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Prepared by Nurjuliana Juhari Mohamad Nazri Abdul Halim School of Microelectronic Engineering

CHAPTER 1: UNITS & DIMENSION. Edited by NURJULIANA JUHARI. Prepared by Nurjuliana Juhari Mohamad Nazri Abdul Halim School of Microelectronic Engineering. Objectives:.

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Prepared by Nurjuliana Juhari Mohamad Nazri Abdul Halim School of Microelectronic Engineering

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  1. CHAPTER 1: UNITS & DIMENSION Edited by NURJULIANA JUHARI Prepared by Nurjuliana Juhari Mohamad Nazri Abdul Halim School of Microelectronic Engineering

  2. Objectives: • Ability to define and understand base and derived quantities, distinguish standard units and sistem of unit, and fundamental quantities. • Ability to understand and apply converting units within a system or from one system of unit to another • Ability to understand and apply Dimensional Analysis.

  3. PHYSICAL QUANTITIES • Physics is based on quantities known as physical quantities. Eg: length, mass, time, force and pressure. • Generally, physical quantity is a quantity that can be measured. • Physical quantities are divided into 2 groups:

  4. Definition Units • "Units" is a physical quantity can be counted or measured using standard size. Measured units are specific values of dimensions defined by law or custom. Many different units can be used for a single dimension, as inches, miles, and centimeters are all units used to measure the dimension length. • Every measurement or quantitative statement requires a unit.  If you say you’re driving a car 30 that doesn't mean anything. Am you’re driving it 30 miles/hour, 30 km/hour, or 30 ft/sec.  30 only means something when you’re attach a unit to it.  What is the speed of light in a vacuum? 186,000 miles/sec or 3 x 108 m/s.  The number depends on the units.

  5. Standard Unit • The elements of substances and motion. • If unit becomes officially accepted, it’s called Standard Unit • Group of Unit and Combination is called SYSTEM OF UNITS. • All things in classical mechanics can be expressed in terms of the fundamental quantities: • Length (L) , MASS (M), TIME (T) • Some examples of more complicated quantities: • Speed has the quantity of L / T (i.e. kmph or mph). • Acceleration has the quantity of L/T2. • Force has the quantity of ML/ T2 (as you will learn).

  6. SI = International Systems of Units • Includes: > base quantities – kilogram and meter >derived quantities – m/h, m/s, N, J, etc. The International System of Units (abbreviated SI from the French Le Système international d'unités) is the modern form of the metric system. It is the world's most widely used system of units, both in everyday commerce and in science. Based on International Committee for Weights and Measures (CIPM), 1954 & 1960 – seven base unit recommended.

  7. Units • British Units: • L = inches, feet, miles, • M = slugs (pounds), • T = seconds • SI (Système International) Units also called Metric System • We will use mostly SI units, but you may run across some problems using British units. You should know how to convert back & forth.

  8. Base Quantities • By convention, physical quantities are organized in a dimensional system built upon base quantities, each of which is regarded as having its own dimension. In the SI system of units, there are seven (7) base units, but other conventions may have a different number of fundamental units.

  9. The base quantities according to the International System of Quantities (ISQ) and their dimensions are listed in the following table:

  10. SI derived units/Derived Quantities • SI derived units are part of the SI system of measurement units and are derived from the seven SI base units.

  11. Example 1: • Force, F =ma Solution: m – mass (kg); a – acceleration (m/s2) Thus, F = kgms-2 (b) Momentum, p =mv Solution: m – mass (kg); v – velocity (m/s) Thus,p = kgms-1

  12. Systems of Units

  13. Other systems of Units • Convert one system of unit to another. • This done by using conversion factors. • C.G.S (cm gram sec) • Eg 2: 1gcm-3 to MKS units?

  14. SI Prefixes

  15. Dimension: Definition A “Dimension" can be measured or derived. The "fundamental dimensions" (length, time, mass, temperature, amount) are distinct and are sufficient to define all the others. We also use many derived dimensions (velocity, volume, density, etc.) for convenience.

  16. Dimension • In common usage, a dimension (Latin, "measured out") is a parameter or measurement required to define thecharacteristics of an object - i.e.length, width, and height or size and shape. • In mathematics, dimensions are the parameters required to describe the position and relevant characteristics of any object within a conceptual space —where the dimensions of a space are the total number of different parameters used for all possible objects considered in the model.

  17. Units & Dimension • Many people aren’t sure of the difference. Let’s try and get a set • of definitions we can use. Consider: • 110mg of sodium • 24 hands high • 5 gal of gasoline

  18. Dimensional Analysis • PURPOSES: 1) TO CHECK THE EQUATION 2) ANALYSIS DIMENSION TO BUILD FORMULA • This is a very important tool to check your work • It’s also very easy! • Example 3 (to check equation): Doing a problem you get the answer for distance d = v t 2( velocity x time2 ) Dimension on left side [d] = L Dimension on right side [vt2] = L / T x T2 = L x T • Left units and right units don’t match, so answer must be wrong !!

  19. Example 4 (analysis to build formula) Velocity of volume, ν in the solid is depends to the density, ρ and pressure, E where ν=ρa Eb. Determine a and b and state the final expression for the equation above.

  20. F = mvR (a) (b) (c) Remember: Force has dimensions of ML/T2 Example 5 • The force (F) to keep an object moving in a circle can be described in terms of the velocity, v, (dimension L/T) of the object, its mass, m, (dimension M), and the radius of the circle, R, (dimension L). • Which of the following formulas for Fcould be correct ?

  21. F = mvR (a) (b) (c) Solution Consider for RHS, since [LHS] = MLT-2 For (a); [mvR] = MLT-1L=ML2T-1 (incorrect) For (b); [mv2R-2] = ML2T-2L-2 = MT-2 (incorrect) For (c); [mv2R-1] = ML2T-2L-1 = MLT-2 (correct) Answer is (c)

  22. (a) (b) (c) Example 6: • There is a famous Einstein's equation connecting energy and mass (relativistic). Using dimensional analysis find which is the correct form of this equation : Solution: Dimension for energy, [E], with unit Joule (J) = L2MT-2 c – light velocity, [c] = LT-1 For (a): [mc] = MLT-1 (incorrect) For (b): [mc2] = ML2T-2 (correct)

  23. Unit Conversions • Because units in different systems, or even different units in the same system, can be used to express the same quantity, it sometimes necessary to CONVERT the units of a quantity from one unit to another. • Mathematically, to change units we use conversion factors. • As example, in British Unit, 1 yard = 3 ft

  24. Converting between different systems of units • Useful Conversion factors: • 1 inch = 2.54 cm • 1 m = 3.28 ft • 1 mile = 5280 ft • 1 mile = 1.61 km • Example: convert miles per hour to meters per second:

  25. Example: • A hall bulletin board has an area of 2.5 m2. What is area in cm2? Solution: The problem is conversion of area units (in the same SI unit: mks  cgs). We know that 1m = 100cm. So, Thus, 1m2 = 104cm2 (conversion factor) Hence,

  26. How about in square inch, (inch)2? Solution: From conversion factor, 1inch = 2.54cm Thus, 1inch2 = 6.4516cm2 (conversion factor) Hence,

  27. When on travel in Kedah you rent a small car which consumes 6 liters of gasoline per 100 km. What is the MPG (mile per gallons) of the car ? Exercise 1: 1L=1000cm3=0.3531ft3 1ft3=0.02832m3=7.481 gal

  28. Significant Figures • The number of digits that matter in a measurement or calculation. • When writing a number, all non-zero digits are significant. • Zeros may or may not be significant. • those used to position the decimal point are not significant. • those used to position powers of ten ordinals may or may not be significant. • in scientific notation all digits are significant • Examples: • 2 1 sig fig • 40 ambiguous, could be 1 or 2 sig figs • 4.0 x 101 2 sig figs • 0.0031 2 sig figs • 3.03 3 sig figs

  29. Significant Figures • When multiplying or dividing, the answer should have the same number of significant figures as the least accurate of the quantities in the calculation. • When adding or subtracting, the number of digits to the right of the decimal point should equal that of the term in the sum or difference that has the smallest number of digits to the right of the decimal point. • Examples: • 2 x 3.1 = 6 • 3.1 + 0.004 = 3.1 • 4.0 x 101 2.04 x 102 =1.6 X 10-1

  30. Summary • Unitscan be counted or measured. • The International System of Units (SI) is the modern form of the metric system. Two system of SI units are mks system and cgs system. • The SI includes base quantities and derived quantities. • A Dimension is a parameter or measurement required to define the characteristics of an object. • Unit conversion - units in different systems, or even different units in the same system. • Significant figures - The number of digits that matter in a measurement or calculation

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