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Chapter 1 Introduction Measurement and unit Physical quantity Its any quantity that can be measured e.g length Unit A physical quantity is expressed as a number with units e.g length: L=2m, time; t=1s. S.I Units S.I stands for systeme internationale in French, meaning “international system”. SI units are agreed international standard units. Base quantities are measured in SI units Basic physical quantities are quantities which are not expressed in terms of other quantities. They are also called fundamental physical quantities. The SI units of base quantities are called basic units or fundamental units. Table 1: Base Physical Quantities Quantity SI Unit Name Unit Symbol Mass kilogram kg Length meter m Time second s Electric current Ampere A Temperature Kelvin K Amount of substance mole mol Luminous intensity candela cd Derived quantities are expressed in terms of basic quantities by multiplication or division. Table 2 shows some examples of derived quantities. Table 2: Derived Quantities Derived Quantity Unit Name Unit Symbol Force Newton N Acceleration Meter per second square Pressure Pascal Work Joule Power Watt Density Mass per volume m/s 2 Unit Prefixes For larger and smaller units, we write them in a short hand by multiply by powers of 10 as in Figure 3. Table 3:Unit Prefixes Prefix Symbol Peta P Tera Factor Prefix Symbol 1015 deci d 10 1 T 1012 centi c 10 2 Giga G 10 9 milli m 10 3 Mega M 10 6 micro µ 10 6 Kilo k 10 3 nano n 10 9 Hector h 10 2 pico p 10 12 Deca da fento f 10 15 10 Factor Changing units Replace the unit you are given with the equivalent quantity in the unit you want Example : Write 72km/hr in m/s 72km 72000m 1hr 60 X 60 s 3600s 72000m therefore,72km / hr 20m / s 3600 s Dimensional analysis It deals with the physical quantity in question regardless of its unit It describes any quantity in terms of fundamental quantities. Dimensions of a physical quantity are the powers to which the fundamental quantities must be raised This is the way of dealing with Table 4: Some Physical quantities with their dimensions Quantity Dimension Mass M Length L Time T Acceleration LT 2 Force ML/T2 work We write [X] meaning ‘‘dimension of quantity X’’ Application of Dimensional analysis 1. Checking if a given equation is consistent/valid Dimensionally consistent equation has both sides the same dimensions/same SI units Let us consider the following equation: 1 2 s ut at 2 Where S is the displacement, u is the initial velocity, a is the acceleration and t is the time taken. Show that this equation is dimensionally consistent/correct/valid. We rewrite the equation, all quantities replaced by their dimensions, we get: s = ut + 1 at 2 2 → L = LT−1 × T + LT−2 × T 2 L=L+L L = 2L We finally get the same dimension of length both sides of the equation. Therefore, the equation is dimensionally correct. 2. We also use dimensional analysis to derive an equation If we know that a quantity is related in same way to other quantities, then it is possible to derive an equation expressing the relationship Example: Suppose we are told that the acceleration a of a particle moving with uniform speed v in a circle of radius r is proportional to some power of r, say r n, and some power of v, say v m. Determine the values of n and m and write the simplest form of an equation for the acceleration We write the relation as :a ∝ r n v m a = kr n v m where k is the proportionality constant Writing the equation dimensionally, we get LT−2 = k[L]n [LT −1 ]m k is dimensionless, so we can take it out of the equation LT−2 = [L]n LT −1 LT−2 = Ln Lm T −m m equating the powers, we get 1 = n + m and − 2 = −m from which, we get m = 2 and n = −1 Therefore, the derived equation is v2 a=k r Decimal point, scientific notation, precision, accuracy and order of magnitude Decimal point: the number 7.32 has two decimal places and 1500 has no decimal point. Scientific notation: this is writing numbers in powers of 10. For example, 6342147.14 can be written in scientific notation as 6.34214714 X 10 6 When counting decimal places from right to left, the exponent is positive When counting decimal places from left to right, the exponent is negative Significant figures (s.f) are the number of digits in scientific notation e.g 4.71 has 3 s.f. Uncertainty in measurement: in any measurement there is always an error, and this is called uncertainty Accuracy refers to closeness of a measured value to a standard or known value. For example, if in a lab you obtain a weight measurement of 3.2 kg for a given substance, but the actual or the known weight is 10 kg, then your measurement is not accurate. Precision refers to the closeness of two or more measurements to each other. For example, if you weigh a given substance five times and get 3.2 kg each time, then your measurement is very precise Order of magnitude: it is a rough approximation of a quantity to the closest power of 10 e.g 8~10 this reds 8 is of the order of 10 Examples 123~100, and 0.07~0.1 To find order of magnitude, round up or down to nearest power of 10 e.g 7.62 × 104 ~105 , 3.2 × 102 ~103 , 8.417 × 10−6 ~10−5 , 1.5 × 10−8 ~10−8 Coordinate Systems Used to describe the position of a point in space Coordinate system consists of a fixed reference point called the origin specific axes with scales and labels instructions on how to label a point relative to the origin and the axes Types of Coordinate Systems Cartesian Plane polar Cartesian coordinate system Also called rectangular coordinate system x- and y- axes Points are labeled (x,y) Plane polar coordinate system Origin and reference line are noted Point is distance r from the origin in the direction of angle , ccw from reference line Points are labeled (r,) Trigonometry Review opposite side sin hypotenuse adjacent side cos hypotenuse opposite side tan adjacent side More Trigonometry Pythagorean Theorem 2 2 2 r x y To find an angle, you need the inverse trig function 1 0.707 45 for example, sin Be sure your calculator is set appropriately for degrees or radians