Engineering Science DMT 111 CHAPTER 1: UNITS & DIMENSION

Topics: Measurement and Units Measurement and Units  Fundamental units  Systems of units  Converting between systems of units  Dimensional Analysis  Significant digits

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: Physical quantitiesDefinition Base quantityContains of length, mass, time, electrical current, temperature, quantity of matter & luminous intensity Derived quantitiesThe quantity which derived from base quantity. Eg: force, energy, pressure, etc

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 10 8 m/s. The number depends on the units.

The elements of substances and motion. fundamental quantities: All things in classical mechanics can be expressed in terms of the fundamental quantities:  Length, L  Mass, M  Time, T Standard Quantities l Some examples of more complicated quantities: ç Speed has the quantity of L / T (i.e. kmph or mph). ç Acceleration has the quantity of L/T 2. ç Force has the quantity of ML / T 2 (as you will learn).

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.

Units SI (Système International) Units: SI (Système International) Units: We will use mostly SI units, but you may run across some problems using British units. You should know how to convert back & forth. MKS systemCGS SYSTEM L = meters (m),centimeter (cm) M = kilograms (kg),gram (g) T = seconds (s)seconds (s)  British Units: L = inches, feet, miles, M = slugs (pounds), T = seconds

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.

The base quantities according to the International System of Quantities (ISQ) and their dimensions are listed in the following table: NameSymbol of quantity Symbol of dimension SI base unit Length l Lmeter (m) Time t Tsecond (s) Mass m Mkilogram (kg) Electrical current I IAmpere (A) Thermodynamic temperature T  Kelvin (K) Amount of substance n Nmole Luminous intensity IvIv JCandela (c)

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.

QuantityNameSymbol of quantity SI unitSymbol of dimension Force, Weight NewtonN mkg/s 2 LMT -2 Energy, Work, Heat JouleJ m 2 kg/s 2 L 2 MT -2 Power, radian flux WattW m 2 kg/s 3 L 2 MT -3 Frequency HertzHz s -1 T -1 Pressure, Stress PascalPa m -1 kg/s 2 L -1 MT -2 Electric charge or flux CoulombC AsAT Electrical potential difference, Electromotive force VoltV m 2 ∙kg∙s −3 ∙A − 1 L 2 MT -3 A -1

QuantityNameSymbol of quantity SI unitSymbol of dimension Electric resistance, Impedance, Reactance OhmΩ m 2 kgs −3 A −2 L 2 MT -3 A -2 Electric capacitance FaradF m −2 kg −1 s 4 A 2 L -2 M -1 T 4 A 2 Magnetic flux density, magnetic induction TeslaT kgs −2 A −1 MT -2 A -1 Magnetic flux WeberW m 2 kgs −2 A −1 L 2 MT -2 A -1 Inductance HenryH m 2 kgs −2 A −2 L 2 MT -2 A -2

Example: (a) Force, F =ma Check the dimension!! m – mass (kg); a – acceleration (m/s 2 ) Solution: Thus, dimension for force, [F] = kgms -2 = MLT -2 (b) Momentum, p =mv Solution: m – mass (kg); v – velocity (m/s) Thus, dimension for momentum, [p] = kgms -1 = MLT -1

Systems of Units

Other systems of Units Convert one system of unit to another. This done by using conversion factors. C.G.S (cm gram sec) Eg: 1gcm -3 to SI units? SI to C.G.SC.G.S to SI 1m= 100cm1cm= m 1kg= 1000g1g= kg 1m 2 =10 4 cm 2 1cm 3 = m 2 1m 3 =10 6 cm 3 1cm 3 =10 -6 m 3

SI Prefixes Name yottazettaexapetateragigamegakilohectodeca SymbolYZEPTGMkhda Factor Name decicentimillimicronanopicofemtoattozeptoyocto Symboldcmµnpfazy Factor

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.

Dimension In common usage, a dimension (Latin, "measured out") is a parameter or measurement required to define the characteristics 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.

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

This is a very important tool to check your work  It’s also very easy! Example: Example: Doing a problem you get the answer for distance d = v t 2 ( velocity x time 2 ) Dimension on left side [d] = L Dimension on right side [vt 2 ] = L / T x T 2 = L x T Left units and right units don’t match, so answer must be wrong !! Left units and right units don’t match, so answer must be wrong !! Dimensional Analysis

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). 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 F could be correct ? Remember: Force has dimensions of ML/T 2 (a)(b)(c) F = mvR

Solution Consider for RHS, since [LHS] = MLT -2 For (a); [mvR] = MLT -1 L=ML 2 T -1 (incorrect) For (b); [mv 2 R -2 ] = ML 2 T -2 L -2 = MT -2 (incorrect) For (c); [mv 2 R -1 ] = ML 2 T -2 L -1 = MLT -2 (correct) Answer is (c) (a)(b)(c) F = mvR

Example: There is a famous Einstein's equation connecting energy and mass (relativistic). Using dimensional analysis find which is the correct form of this equation : (a)(b)(c) Solution: Dimension for energy, [E], with unit Joule (J) = L 2 MT -2 c – light velocity, [c] = LT -1 For (a): [mc] = MLT -1 (incorrect) For (b): [mc 2 ] = ML 2 T -2 (correct)

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

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:

Example: A hall bulletin board has an area of 4.5 m 2. What is area in cm 2 ? Solution: The problem is conversion of area units (in the same SI unit: mks  cgs). We know that 1m = 100cm. So, Thus, 1m 2 = 10 4 cm 2 (conversion factor) Hence,

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

Exercise 1: 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 ?

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:  21 sig fig  40ambiguous, could be 1 or 2 sig figs  4.0 x sig figs  sig figs  sig figs

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  4.0 x 10 1  2.04 x 10 2 = 1.6 X 10 -1

Summary Units can 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