# Units and Measurements Class 11 Notes

## Table of Contents

## Introduction

System of units

Dimensions of physical quantities

Dimensional analysis and its applications

Dimensional formulae and dimensional equations

Significant figures

Errors Analysis

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## ►INTRODUCTION

**Quantity :**Anything we can see directly or indirectly around us is called a quantity for the universe.

**Types of Quantities**

**i. Physical Quantities**

- All the quantities which can be measured are called physical quantities.
- Example; Length, mass, time, speed, force e.t.c.

**ii. Non-Physical Quantities**

- Those quantities which cannot be measured are called non-physical quantities.
- Example; Car, bus, building, town, happiness e.t.c.

**NOTE:**

- Physical quantities are measurable.
- Non-physical quantities are non-measurable.

** ► CLASSIFICATION OF PHYSICAL QUANTITIES:**

I. Base / Fundamental physical quantity

II. Supplementary physical quantity

III. Derived physical quantity

**I. Base / Fundamental physical quantity:**

Those physical quantities which do not depend on any other physical quantities are called base or fundamental physical quantities.

__There are 7 base physical quantities:__

- Length
- Mass
- Time
- Temperature
- Electric current
- Amount of substance
- Luminous intensity

**NOTE:**

- Luminous intensity means intensity or brightness of light.

**II. Supplementary physical quantities :**

Plane angle and solid angle are taken as supplementary physical quantities.

III. Derived physical quantities :

Those physical quantities which are made with the help of base physical quantities are called derived physical quantities.

Example: Volume, density, work, velocity e.t.c.

i.e, velocity is made with the help of base physical quantities length and time.

## ► MEASUREMENT :

The comparison of any physical quantity with its standard unit is called Measurement.

The measurement of the quantity is expressed by two terms :

- Number
- Letter/ group of letters/ symbol

Here,

- Number term describes the amount of the given physical quantity which is called its magnitude.

Example: in 50 kg mass ( 50 - magnitude)

- Letter/ group of letters/ symbol make the given quantity identified, that which quantity it is.

Now, we cannot identify the given quantity that which quantity it is, but If we write the quantity as 51 kg, thus now we find it as mass which is identified by its unit (Letter / group of letters / symbol)

IN SHORT

- Answer to how much - magnitude
- Answer to which one - unit

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

Merit Points / Points Regarding Units :

- The unit is always written in singular form.

Example; **foot **(correct)** ; ** feet (wrong)

- No punctuation marks are used after unit.

Example; **sec **(correct) ; ** ** **secs.** (wrong)

- If a unit is named after a person, the symbol is not written with capital initial letter.

Example: **newton **(correct) ; ** ** **Newton **(wrong)

- If unit is named after a person, the symbol used is a capital letter.

Example: for newton **"N" **(correct) ; ** **** ****“n”. **(wrong)

- Two or more physical quantities are added or subtracted when their units and dimensions are same.
- After multiplication or division, the resultant quantity may have different unit.

**► **TYPES OF UNIT

I. Fundamental Unit

II. Supplementary Unit

III. Derived Unit

IV. Practical Unit

I. FUNDAMENTAL UNIT :

The units used for fundamental quantities are called fundamental units.

Example: meter, centimeter, foot, gram e.t.c.

**SYSTEM OF UNITS**

A system of units is a collection of fundamental and derived units for all kinds of physical quantities. The common systems are given below:

**i. FPS system :**

- In this system length, mass and time have been taken as fundamental quantities, and the corresponding fundamental units are foot(f), pound(P) and second(s).
- In this system force is a derived quantity with unit poundal.
- FPS unit of force: pound ft/s2- poundal

**ii. CGS system :**

- In this system length, mass and time have been taken as the fundamental quantities and corresponding fundamental units are centimeter (cm), gram (g) and second (s) respectively.
- The system is also called Gaussian system of units.

**iii. MKS system :**

- In this system also length, mass and time have been taken as fundamental quantities, and the corresponding fundamental units are metre(m), kilogram(kg) and second(s).
- The system is also called Giorgi system.

- M.K.S. – system is internationally accepted system and further it is called as SI - system for length, mass and time.
- Full form of S.I system - Standard International System of Unit .
- Given systems of unit give the unit for length mass, time and for those quantities which are derived from these three.
- But in SI - system we have unit for all the physical quantities either it is base, derived or supplementary.
- There are seven fundamental quantities in this system. These quantities and their units are given in the following table

SI - UNIT AND SYMBOL OF BASE PHYSICAL QUANTITIES

PHYSICAL QUANTITY | SI-UNIT | SYMBOL |

Length | Meter | m |

Mass | Kilogram | Kg |

Time | Second | s |

Temperature | kelvin | K |

Electric Current | ampere | A |

Amount of substance | mole | mol |

Luminous intensity | candela | Cd |

II. SUPPLEMENTARY UNITS :

The unit used for supplementary physical quantities are called supplementary units.

SI UNIT WITH SYMBOL FOR SUPPLEMENTARY PHYSICAL QUANTITIES

III. DERIVED UNITS:

The units used for derived physical quantities are called derived units.

► **How can we write the units for Derived physical quantities**

__To find the derived unit, follow 4 steps:-__

1. Write the formula of the derived quantity.

2. Convert the formula in terms of fundamental physical quantities.

3. Write the unit for each term in proper system.

4. Make proper algebraic combination and get the result.

Question:

Find the SI unit of force.

Solution:

Step 1, F = ma

Test Yourself 01:

**Write the SI unit for the given physical quantities :**

1. Area = length x breadth

2. Volume = length x breadth x height

3. Mass density = mass / volume

4. Velocity = displacement / time

5. Acceleration = velocity / time

6. Force = mass x acceleration

7. Work = force x displacement

8. Power = work / time

9. Pressure = Force / Area

10. Momentum = mass x velocity

IV. PRACTICAL UNITS:

A large number of units are used in general life for measurement of different quantities in comfortable manner.

SOME PRACTICAL UNITS ARE LISTED BELOW

S.no. | Practical units of Length | Practical units of Mass | Practical units of Time |

1. | 1 light year = m | 1 quintal = | 1 year = solar days. |

2. | 1 astronomical unit or 1 AU | 1 metric ton = kg | 1 lunar month = 86400s |

3. | 1 parsec = 3.26 light year | 1 atomic mass unit (amu) = | Tropical year: It is that year in which solar eclipse occurs. |

4. | 1 angstrom = m | 1 pound = 0.4537 kg | Leap year : It is that year in which the month of February is of 29 days. |

5. | 1 fermi = m |

→ PREFIXES: (or) Abbreviations for multiples and sub–multiples of 10.

SI- PREFIXES

## ► DIMENSION OF PHYSICAL QUANTITIES:

The dimension of a physical quantity are the powers to which the base quantities are raised to represent that quantity.

Dimension of a quantity is represented by a square bracket round it. [PHYSICAL QUANTITY]

DIMENSION OF BASE / FUNDAMENTAL PHYSICAL QUANTITY :

Time | [T] |

Temperature | [K] |

Electric current | [A] / [I] |

Amount of substance | [mol] |

Luminous intensity | [Cd] |

MERIT POINT:

- In dimensional representation, magnitude of a physical quantity is not considered.
- Example- Dimension of 10kg mass : [M]

DIMENSION OF SUPPLEMENTARY PHYSICAL QUANTITES:

Supplementary physical quantities are Dimensionless.

DIMENSION OF A DERIVED PHYSICAL QUANTITES:

__► How can we write the dimensional formula for derived physical quantities?__

To find the derived dimension, follow 4 steps:-

1. Write the formula of the derived quantity.

2. Convert the formula in terms of fundamental physical quantities.

3. Write the dimension for each term.

4. Make proper algebraic combination and get the result.

Question:

Find the Dimensional formula of force.

Solution:

Dimensional formula of force = [MLT-2]

**Find the dimension of the following;**

1. Area = length x breadth

2. Volume = length x breadth x height

3. Mass density = mass / volume

4. Velocity = displacement / time

5. Acceleration = velocity / time

6. Force = mass x acceleration

7. Work = force x displacement

8. Power = work / time

9. Pressure = Force / Area

10. Momentum = mass x velocity

**→ Characteristics of Dimensions**

- Dimensions do not change with change in unit.
- Quantities with similar dimensions can be added or subtracted from each other.
- Dimensions can be obtained from the units of the physical quantities and vice versa.
- Two different quantities can have the same dimension.
- When two dimensions are multiplied or divided, it will form the dimension of the third quantity.

**→ Uses of Dimensional Analysis :**

1. Checking the correctness of an equation

- We can check an equation that it may be correct or not, by using a principle based on dimensions called principle of homogeneity.

#### PRINCIPLE OF HOMOGENEITY

According to this principle, dimension of each term of correct equation be same.

Means:- If, A = B +C

then, it will be correct

if, [A] = [B] = [C]

Question:

Check the equation,

Here,

x = distance

u = velocity / speed

= acceleration

t = time

2. To derive the relation among physical quantities.

Principle of homogeneity is powerful tool to establish the relation among various physical quantities.

Example,

Derive an expression for time period (t) of a simple pendulum, which may depend upon: mass of bob (m), length of pendulum (l) and acceleration due to gravity (g).

3. To convert a physical quantity from one system of unit into other

Let, a physical quantity is measured two times in different-different units.

DIMENSIONAL FORMULAE AND EQUATIONS :

► DIMENSIONAL FORMULA

It is an expression which shows how and which of the fundamental units are required to represent the unit of physical quantity.

DERIVED QUANTITES WITH UNITS, SYMBOL AND DIMENSIONAL FORMULA

→ 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.
- It fails to derive a relation which contains two or more than two quantities of same nature.
- it doesn’t tell whether the quantity is vector or scalar

The reliable digits plus the first uncertain digit are known as significant digits or significant figures.

► RULES TO COUNT SIGNIFICANT FIGURES:

- All the non-zero digits are significant.

For example,

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

For example,

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

For example,

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

For example,

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

For example,

- Power of 10 (like ) is not taken as significant figure.

For example

- Pure number and constants have infinite number of significant figures.

For example :Refractive index of diamond, 2.42 (has infinite number of significant)

NOTE:

- There is no effect on no. of significant figures with change in units.

Example: 20m (1 S.F) = 2000cm (1 S.F)

3.0 Km (2 S.F) = 3000m (1 S.F)

3.0 Km (2 S.F) = 3.0 x m (2 S.F)

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

► ROUNDING OFF :

The result of computation with approximate numbers, which contain more than one

uncertain digit should be rounded off.

Rules to rounding off

Rule 1.

If the digit to be dropped is less than 5, then the preceding digit is left unchanged.

For example, 7.82 = 7.8 , rounded off to 2 S.F.

Rule 2.

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

For example, 6.87 = 6.9, rounded off to 2 S.F.

12.78 = 12.8 , rounded off to 2 S.F.

Rule 3. If the digit to be dropped is 5 followed by digit other than zero, then the president digit is raised by 1.

For example, 16.351 = 16.4, rounded off to 3 S.F.

Again 6.758 = 6.8, rounded off to 2 S.F.

Rule 4.

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

For example, 3.250 becomes 3.2 on rounding off to 2 S.F

again 12.650 becomes 12.6 on rounding off.

Rule 5. If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by 1 if it is odd.

for example 3.750 =3.8,rounded off to 3 S.F.

16.150 = 16.2, rounded off to 3 S.F.

► ARITHMETIC OPERATIONS WITH SIGNIFICANT FIGURES

Addition and Subtraction

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

Example :

Multiplication and division

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

Example :

If any measurement, there is always some difference between the measured value and the true value of a quantity, which is called error.

Error = Measured value – True value

- Let a quantity is measured ‘n’ times and we get measured values – a1, a2, a3 ……… an

True value = mean value

ABSOLUTE ERROR :

- The magnitude of the difference between the measured value and the true value of a quantity is called the absolute error of the measurement.
- Or, +ve value of the error in a measurement.

MEAN ABSOLUTE ERROR :

- Mean value of all absolute errors obtained in several times of measurement of a quantity.
- It is denoted by

RELATIVE ERROR :

- The ratio of mean absolute error to the mean value or true value of quantity is called relative error in it.
- i.e Relative error =

PERCENTAGE ERROR :

- The percent value of relative error is called % error.

- % error =

We measure the period of oscillation of a simple pendulum. In successive measurement, the readings turns out to be 2.63 sec, 2.565, 2.425, 2.71 and 2.80s. Calculate the absolute errors, relative error or % error.

LEAST COUNT ERROR :

- The minimum value that can be measured accurately by an instrument is called the least count.

Examples

- The least count of a metre scale is millimetre marked as 1 mm.
- The least count of a watch having second's hand is 1 s.

PROPAGATION OF ERROR :

1. Error in sum of the quantities:

Suppose x = a + b

Let △ a = absolute error in measurement of a

△ b = absolute error in measurement of b

△ x = absolute error in calculation of x i.e. sum of a and b.

The maximum absolute error in x is △ x =±(△ a +△ b)

Percentage error in the value of x=

2. Error in difference of the quantities:

Suppose x = a – b

Let △ a = absolute error in measurement of a,

△ b = absolute error in measurement of b

△ x = absolute error in calculation of x i.e. difference of a and b.

The maximum absolute error in x is △ x =±(△ a +△ b)

Percentage error in the value of x=

3. Error in product of quantities:

Suppose x = a × b

Let △ a = absolute error in measurement of a,

△ b = absolute error in measurement of b

△ x = absolute error in calculation of x i.e. product of a and b.

The maximum fractional error in x is

Percentage error in the value of x = (Percentage error in value of a) + (Percentage error in value of b)

4. Error in division of quantities :

Suppose x=

Let △ a = absolute error in measurement of a,

△ b = absolute error in measurement of b

△ x = absolute error in calculation of x i.e. division of a and b.

The maximum fractional error in x is

Percentage error in the value of x = (Percentage error in value of a) + (Percentage error in value of b)

5. Error in quantity raised to some power :

Suppose

Let △ a = absolute error in measurement of a,

△ b = absolute error in measurement of b

△ x = absolute error in calculation of x

The maximum fractional error in x is

Percentage error in the value of x = n (Percentage error in value of a) + m (Percentage error in value of b)

RANGES AND ORDERS OF LENGTH, MASS AND TIME

PARALLAX METHOD :

Large distances such as the distance of a planet or a star from the earth cannot be measured directly with a metre scale. An important method in such cases is the parallax method.

To measure the distance D of a far away planet S by the parallax method, we observe it from two different positions (observatories) A and B on the Earth, separated by distance AB = b at the same time as shown in figure. We measure the angle between the two directions along which the planet is viewed at these two points. The ∠ASB in Figure represented by symbol Īø is called the parallax angle or parallactic angle.

As the planet is very far away, 1, b / D<< and therefore, Īø is very small. Then we approximately take AB as an arc of length b of a circle with centre at S and the distance D as the radius AS = BS so that

AB = b = D Īø where Īø is in radians.

Having determined D, we can employ a similar method to determine the size or angular diameter of the planet. If d is the diameter of the planet and Ī± the angular size of the planet (the angle subtended by d at the earth), we have

Ī± = d/D

The angle Ī± can be measured from the same location on the earth. It is the angle between the two directions when two diametrically opposite points of the planet are viewed through the telescope. Since D is known, the diameter d of the planet.

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## SUMMARY

- Units and measurements are essential for scientific research, engineering, manufacturing, commerce, and everyday life. They enable accurate and consistent communication, facilitate comparison and replication of results, ensure quality control, and support technological advancements.

- Definition of Units: Units are standardized quantities used to measure and express different physical quantities. It ensures the consistency and accuracy in measurements.

- SI Units: The International System of Units (SI) is the most widely used system of measurement. It provides a consistent and coherent set of units for physical quantities. The SI units include the meter (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for temperature, mole (mol) for amount of substance, and candela (cd) for luminous intensity.

- Derived Units: Derived units are formed by combining base units through multiplication or division. For example, the unit of speed is derived by dividing the unit of length (meter) by the unit of time (second), resulting in meters per second (m/s)

- Measurement Tools: Various instruments and devices are used for measurement in different fields. Examples include rulers and tape measures for length, balances and scales for mass, clocks and timers for time, thermometers for temperature, and voltmeters and ammeters for electrical quantities.

- Conversion: Conversion is the process of changing a measurement from one unit to another. It involves multiplying or dividing by conversion factors that relate the two units. For example, to convert kilometers to miles, you would multiply the number of kilometers by the conversion factor of 0.62137119.