his eyes below the meniscus, he will introduce an error due to parallax.

Note : Systematic errors can be minimised by improving experimental techniques, selecting better instruments and removing personal bias as far as possible.

#### ▶Random Errors

The errors which cannot be associated with any systematic or constant cause are called random errors. These errors occur irregularly and can randomly have any sign i.e. positive or negative. The magnitude or the size of errors can also vary randomly. These errors can arise due to unpredictable fluctuations in experimental conditions e.g. random changes in pressure, temperature, voltage supply etc.

Least Count Error : The smallest value of the measurement that can be directly taken from a measuring instrument is called its least count. It is also called the resolution of the instrument. Thus a least count error is related to the precision provided by the measuring instrument.

### ▶Absolute, Relative and Percentage Errors

#### ▶Absolute Error

Let the measurement of a physical quantity is taken carefully n number of times. If these measured values are a1, a2 ... an, then their arithmetic mean is considered to be the most accurate value (the value closest to the true value) of the quantity.

Thus, true value

a_{mean}=\frac{a_{1}+a_{2}+a_{3}...+a_{n}}{n}

The magnitude of the difference between the individual measured value and the true value of the quantity is called the absolute error of the measurement.

|Δa1| = |a1 – amean|
|Δa2| = |a2 – amean|
...........................
...........................
|Δan| = |an – amean|

#### ▶Relative Error

The relative error is the ratio of the mean absolute error Δamean to the mean value amean of a measured quantity. It is also called fractional error sometimes.

∴ Relative-error=\frac{\triangle a_{mean}}{a_{mean}}

#### ▶Percentage Error

The relative error expressed in percent gives the percentage error. It is denoted by δa.

%error=\frac{\triangle a_{mean}}{a_{mean}}\times100

## ▶Significant Figures

All those certain digits and the one uncertain digit which are used in measuring of any physical quantity are called the significant digits or significant figures in a measured value.

### ▶Rules to Find Significant Figures

1. All non-zero digits are significant : e.g., 2.483 contains four significant figures.

2. All zeroes appearing between two non-zero digits are significant : e.g., 200.9 has four significant figures.

3. The trailing zeroes in a number without a decimal point are insignificant : e.g., in, 2304000. There are four significant figures only. The three zeroes appearing at the end are not significant.

4. The trailing zeroes in a number having a decimal point are significant : e.g., the number 308.600 has six significant figures.

5. 5. If a number is less than one, the zeroe(s) on the left of the first non-zero digit are not significant : e.g., In the number 0.002783, there are four significant figures only. The three zeroes appearing in the beginning are not significant.

6. For a measurement reported in scientific notation i.e., in the form a × 10b, all the digits appearing in the base number ‘a’ are significant. The power of 10 is irrelevant in the determination of significant figures.

Example : How many significant figures are there in the measured values : (i) 227.2 g, (ii) 3600 g and (iii) 0.00602 g?

Solution : (i) 227.2 g has all the non-zero digits. Hence, it has four significant figures.

(ii) According to rule number 3, trailing zeroes are not significant. Hence, 3600 g has 2 significant figures.

(iii) 0.00602 g. According to the rule number 5, the zeroes at the beginning are not significant. Hence, 3 significant figures.

### ▶Rounding Off the Uncertain Digits

1. If the insignificant digit to be dropped is more than 5, the preceding digit is raised by 1. Let the insignificant digit in the number 3.78 be 8 (circled). Since 8 > 5, we raise the preceding digit 7 by 1. Hence, the number becomes 3.8.

2. If the insignificant digit to be dropped is less than 5, the preceding digits is left unchanged. Let the insignificant digit in the number 3.74 be 4 (circled). Since 4 < 5, we keep the preceding digit 7 unchanged. Hence the number becomes 3.7.

3. If the insignificant digit to be dropped is 5, the preceding digit is raised by 1 if it is odd, and is left unchanged if it is even. Let 5 (circled) be the insignificant digit in the numbers 3.74 5 and 3.77 5 . In the first number, since the preceding digit 4 is even, it remains as such and the number becomes 3.74. In the second number, the preceding digit 7 is odd, hence it is raised by 1 and the number is written as 3.78.

4. When a complex multi-step calculation is involved, all the numbers occurring in the intermediate steps should retain a digit more than the significant digits present in them. The final answer at the end of the calculation, can then be rounded off to the appropriate significant figures.

5. The exact numbers like π, 2, 3, 4 etc. that appear in formulae and are known to have infinite significant figures, can be rounded off to a limited number of significant figures as per the requirement.

## ▶Dimensions of Physical Quantities

The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity. The seven dimensions of physical world are represented as follows

[M]   for mass

[L]    for length

[T]    for time

[A]   for electric current

[K]    for thermodynamic temperature

[Cd]   for luminous intensity

[mol]   for amount of substance

### ▶Applications of Dimensions

The concept of dimensions and dimensional formulae are put to the following uses:

• Checking the results obtained
• Conversion from one system of units to another
• Deriving relationships between physical quantities
• Scaling and studying of models.

### ▶Limitations of Dimensional Analysis

The method of dimensions has the following limitations:

• by this method the value of dimensionless constant cannot be calculated.
• by this method the equation containing trigonometric, exponential and logarithmic terms cannot be analyzed.
• if a physical quantity in mechanics depends on more than three factors, then relation among them cannot be established because we can have only three equations by equalizing the powers of M, L and T.
• it doesn’t tell whether the quantity is vector or scalar.

## ▶Light Year (LY)

The distance which is covered by light in one year (according to Earth) is called light year. It is used in measuring of large distance. As, distance of two planet in the universe.

1 Light Year = (Speed of light) . (1 Year)

1 LY = 3 x 108 x 365 x 24 x 60 x 60

1 LY = 94608000 x 108 m

1 LY = 9.46 x 1015 m

## ▶Summary

• Physics is a quantitative science, based on measurement of physical quantities. Certain physical quantities have been chosen as fundamental or base quantities (such as length, mass, time, electric current, temperature, amount of substance, and luminous intensity).

• Each base quantity is defined in terms of a certain basic, arbitrarily chosen but properly standardised reference standard called unit (such as metre, kilogram, second, ampere, kelvin, mole and candela). The units for the fundamental or base quantities are called fundamental or base units.

• Other physical quantities, derived from the base quantities, can be expressed as a combination of the base units and are called derived units. A complete set of units, both fundamental and derived, is called a system of units.

• The International System of Units (SI) based on seven base units is at present internationally accepted unit system and is widely used throughout the world.

• The SI units are used in all physical measurements, for both the base quantities and the derived quantities obtained from them. Certain derived units are expressed by means of SI units with special names (such as joule, newton, watt, etc).

• The SI units have well defined and internationally accepted unit symbols (such as m for metre, kg for kilogram, s for second, A for ampere, N for newton etc.).

• Physical measurements are usually expressed for small and large quantities in scientific notation, with powers of 10. Scientific notation and the prefixes are used to simplify measurement notation and numerical computation, giving indication to the precision of the numbers.

• In computing any physical quantity, the units for derived quantities involved in the relationship(s) are treated as though they were algebraic quantities till the desired units are obtained.

• Direct and indirect methods can be used for the measurement of physical quantities. In measured quantities, while expressing the result, the accuracy and precision of measuring instruments along with errors in measurements should be taken into account.

• In measured and computed quantities proper significant figures only should be retained. Rules for determining the number of significant figures, carrying out arithmetic operations with them, and ‘rounding off ‘ the uncertain digits must be followed.

• The dimensions of base quantities and combination of these dimensions describe the nature of physical quantities. Dimensional analysis can be used to check the dimensional consistency of equations, deducing relations among the physical quantities, etc. A dimensionally consistent equation need not be actually an exact (correct) equation, but a dimensionally wrong or inconsistent equation must be wrong.