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General Physics
(PHYS 101)
Introduction
Dr. Said Moh’d Azar
Physics


Fundamental Science (‫)علم اساسي‬






Concerned with the fundamental principles of the Universe
Foundation of other physical sciences (‫اساس للعلوم الفيزيائية‬
‫)الرخرى‬
Has simplicity of fundamental concepts (‫ميهافملل طيسبتبسيط للمفاهيم‬
‫)الساسية‬

Divided into six major areas (‫ميهافملل طيسبتقسم الى ستة فروع‬
‫)اساسية‬







Classical Mechanics (‫)الميكانيكا الكلسيكية‬
Relativity (‫)النسبية‬
Thermodynamics (‫)الديناميكا الحرارية‬
Electromagnetism (‫)الكهرمغناطيسية‬
Optics (‫)البصريات‬
Quantum Mechanics (‫)ميكانيكا الكم‬
Classical Physics (‫الفيزياء‬

‫)الكلسيكية‬

 Mechanics

and electromagnetism are basic
to all other branches of classical and modern
physics
 Classical physics



Developed before 1900
Our study will start with Classical Mechanics


Also called Newtonian Mechanics or Mechanics

 Modern


physics (‫)الفيزياء الحديثة‬

From about 1900 to the present
Objectives of Physics (‫اهداف‬

‫)الفيزياء‬
 To

find the limited number of fundamental
laws that govern natural phenomena
 To use these laws to develop theories that
can predict the results of future experiments
 Express the laws in the language of
mathematics


Mathematics provides the bridge between theory
and experiment
Theory and Experiments
(‫)النظرية و التجربة‬
 Should

complement each other
 When a discrepancy occurs, theory may be
modified


Theory may apply to limited conditions




Example: Newtonian Mechanics is confined to objects
traveling slowly with respect to the speed of light

Try to develop a more general theory
Syllabus
Week

Date

Chapter and Subject

1st

26/ 01/ 2014

No Class

2nd

02/ 02/ 2014

Introduction and Syllabus

3rd

09/ 02/ 2014

Ch#1:Motion in a straight line

4th

16/ 02/ 2014

Ch#2: Motion in Two
Dimensions and Vectors

5th

23/ 02/ 2014

Ch#3: Newton’s Laws of Motion

6th

02/ 03/ 2014

Ch#4: Statics: Torques, and
Equilibrium

7th

09/ 03/ 2014

8

16/ 03/ 2014

Assignments
and Notes

th

Mid-term Exam

Assignment II

Quiz I
Assignment II
Continue
9th

23/03/ 2014

10th

30/03/ 2014

Ch#6: Work, Energy and Power

11th

06/04/ 2014

Ch#13: Fluid Mechanics

12th

13/04/ 2014

13th

20/04/ 2014

14th

27/04/ 2014

15th

04/05/ 2014

16th

11/05/ 2014

17th

18/ 05/ 2014

Ch#16: Electric Forces, Fields, and
Potentials
Ch#17: Direct currents and Ohm’s
Law
Ch#24: Mirrors, Lenses, and
Imaging Systems
Ch#30: Nuclear Physics:
Radioactivity and Half-Life
Ch#31: Ionizing Radiation: The
Interaction of Radiation with Matter
Revision

18th

25/05 / 2014

19th

01/ 06/ 2014

Spring Holiday

Final Exam

Assignment III

Assignment VI
Quiz II
Assignment V
Practical Lab Experiments
Week

Date

Experiment
No.

Experiment title

1st

26/01 / 2014

2nd

02/ 02/ 2014

3rd

09/ 02/ 2014

Exp(1)

Introduction
Measurements in physics

4th

16/ 02/ 2014

Exp(2)

Errors and Uncertainty

5th

23/ 02/ 2014

Exp(3)

Simple Pendulum

6th

02/03/ 2014

Exp(4)

Force Table

7th & 8th & 9th

09/03/ 2014

No Lab

Mid term and Spring Holiday
Continue
10th
11th
12th
13th

30/03/ 2014
06/04/ 2014
13/04/ 2014

14th
15th

27/04/ 2014
04/05/ 2014

20/04/ 2014

Exp(5)
Exp(6)
Exp(7)
Exp(8)

Hook’s Law
Ohm’s Law
Quiz 2
Viscosity
Prism and
refraction
Labs compensation
Final Lab Exam
Classical Physics Overview




Classical physics includes principles in many
branches developed before 1900
Mechanics




Major developments by Newton, and continuing through
the 18th century

Thermodynamics, optics and electromagnetism



Developed in the latter part of the 19th century
Apparatus for controlled experiments became available
Modern Physics
 Began

near the end of the 19th century
 Phenomena that could not be explained by
classical physics
 Includes theories of relativity and quantum
mechanics
Special Relativity (‫النسبية‬
‫)الخاصة‬
 Correctly

describes motion of objects moving
near the speed of light
 Modifies the traditional concepts of space,
time, and energy
 Shows the speed of light is the upper limit for
the speed of an object
 Shows mass and energy are related
Quantum Mechanics
 Formulated

to describe physical phenomena
at the atomic level
 Led to the development of many practical
devices
Measurements (‫)القياسات‬
 Used

to describe natural phenomena
 Needs defined standards
 Characteristics of standards for
measurements (‫)خصائص القياسات المعيارية‬




Readily accessible (‫)بسهولة الوصول اليها‬
Possess some property that can be measured
reliably (‫)موثوقة‬
Must yield the same results when used by anyone
anywhere (‫نتيجة القياس نفسها بغض النظر عن الشخص او‬
‫)المكان‬



Cannot change with time (‫)لتتتغير مع الزمن‬
Standards of Fundamental
Quantities(‫معايير الكميات‬
‫)الساسية‬
 Standardized


Agreed upon by some authority, usually a
governmental body

 SI



systems

– Systéme International

Agreed to in 1960 by an international committee
Main system used in this text
Fundamental Quantities and
Their Units (‫الكميات الساسية و‬
‫)وحداتتها‬

Quantity

SI Unit

Length (‫)الطول‬

Meter (m)

Mass (‫)الكتلة‬

Kilogram (kg)

Time (‫)الزمن‬

Second (sec)

Temperature (‫)درجة الحرارة‬

Kelvin (K)

Electric Current (‫)التيار الكهربائي‬

Ampere (A)

Luminous Intensity (‫)شدة الاضاءة‬

Candela

Amount of Substance (‫ةيمكمية‬
‫)المادة‬

mole
Quantities Used in Mechanics
 In

mechanics, three basic quantities (‫ةيمكميات‬
‫)اساسية‬are used




Length (‫)الطول‬
Mass (‫)الكتلة‬
Time (‫)الزمن‬

 Will


also use derived quantities (‫)ةيمكميات مشتقة‬

These are other quantities that can be expressed
in terms of the basic quantities


Example: Area is the product of two lengths



Area is a derived quantity
Length is the fundamental quantity
Length (‫)الطول‬
 Length

is the distance between two points in

space
 Units


SI – meter, m

 Defined

in terms of a meter – the distance
traveled by light in a vacuum during a given
time
Mass (‫)الكتلة‬
 Units


SI – kilogram, kg

 Defined

in terms of a kilogram, based on a
specific cylinder kept at the International
Bureau of Standards
Standard Kilogram
Time
 Units


seconds, s

 Defined

in terms of the oscillation of radiation
from a cesium atom
Reasonableness of Results
(‫) معقولية النتائج‬
 When

solving a problem, you need to check
your answer to see if it seems reasonable
 Reviewing the tables of approximate values
for length, mass, and time will help you test
for reasonableness
Number Notation (‫تتدوين‬
‫)الرقام‬
 When

writing out numbers with many digits,
spacing in groups of three will be used



No commas
Standard international notation
(‫)المعيار الدولي لتدوين الرقام‬

 Examples:



25 100
5.123 456 789 12
US Customary System
(‫) نظام الوحدات المريكي‬


Still used in the US, but text will use SI
Quantity

Unit

Length

foot

Mass

slug

Time

second
Prefixes (‫البادءات او الختصارات‬
)
 Prefixes

correspond to powers of 10
 Each prefix has a specific name
 Each prefix has a specific abbreviation
Prefixes, cont.




The prefixes can be used with any basic units
They are multipliers of the basic unit
Examples:



1 mm = 10-3 m
1 mg = 10-3 g
Basic Quantities and Their
Dimension
(‫) الكميات الساسية و ابعادها‬
 Dimension

has a specific meaning – it
denotes the physical nature of a quantity
 Dimensions are denoted with square
brackets




Length [L]
Mass [M]
Time [T]
Dimensions and Units



Each dimension can have many actual units
Table 1.5 for the dimensions and units of some
derived quantities
Dimensional Analysis
(‫)تحليل البعاد‬



Technique to check the correctness of an equation
or to assist in deriving an equation
Dimensions (length, mass, time, combinations) can
be treated as algebraic quantities







add, subtract, multiply, divide (‫)الجمع,الطرح,الضرب,القسمة‬

Both sides of equation must have the same
dimensions
Any relationship can be correct only if the
dimensions on both sides of the equation are the
same
Cannot give numerical factors: this is its limitation
Dimensional Analysis,
example
Dimensional Analysis to
Determine a Power Law
Symbols (‫)الرموز‬




The symbol used in an equation is not necessarily
the symbol used for its dimension
Some quantities have one symbol used consistently




Some quantities have many symbols used,
depending upon the specific situation






For example, time is t virtually all the time

For example, lengths may be x, y, z, r, d, h, etc.

The dimensions will be given with a capitalized,
nonitalicized letter
The algebraic symbol will be italicized
Conversion of Units (‫تحول وحدات‬
‫)القياس‬

 When

units are not consistent, you may need
to convert to appropriate ones

 See

Appendix A for an extensive list of
conversion factors (‫)معاملت التحويل‬

 Units

can be treated like algebraic quantities
that can cancel each other out
‫البعاد من الصغير جدا إلى الكبير‬
‫جدا‬
Conversion (‫)تحويل الوحدات‬





Always include units for every quantity, you can
carry the units through the entire calculation
Multiply original value by a ratio equal to one
Example
15.0 in = ? cm



 2.54 cm 
15.0 in 
÷ = 38.1cm
 1in 
Note the value inside the parentheses is equal to 1 since 1
in. is defined as 2.54 cm
Order of Magnitude ()
 Approximation

based on a number of

assumptions


may need to modify assumptions if more precise
results are needed

 Order

of magnitude is the power of 10 that
applies
Order of Magnitude – Process


Estimate a number and express it in scientific
notation





The multiplier of the power of 10 needs to be between 1
and 10

Divide the number by the power of 10
Compare the remaining value to 3.162 ( 10 )




If the remainder is less than 3.162, the order of magnitude
is the power of 10 in the scientific notation
If the remainder is greater than 3.162, the order of
magnitude is one more than the power of 10 in the
scientific notation
Using Order of Magnitude
 Estimating

too high for one number is often
canceled by estimating too low for another
number


The resulting order of magnitude is generally
reliable within about a factor of 10

 Working

the problem allows you to drop
digits, make reasonable approximations and
simplify approximations
 With practice, your results will become better
and better
Uncertainty in Measurements
(‫) عدم التيقن في القياس‬
 There

is uncertainty in every measurement –
this uncertainty carries over through the
calculations




May be due to the apparatus, the experimenter,
and/or the number of measurements made
Need a technique to account for this uncertainty

 We

will use rules for significant figures to
approximate the uncertainty in results of
calculations
Significant Figures (‫الرقام‬

‫)المعنوية‬


A significant figure is one that is reliably known



Zeros may or may not be significant





Those used to position the decimal point are not significant
To remove ambiguity, use scientific notation

In a measurement, the significant figures include the
first estimated digit
Significant Figures, examples


0.0075 m has 2 significant figures





10.0 m has 3 significant figures




The leading zeros are placeholders only
Can write in scientific notation to show more clearly:
7.5 x 10-3 m for 2 significant figures
The decimal point gives information about the reliability of
the measurement

1500 m is ambiguous




Use 1.5 x 103 m for 2 significant figures
Use 1.50 x 103 m for 3 significant figures
Use 1.500 x 103 m for 4 significant figures
Operations with Significant
Figures – Multiplying or Dividing
 When

multiplying or dividing, the number of
significant figures in the final answer is the
same as the number of significant figures in
the quantity having the lowest number of
significant figures.
 Example: 25.57 m x 2.45 m = 62.6 m2


The 2.45 m limits your result to 3 significant
figures
Operations with Significant
Figures – Adding or Subtracting
 When

adding or subtracting, the number of
decimal places in the result should equal the
smallest number of decimal places in any
term in the sum.
 Example: 135 cm + 3.25 cm = 138 cm


The 135 cm limits your answer to the units
decimal value
Operations With Significant
Figures – Summary






The rule for addition and subtraction are different
than the rule for multiplication and division
For adding and subtracting, the number of decimal
places is the important consideration
For multiplying and dividing, the number of
significant figures is the important consideration
Rounding (‫)التقريب‬





Last retained digit is increased by 1 if the last digit
dropped is greater than 5
Last retained digit remains as it is if the last digit
dropped is less than 5
If the last digit dropped is equal to 5, the retained
digit should be rounded to the nearest even number
Saving rounding until the final result will help
eliminate accumulation of errors

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Introduction

  • 2. Physics  Fundamental Science (‫)علم اساسي‬     Concerned with the fundamental principles of the Universe Foundation of other physical sciences (‫اساس للعلوم الفيزيائية‬ ‫)الرخرى‬ Has simplicity of fundamental concepts (‫ميهافملل طيسبتبسيط للمفاهيم‬ ‫)الساسية‬ Divided into six major areas (‫ميهافملل طيسبتقسم الى ستة فروع‬ ‫)اساسية‬       Classical Mechanics (‫)الميكانيكا الكلسيكية‬ Relativity (‫)النسبية‬ Thermodynamics (‫)الديناميكا الحرارية‬ Electromagnetism (‫)الكهرمغناطيسية‬ Optics (‫)البصريات‬ Quantum Mechanics (‫)ميكانيكا الكم‬
  • 3. Classical Physics (‫الفيزياء‬ ‫)الكلسيكية‬  Mechanics and electromagnetism are basic to all other branches of classical and modern physics  Classical physics   Developed before 1900 Our study will start with Classical Mechanics  Also called Newtonian Mechanics or Mechanics  Modern  physics (‫)الفيزياء الحديثة‬ From about 1900 to the present
  • 4. Objectives of Physics (‫اهداف‬ ‫)الفيزياء‬  To find the limited number of fundamental laws that govern natural phenomena  To use these laws to develop theories that can predict the results of future experiments  Express the laws in the language of mathematics  Mathematics provides the bridge between theory and experiment
  • 5. Theory and Experiments (‫)النظرية و التجربة‬  Should complement each other  When a discrepancy occurs, theory may be modified  Theory may apply to limited conditions   Example: Newtonian Mechanics is confined to objects traveling slowly with respect to the speed of light Try to develop a more general theory
  • 6. Syllabus Week Date Chapter and Subject 1st 26/ 01/ 2014 No Class 2nd 02/ 02/ 2014 Introduction and Syllabus 3rd 09/ 02/ 2014 Ch#1:Motion in a straight line 4th 16/ 02/ 2014 Ch#2: Motion in Two Dimensions and Vectors 5th 23/ 02/ 2014 Ch#3: Newton’s Laws of Motion 6th 02/ 03/ 2014 Ch#4: Statics: Torques, and Equilibrium 7th 09/ 03/ 2014 8 16/ 03/ 2014 Assignments and Notes th Mid-term Exam Assignment II Quiz I Assignment II
  • 7. Continue 9th 23/03/ 2014 10th 30/03/ 2014 Ch#6: Work, Energy and Power 11th 06/04/ 2014 Ch#13: Fluid Mechanics 12th 13/04/ 2014 13th 20/04/ 2014 14th 27/04/ 2014 15th 04/05/ 2014 16th 11/05/ 2014 17th 18/ 05/ 2014 Ch#16: Electric Forces, Fields, and Potentials Ch#17: Direct currents and Ohm’s Law Ch#24: Mirrors, Lenses, and Imaging Systems Ch#30: Nuclear Physics: Radioactivity and Half-Life Ch#31: Ionizing Radiation: The Interaction of Radiation with Matter Revision 18th 25/05 / 2014 19th 01/ 06/ 2014 Spring Holiday Final Exam Assignment III Assignment VI Quiz II Assignment V
  • 8. Practical Lab Experiments Week Date Experiment No. Experiment title 1st 26/01 / 2014 2nd 02/ 02/ 2014 3rd 09/ 02/ 2014 Exp(1) Introduction Measurements in physics 4th 16/ 02/ 2014 Exp(2) Errors and Uncertainty 5th 23/ 02/ 2014 Exp(3) Simple Pendulum 6th 02/03/ 2014 Exp(4) Force Table 7th & 8th & 9th 09/03/ 2014 No Lab Mid term and Spring Holiday
  • 9. Continue 10th 11th 12th 13th 30/03/ 2014 06/04/ 2014 13/04/ 2014 14th 15th 27/04/ 2014 04/05/ 2014 20/04/ 2014 Exp(5) Exp(6) Exp(7) Exp(8) Hook’s Law Ohm’s Law Quiz 2 Viscosity Prism and refraction Labs compensation Final Lab Exam
  • 10. Classical Physics Overview   Classical physics includes principles in many branches developed before 1900 Mechanics   Major developments by Newton, and continuing through the 18th century Thermodynamics, optics and electromagnetism   Developed in the latter part of the 19th century Apparatus for controlled experiments became available
  • 11. Modern Physics  Began near the end of the 19th century  Phenomena that could not be explained by classical physics  Includes theories of relativity and quantum mechanics
  • 12. Special Relativity (‫النسبية‬ ‫)الخاصة‬  Correctly describes motion of objects moving near the speed of light  Modifies the traditional concepts of space, time, and energy  Shows the speed of light is the upper limit for the speed of an object  Shows mass and energy are related
  • 13. Quantum Mechanics  Formulated to describe physical phenomena at the atomic level  Led to the development of many practical devices
  • 14. Measurements (‫)القياسات‬  Used to describe natural phenomena  Needs defined standards  Characteristics of standards for measurements (‫)خصائص القياسات المعيارية‬    Readily accessible (‫)بسهولة الوصول اليها‬ Possess some property that can be measured reliably (‫)موثوقة‬ Must yield the same results when used by anyone anywhere (‫نتيجة القياس نفسها بغض النظر عن الشخص او‬ ‫)المكان‬  Cannot change with time (‫)لتتتغير مع الزمن‬
  • 15. Standards of Fundamental Quantities(‫معايير الكميات‬ ‫)الساسية‬  Standardized  Agreed upon by some authority, usually a governmental body  SI   systems – Systéme International Agreed to in 1960 by an international committee Main system used in this text
  • 16. Fundamental Quantities and Their Units (‫الكميات الساسية و‬ ‫)وحداتتها‬ Quantity SI Unit Length (‫)الطول‬ Meter (m) Mass (‫)الكتلة‬ Kilogram (kg) Time (‫)الزمن‬ Second (sec) Temperature (‫)درجة الحرارة‬ Kelvin (K) Electric Current (‫)التيار الكهربائي‬ Ampere (A) Luminous Intensity (‫)شدة الاضاءة‬ Candela Amount of Substance (‫ةيمكمية‬ ‫)المادة‬ mole
  • 17. Quantities Used in Mechanics  In mechanics, three basic quantities (‫ةيمكميات‬ ‫)اساسية‬are used    Length (‫)الطول‬ Mass (‫)الكتلة‬ Time (‫)الزمن‬  Will  also use derived quantities (‫)ةيمكميات مشتقة‬ These are other quantities that can be expressed in terms of the basic quantities  Example: Area is the product of two lengths   Area is a derived quantity Length is the fundamental quantity
  • 18. Length (‫)الطول‬  Length is the distance between two points in space  Units  SI – meter, m  Defined in terms of a meter – the distance traveled by light in a vacuum during a given time
  • 19. Mass (‫)الكتلة‬  Units  SI – kilogram, kg  Defined in terms of a kilogram, based on a specific cylinder kept at the International Bureau of Standards
  • 21. Time  Units  seconds, s  Defined in terms of the oscillation of radiation from a cesium atom
  • 22. Reasonableness of Results (‫) معقولية النتائج‬  When solving a problem, you need to check your answer to see if it seems reasonable  Reviewing the tables of approximate values for length, mass, and time will help you test for reasonableness
  • 23. Number Notation (‫تتدوين‬ ‫)الرقام‬  When writing out numbers with many digits, spacing in groups of three will be used   No commas Standard international notation (‫)المعيار الدولي لتدوين الرقام‬  Examples:   25 100 5.123 456 789 12
  • 24. US Customary System (‫) نظام الوحدات المريكي‬  Still used in the US, but text will use SI Quantity Unit Length foot Mass slug Time second
  • 25. Prefixes (‫البادءات او الختصارات‬ )  Prefixes correspond to powers of 10  Each prefix has a specific name  Each prefix has a specific abbreviation
  • 26. Prefixes, cont.    The prefixes can be used with any basic units They are multipliers of the basic unit Examples:   1 mm = 10-3 m 1 mg = 10-3 g
  • 27. Basic Quantities and Their Dimension (‫) الكميات الساسية و ابعادها‬  Dimension has a specific meaning – it denotes the physical nature of a quantity  Dimensions are denoted with square brackets    Length [L] Mass [M] Time [T]
  • 28. Dimensions and Units   Each dimension can have many actual units Table 1.5 for the dimensions and units of some derived quantities
  • 29. Dimensional Analysis (‫)تحليل البعاد‬   Technique to check the correctness of an equation or to assist in deriving an equation Dimensions (length, mass, time, combinations) can be treated as algebraic quantities     add, subtract, multiply, divide (‫)الجمع,الطرح,الضرب,القسمة‬ Both sides of equation must have the same dimensions Any relationship can be correct only if the dimensions on both sides of the equation are the same Cannot give numerical factors: this is its limitation
  • 32. Symbols (‫)الرموز‬   The symbol used in an equation is not necessarily the symbol used for its dimension Some quantities have one symbol used consistently   Some quantities have many symbols used, depending upon the specific situation    For example, time is t virtually all the time For example, lengths may be x, y, z, r, d, h, etc. The dimensions will be given with a capitalized, nonitalicized letter The algebraic symbol will be italicized
  • 33. Conversion of Units (‫تحول وحدات‬ ‫)القياس‬  When units are not consistent, you may need to convert to appropriate ones  See Appendix A for an extensive list of conversion factors (‫)معاملت التحويل‬  Units can be treated like algebraic quantities that can cancel each other out
  • 34.
  • 35.
  • 36. ‫البعاد من الصغير جدا إلى الكبير‬ ‫جدا‬
  • 37. Conversion (‫)تحويل الوحدات‬    Always include units for every quantity, you can carry the units through the entire calculation Multiply original value by a ratio equal to one Example 15.0 in = ? cm   2.54 cm  15.0 in  ÷ = 38.1cm  1in  Note the value inside the parentheses is equal to 1 since 1 in. is defined as 2.54 cm
  • 38. Order of Magnitude ()  Approximation based on a number of assumptions  may need to modify assumptions if more precise results are needed  Order of magnitude is the power of 10 that applies
  • 39. Order of Magnitude – Process  Estimate a number and express it in scientific notation    The multiplier of the power of 10 needs to be between 1 and 10 Divide the number by the power of 10 Compare the remaining value to 3.162 ( 10 )   If the remainder is less than 3.162, the order of magnitude is the power of 10 in the scientific notation If the remainder is greater than 3.162, the order of magnitude is one more than the power of 10 in the scientific notation
  • 40. Using Order of Magnitude  Estimating too high for one number is often canceled by estimating too low for another number  The resulting order of magnitude is generally reliable within about a factor of 10  Working the problem allows you to drop digits, make reasonable approximations and simplify approximations  With practice, your results will become better and better
  • 41. Uncertainty in Measurements (‫) عدم التيقن في القياس‬  There is uncertainty in every measurement – this uncertainty carries over through the calculations   May be due to the apparatus, the experimenter, and/or the number of measurements made Need a technique to account for this uncertainty  We will use rules for significant figures to approximate the uncertainty in results of calculations
  • 42. Significant Figures (‫الرقام‬ ‫)المعنوية‬  A significant figure is one that is reliably known  Zeros may or may not be significant    Those used to position the decimal point are not significant To remove ambiguity, use scientific notation In a measurement, the significant figures include the first estimated digit
  • 43. Significant Figures, examples  0.0075 m has 2 significant figures    10.0 m has 3 significant figures   The leading zeros are placeholders only Can write in scientific notation to show more clearly: 7.5 x 10-3 m for 2 significant figures The decimal point gives information about the reliability of the measurement 1500 m is ambiguous    Use 1.5 x 103 m for 2 significant figures Use 1.50 x 103 m for 3 significant figures Use 1.500 x 103 m for 4 significant figures
  • 44. Operations with Significant Figures – Multiplying or Dividing  When multiplying or dividing, the number of significant figures in the final answer is the same as the number of significant figures in the quantity having the lowest number of significant figures.  Example: 25.57 m x 2.45 m = 62.6 m2  The 2.45 m limits your result to 3 significant figures
  • 45. Operations with Significant Figures – Adding or Subtracting  When adding or subtracting, the number of decimal places in the result should equal the smallest number of decimal places in any term in the sum.  Example: 135 cm + 3.25 cm = 138 cm  The 135 cm limits your answer to the units decimal value
  • 46. Operations With Significant Figures – Summary    The rule for addition and subtraction are different than the rule for multiplication and division For adding and subtracting, the number of decimal places is the important consideration For multiplying and dividing, the number of significant figures is the important consideration
  • 47. Rounding (‫)التقريب‬     Last retained digit is increased by 1 if the last digit dropped is greater than 5 Last retained digit remains as it is if the last digit dropped is less than 5 If the last digit dropped is equal to 5, the retained digit should be rounded to the nearest even number Saving rounding until the final result will help eliminate accumulation of errors