11P02

Units and Measurements

Introduction

Learning Objective

• Physical Quantities and Units
• Measurements
• Errors in Measurements
• Significant Figures
• Dimensional Analysis and its Application

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11P02.1

Physical Quantities and Units

11P02.1 Physical Quantities and Units

Learning Objective

• Introduction to Physical Quantities
• Units
• System of Units

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CV-1

Introduction to Physical Quantities

11P02.1 Physical Quantities and Units

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Physical Quantity

11P02.1 Physical Quantities and Units

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Can be Measured

Physical Quantity

11P02.1 Physical Quantities and Units

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Can be Measured

Is Mass a Physical Quantity?

Yes, Mass is a physical quantity we can measure mass by balance and weighing machines

Physical Quantity

Is Fever a Physical Quantity?

No, Doctor did not measure fever he measures temperature. Temperature is a Physical Quantity which is measured by the thermometer

11P02.1 Physical Quantities and Units

Physical Quantity

11P02.1 Physical Quantities and Units

Physical Quantity

Fundamental Quantity

11P02.1 Physical Quantities and Units

Physical Quantity

Derived Quantity

Fundamental Quantity

11P02.1 Physical Quantities and Units

Physical Quantity

Derived Quantity

Fundamental Quantity

Example: Mass, length,time etc.

11P02.1 Physical Quantities and Units

Physical Quantity

Derived Quantity

Fundamental Quantity

Example: Velocity, Force,work etc.

Example: Mass, length,time etc.

11P02.1 Physical Quantities and Units

Physical Quantity

Derived Quantity

Fundamental Quantity

Example: Velocity, Force,work etc.

Example: Mass, length,time etc.

CV-2

Introduction to Measurements

11P02.1 Physical Quantities and Units

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Measuring a Physical Quantity

11P02.1 Physical Quantities and Units

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Measuring a Physical Quantity

11P02.1 Physical Quantities and Units

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Measuring a Physical Quantity

Comparison

11P02.1 Physical Quantities and Units

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Measuring a Physical Quantity

Comparison

11P02.1 Physical Quantities and Units

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Unit

Measuring a Physical Quantity

Comparison

11P02.1 Physical Quantities and Units

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Comparison with Reference Standard

11P02.1 Physical Quantities and Units

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Representing a Physical Quantity

11P02.1 Physical Quantities and Units

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Physical Quantity = Magnitude(n) × Unit(u) = n u

11P02.1 Physical Quantities and Units

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Example:Length = 28 m

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Magnitude

Unit

Physical Quantity = Magnitude(n) × Unit(u) = n u

11P02.1 Physical Quantities and Units

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11P02.1 Physical Quantities and Units

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CV-3

Introduction to Units

11P02.1 Physical Quantities and Units

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Units

11P02.1 Physical Quantities and Units

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Units

Fundamental/Base Units

11P02.1 Physical Quantities and Units

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Units

Fundamental/Base Units

Metre, Kilogram etc.

11P02.1 Physical Quantities and Units

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Units

Fundamental/Base Units

Derived Units

Metre, Kilogram etc.

11P02.1 Physical Quantities and Units

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Units

Fundamental/Base Units

Derived Units

Metre, Kilogram etc.

Newton (kg ms^-2), ms^-1 etc.

11P02.1 Physical Quantities and Units

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System of Units

11P02.1 Physical Quantities and Units

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System of Units

Complete set of units

11P02.1 Physical Quantities and Units

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System of Units

MKS

Complete set of units

11P02.1 Physical Quantities and Units

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System of Units

MKS

CGS

Complete set of units

11P02.1 Physical Quantities and Units

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System of Units

MKS

FPS

CGS

Complete set of units

11P02.1 Physical Quantities and Units

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Metre, kilogram and second respectively

System of Units

MKS

FPS

CGS

Base units for length, mass and time

Complete set of units

11P02.1 Physical Quantities and Units

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Metre, kilogram and second respectively

System of Units

MKS

FPS

Centimetre, gram and second respectively

CGS

Base units for length, mass and time

Complete set of units

11P02.1 Physical Quantities and Units

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Metre, kilogram and second respectively

System of Units

MKS

FPS

Centimetre, gram and second respectively

Foot, pound and second respectively

CGS

Base units for length, mass and time

Complete set of units

11P02.1 Physical Quantities and Units

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SI System

11P02.1 Physical Quantities and Units

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International System of units

SI System

11P02.1 Physical Quantities and Units

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International System of units

SI System

Seven Fundamental Units and two Supplementary Units

11P02.1 Physical Quantities and Units

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International System of units

SI System

Seven Fundamental Units and two Supplementary Units

Fundamental unit

11P02.1 Physical Quantities and Units

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International System of units

SI System

Seven Fundamental Units and two Supplementary Units

Fundamental unit

Supplimentry Units

11P02.1 Physical Quantities and Units

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Plane angle

11P02.1 Physical Quantities and Units

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Solid angle

11P02.1 Physical Quantities and Units

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11P02.2

Measurements

CV-1

Measurements of Length

11P02.2 Measurements

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Measurement of Length

Direct Method

Indirect Method

1. Measurements of Very Large Distances (Distance of a planet from earth)
2. Measurements of very small Distances (Diameter of a Molecule)

1. By Meter Scale

(Accuracy of 10^-3 m)

2. By Vernier Callipers (Accuracy of 10^-4 m)
3. By Spherometer or Screw Gauge(Accuracy of 10^-5 m)

11P02.2 Measurements

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Measurements of Large distances such as the distance of a planet or a star from the earth

Parallax Method

11P02.2 Measurements

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Right

B

base

A

Left

11P02.2 Measurements

Parallax Method:

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D

D

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O₂

O₁

Parallax angle

Basis

11P02.2 Measurements

θ

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D

D

b

O₂

O₁

Parallax Method:

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11P02.2 Measurements

Parallax Method:

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θ

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D

D

b

O₂

O₁

11P02.2 Measurements

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Estimation of Diameter of a Star by Parallax Method:

d

O

D

D

α

11P02.2 Measurements

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Estimation of Diameter of a Star by Parallax Method:

d

O

D

D

α

11P02.2 Measurements

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Estimation of Diameter of a Star by Parallax Method:

d

O

D

D

α

CV-2

Measurements of Very Small Distances

11P02.2 Measurements

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Estimation of Very Small Distances: Size of a Oleic Acid Molecule

11P02.2 Measurements

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Estimation of Very Small Distances: Size of a Molecule

11P02.2 Measurements

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Estimation of Very Small Distances: Size of a Molecule

Lycopodium Powder

11P02.2 Measurements

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Estimation of Very Small Distances: Size of a Molecule

11P02.2 Measurements

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Estimation of Very Small Distances: Size of a Molecule

11P02.2 Measurements

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Estimation of Very Small Distances: Size of a Molecule

CV-3

Measurements of Mass and Time

11P02.2 Measurements

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Measurements of Mass:

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• Mass of commonly available objects can be determined by a common balance.

11P02.2 Measurements

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Measurements of Mass:

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• Mass of commonly available objects can be determined by a common balance.

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• Large masses in the universe like planets, stars, etc., can be measured by using gravitational method (Chapter 8).

11P02.2 Measurements

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Measurements of Mass:

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• Mass of commonly available objects can be determined by a common balance.

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• Large masses in the universe like planets, stars, etc., can be measured by using gravitational method (Chapter 8).

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• Measurement of small masses of atomic/subatomic particles etc., we make use of mass spectrograph in which radius of the trajectory is proportional to the mass of a charged particle moving in uniform electric and magnetic field.

11P02.2 Measurements

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Range of Masses:

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11P02.2 Measurements

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Measurement of Time:

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Atomic standard of time

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11P02.2 Measurements

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Measurement of Time:

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Cesium clock

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Atomic standard of time

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11P02.2 Measurements

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Measurement of Time:

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Cesium clock

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Atomic standard of time

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1 second = time needed for 9,192,631,770 vibrations of the radiation corresponding to the transition between the two hyperfine levels of the ground state of cesium-133 atom

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Concept Test

Ready for Challenge

11P02.3

Errors in Measurements

CV-1

Accuracy and Precision

11P02.3 Errors in Measurements

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Errors

Every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error

11P02.3 Errors in Measurements

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Accuracy

Precision

It is a measure of how close is a measured value to the true value

It tells us, to what resolution or limit is the quantity measured

11P02.3 Errors in Measurements

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A

B

True length L =3.678 cm

11P02.3 Errors in Measurements

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A

B

True length L =3.678 cm

Instrument-1(resolution 0.1 cm)

L₁ = 3.5 cm

11P02.3 Errors in Measurements

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A

B

True length L =3.678 cm

Instrument-2(vernier calliper) (resolution 0.01 cm)L₂ = 3.38 cm

11P02.3 Errors in Measurements

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A

B

True length L =3.678 cm

Instrument-2(vernier calliper) (resolution 0.01 cm)L₂ = 3.38 cm

Instrument-1(resolution 0.1 cm)

L₁ = 3.5 cm

Less precise but more accurate

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11P02.3 Errors in Measurements

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A

B

True length L =3.678 cm

Instrument-2(vernier calliper) (resolution 0.01 cm)L₂ = 3.38 cm

Instrument-1(resolution 0.1 cm)

L₁ = 3.5 cm

Less accurate but more precise

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Less precise but more accurate

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11P02.3 Errors in Measurements

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11P02.3 Errors in Measurements

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11P02.3 Errors in Measurements

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11P02.3 Errors in Measurements

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11P02.3 Errors in Measurements

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Concept Test

Ready for Challenge

Q. Two clocks are being tested against a standard clock located in a national laboratory. At 12:00:00 noon by the standard clock, the readings of the two clocks are:

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If you are doing an experiment that requires precision time interval measurements, which of the two clocks will you prefer ?

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Sol.

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Clock 1

More accurate Less precise

Sol.

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Clock 1

More accurate Less precise

Clock 2

Less accurate More precise

Sol.

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Clock 1

More accurate Less precise

Clock 2

Less accurate More precise

Clock 2 is Preferred Because It is more precise

CV-2

Errors in Measurements

11P02.3 Errors in Measurements

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Theory of Errors

Errors

11P02.3 Errors in Measurements

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Theory of Errors

Errors

Systematic Errors

11P02.3 Errors in Measurements

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Theory of Errors

Errors

Systematic Errors

Random Errors

11P02.3 Errors in Measurements

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Systematic Errors

11P02.3 Errors in Measurements

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Systematic Errors

Errors tend to be in one direction, either positive or negative

11P02.3 Errors in Measurements

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Systematic Errors

Instrumental errors

Source

Errors tend to be in one direction, either positive or negative

11P02.3 Errors in Measurements

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Systematic Errors

Instrumental errors

Error due to Imperfection in experimental technique

Source

Errors tend to be in one direction, either positive or negative

11P02.3 Errors in Measurements

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Systematic Errors

Instrumental errors

Error due to Imperfection in experimental technique

Personal errors

Source

Errors tend to be in one direction, either positive or negative

11P02.3 Errors in Measurements

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Systematic Errors

Instrumental errors

Error due to Imperfection in experimental technique

Personal errors

Source

Can be minimised by improving experimental techniques, selecting better instruments

Errors tend to be in one direction, either positive or negative

11P02.3 Errors in Measurements

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Random errors

11P02.3 Errors in Measurements

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Random errors

Due to random and unpredictable fluctuations in experimental conditions

Random with respect

to sign and size

Can be minimized by repeatedly taking readings

11P02.3 Errors in Measurements

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Least Count

11P02.3 Errors in Measurements

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Smallest value which can be measure by the Instrument

Least Count

11P02.3 Errors in Measurements

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Smallest value which can be measure by the Instrument

Least Count

11P02.3 Errors in Measurements

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In the measurement of the object, it is shown between two division lines, so then the observer needs to take the approximation for the measurement.

When a measurement falls between two divisions, then errors due to approximate measurement made by the observer

Smallest value which can be measure by the Instrument

Least Count

2.6 2.7

Least Count Error

CV-3

Calculation of Errors

11P02.3 Errors in Measurements

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MEASUREMENT OF ERRORS

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Values obtained in several measurements are

Absolute Error

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

11P02.3 Errors in Measurements

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MEASUREMENT OF ERRORS

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Relative error

The relative error is the ratio of the mean absolute error ∆a_mean to the mean value a_mean of the quantity measured

11P02.3 Errors in Measurements

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MEASUREMENT OF ERRORS

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Percentage error

When the relative error is expressed in percent, it is called the percentage error (δa)

Concept Test

Ready for Challenge

Pause Video

Time Duration : 2 Minutes

Errors in Measurements are:

CV-4

Errors in Measurements

11P02.3 Errors in Measurements

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Propagation of Errors

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Errors in Measurement of Mass

Errors in Measurement of Volume

Errors in Density ??

11P02.3 Errors in Measurements

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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

Errors of a Sum or a Difference

11P02.3 Errors in Measurements

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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

Errors of a multiplication or devidation

11P02.3 Errors in Measurements

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

Error in case of a Measured Quantity Raised to a Power

Concept Test

Ready for Challenge

Pause Video

Time Duration : 2 Minutes

11P02.4

Significant Figures

CV-1

Significant Figures

11P02.4 Significant Figures

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Significant Figures: The significant figures are normally those digits in a measured

quantity which are known reliably plus one additional digit that is uncertain.

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Example: The length of an object reported after measurement to be 287.5 cm

287.5

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Known Reliably

Uncertain(Lowest Significant Digit or LSD)

11P02.4 Significant Figures

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Rules for determining the number of Significant Figures:

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I. 00123.4 SD=4

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II. 021400 SD=3

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III. 431.00 SD=5

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IV. 01203.050 SD=7

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11P02.4 Significant Figures

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Rules for determining the number of Significant Figures:

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I. 00123.4 SD=4

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II. 021400 SD=3

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III. 431.00 SD=5

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IV. 01203.050 SD=7

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11P02.4 Significant Figures

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Rules for determining the number of Significant Figures:

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I. 00123.4 SD=4

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II. 021400 SD=3

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III. 431.00 SD=5

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IV. 01203.050 SD=7

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11P02.4 Significant Figures

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Rules for determining the number of Significant Figures:

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I. 00123.4 SD=4

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II. 021400 SD=3

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III. 431.00 SD=5

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IV. 01203.050 SD=7

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11P02.4 Significant Figures

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Rules for determining the number of Significant Figures:

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I. 00123.4 SD=4

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II. 021400 SD=3

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III. 431.00 SD=5

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IV. 01203.050 SD=7

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11P02.4 Significant Figures

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Rounding off the Uncertain Digits:

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We look at the uncertain digit ‘d’

Rule 1: If d<5, drop ‘d’

Rule 2: If d>5, increase the preceding digit by 1 and

drop ‘d’

Rule 3: If d=5, look at the preceding digit ‘c’

(i) If ‘c’ is even drop ‘d’

(ii) If ‘c’ is odd increase ‘c’ by 1 and drop ‘d’

11P02.4 Significant Figures

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Rules for Arithmetic Operations with Significant Figures:

1. Addition or Subtraction:

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

Example: Find sum of 12.4, 3.44, 0.027 and subtract 2.75 by 0.3

Solution:

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11P02.4 Significant Figures

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Concept Test

Ready for Challenge

Q. Of the following, the correct statement may be incorrect

1. A dimensionally correct equation may be incorrect
2. A dimensionally incorrect equation may be correct
3. A dimensionally correct equation may be correct
4. A dimensionally incorrect equation may be incorrect

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Sol. (c) a & c

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11P02.5

Dimensional Analysis and its Application

11P02.5 Dimensional Analysis and its Application

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Dimension:

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5 kg

1 m

1 m

0.5 m

11P02.5 Dimensional Analysis and its Application

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11P02.5 Dimensional Analysis and its Application

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Symbolic Representation of Dimension:

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11P02.5 Dimensional Analysis and its Application

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To Find the Dimensional Formula and Units:

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11P02.5 Dimensional Analysis and its Application

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Checking the Dimensional Consistency of Equations:

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Principle of homogeneity of dimensions

If the dimensions of all the terms

are not same, the equation is wrong

11P02.5 Dimensional Analysis and its Application

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Checking the Dimensional Consistency of Equations:

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Principle of homogeneity of dimensions

If the dimensions of all the terms

are not same, the equation is wrong

if an equation fails consistency test, it is proved wrong, but if it passes, it is not proved right. Thus, a dimensionally correct equation need not be actually an exact (correct) equation, but a dimensionally wrong (incorrect) or inconsistent equation must be wrong

11P02.5 Dimensional Analysis and its Application

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11P02.5 Dimensional Analysis and its Application

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Deducing Relation among the Physical Quantities:

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