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Ying Yi PhD Chapter 1 Introduction and Mathematical Concepts 1 PHYS HCC.

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Presentation on theme: "Ying Yi PhD Chapter 1 Introduction and Mathematical Concepts 1 PHYS HCC."— Presentation transcript:

1 Ying Yi PhD Chapter 1 Introduction and Mathematical Concepts 1 PHYS I @ HCC

2 What does Physics do? PHYS I @ HCC 2 Develop Theory by experiment Predict experiment results More experiments Check prediction

3 Outline PHYS I @ HCC 3 Fundamental quantities & dimension analysis Uncertainty in measurements (Significant figures) Coordinate system (Trigonometry) Addition of vectors

4 Fundamental Quantities and Their Dimension Length [L] Mass [M] Time [T] 4 PHYS I @ HCC Other physical quantities can be constructed from these three

5 Units To communicate the result of a measurement for a quantity, a unit must be defined Defining units allows everyone to relate to the same fundamental amount 5 PHYS I @ HCC

6 Systems of Measurement Standardized systems Agreed upon by some authority, usually a governmental body SI – Systéme International Agreed to in 1960 by an international committee Main system used in this text 6 PHYS I @ HCC

7 Length Units SI – meter, m Defined in terms of a meter – the distance traveled by light in a vacuum during a given time Also establishes the value for the speed of light in a vacuum Example: Radius of earth 6×10 6 m 7 PHYS I @ HCC

8 Mass Units SI – kilogram, kg Defined in terms of kilogram, based on a specific cylinder kept at the International Bureau of Weights and Measures Example: mass of human 70 kg 8 PHYS I @ HCC

9 Standard Kilogram 9 PHYS I @ HCC

10 Time Units seconds, s Defined in terms of the oscillation of radiation from a cesium atom Example: One year 3×10 7 s 10 PHYS I @ HCC

11 Other Systems of Measurements cgs – Gaussian system Named for the first letters of the units it uses for fundamental quantities US Customary Everyday units Often uses weight, in pounds, instead of mass as a fundamental quantity 11 PHYS I @ HCC

12 Prefixes (Table 1.2) Prefixes correspond to powers of 10 Each prefix has a specific name Each prefix has a specific abbreviation 12 PHYS I @ HCC

13 Dimensional Analysis Technique to check the correctness of an equation Dimensions (length, mass, time, combinations) can be treated as algebraic quantities Add, subtract, multiply, divide Both sides of equation must have the same dimensions 13 PHYS I @ HCC

14 Example: Analysis of an equation PHYS I @ HCC 14 Show that the expression v=v 0 +at is dimensionally correct, where v and v 0 represent velocities, a is acceleration, and t is a time interval.

15 Outline PHYS I @ HCC 15 Fundamental quantities & dimension analysis Uncertainty in measurements (Significant figures) Coordinate system (Trigonometry) Addition of vectors

16 Uncertainty in Measurements There is uncertainty in every measurement, this uncertainty carries over through the calculations Need a technique to account for this uncertainty We will use rules for significant figures to approximate the uncertainty in results of calculations 16 PHYS I @ HCC

17 Significant Figures A significant figure is a reliably known digit All non-zero digits are significant Zeros are significant when Between other non-zero digits After the decimal point and another significant figure Can be clarified by using scientific notation 17 PHYS I @ HCC

18 Operations with Significant Figures When multiplying and dividing, the number of significant figures in the final result is the same as the number of significant figures in the least accurate of the factors being combined 18 PHYS I @ HCC When adding or subtracting, round the result to the smallest number of decimal places of any term in the sum

19 Operations with Significant Figures, rounding If the last digit to be dropped is less than 5, drop the digit If the last digit dropped is greater than or equal to 5, raise the last retained digit by 1 19 PHYS I @ HCC

20 Example Carpet Calculations PHYS I @ HCC 20 Several carpet installers make measurements for carpet installation in the different rooms of a restaurant reporting their measurements with inconsistent accuracy, as compiled in Table 1.6. compute the areas for (a) the banquet hall, (b) the meeting room, and (c) the dining room, taking into account significant figures. (d) what total area of carpet is required for these rooms?

21 Conversion of units When units are not consistent, you may need to convert to appropriate ones See the inside of the front cover for an extensive list of conversion factors Units can be treated like algebraic quantities that can “cancel” each other Example: 21 PHYS I @ HCC

22 Examples 1.1 & 1.2 PHYS I @ HCC 22 979.0 m= feet 65 mi/h= m/s

23 Outline PHYS I @ HCC 23 Fundamental quantities & dimension analysis Uncertainty in measurements (Significant figures) Coordinate system (Trigonometry) Addition of vectors

24 Coordinate Systems Used to describe the position of a point in space Coordinate system consists of A fixed reference point called the origin, O Specified axes with scales and labels Instructions on how to label a point relative to the origin and the axes 24 PHYS I @ HCC

25 Types of Coordinate Systems Cartesian Plane polar 25 PHYS I @ HCC

26 26 Cartesian coordinate system Also called rectangular coordinate system x- and y- axes Points are labeled (x,y)

27 PHYS I @ HCC 27 Plane polar coordinate system Origin and reference line are noted Point is distance r from the origin in the direction of angle  from reference line Points are labeled (r,  )

28 Trigonometry Review 28 PHYS I @ HCC

29 More Trigonometry Pythagorean Theorem r 2 = x 2 + y 2 To find an angle, you need the inverse trig function For example, q = sin -1 0.707 = 45.0° 29 PHYS I @ HCC

30 Degrees vs. Radians Be sure your calculator is set for the appropriate angular units for the problem For example: tan -1 0.5774 = 30.00° tan -1 0.5774 = 0.5236 rad 30 PHYS I @ HCC

31 Rectangular  Polar Rectangular to polar Given x and y, use Pythagorean theorem to find r Use x and y and the inverse tangent to find angle Polar to rectangular x = r cos  y = r sin  31 PHYS I @ HCC

32 Example 1.4 Using Trigonometric Function PHYS I @ HCC 32 On a sunny day, a tall building casts a shadow that is 67.2 m long. The angle between the sun’s rays and the ground is Ɵ =50.0°, as Figure 1.6 shows. Determine the height of the building.

33 Group problem PHYS I @ HCC 33 A lakefront drops off gradually at an angle Ɵ, as Figure 1.7 indicates. For safety reasons, it is necessary to know how deep the lake is at various distances from the shore. To provide some information about the depth, a lifeguard rows straight out from the shore a distance of 14.0 m and drops a weighted fishing line. By measuring the length of the line, the lifeguard determines the depth to be 2.25 m. (a) What is the value of Ɵ ? (b) What would be the depth d of the lake at a distance of 22.0 m from the shore?

34 Outline PHYS I @ HCC 34 Fundamental quantities & dimension analysis Uncertainty in measurements (Significant figures) Coordinate system (Trigonometry) Addition of vectors

35 Vector vs. Scalar 35 PHYS I @ HCC Difference: Magnitude and direction Only Magnitude Notation: Examples: Displacement Mass, length

36 Vector Notation When handwritten, use an arrow: When printed, will be in bold print with an arrow: When dealing with just the magnitude of a vector in print, an italic letter will be used: A 36 PHYS I @ HCC

37 Properties of Vectors Equality of Two Vectors Two vectors are equal if they have the same magnitude and the same direction Movement of vectors in a diagram Any vector can be moved parallel to itself without being affected Negative Vectors Two vectors are negative if they have the same magnitude but are 180° apart (opposite directions) Resultant Vector The resultant vector is the sum of a given set of vectors 37 PHYS I @ HCC

38 Adding Vectors Geometrically (Triangle or Polygon Method) Choose a scale Draw the first vector with the appropriate length and direction, with respect to a coordinate system Draw the next vector using tip-to-tail principle to The resultant is drawn from the origin of to the end of the last vector 38 PHYS I @ HCC

39 39 Adding Vectors, multiple vectors When you have many vectors, just keep repeating the process until all are included The resultant is still drawn from the origin of the first vector to the end of the last vector Does it matter if you add them with another order?

40 PHYS I @ HCC 40 Commutative Law of Addition The order in which the vectors are added doesn’t affect the result

41 Application: Archery PHYS I @ HCC 41

42 Vector Subtraction Special case of vector addition Add the negative of the subtracted vector Continue with standard vector addition procedure 42PHYS I @ HCC

43 Multiplying or Dividing a Vector by a Scalar The result of the multiplication or division is a vector The magnitude of the vector is multiplied or divided by the scalar If the scalar is positive, the direction of the result is the same as of the original vector If the scalar is negative, the direction of the result is opposite that of the original vector 43 PHYS I @ HCC

44 44 Components of a Vector A component is a part It is useful to use rectangular components These are the projections of the vector along the x- and y-axes

45 More About Components, cont. The components are the legs of the right triangle whose hypotenuse is The value will be correct only if the angle lies in the first or fourth quadrant In the second or third quadrant, add 180° 45 PHYS I @ HCC

46 46 Other Coordinate Systems It may be convenient to use a coordinate system other than horizontal and vertical Choose axes that are perpendicular to each other Adjust the components accordingly

47 Example 1.8 Finding the component of a vector PHYS I @ HCC 47 A displacement vector has a magnitude of r=175 m and points at an angle of 50.0° relative to the x axis in Figure 1.20. Find the x and y components of this vector.

48 Example 1.9 Vector addition PHYS I @ HCC 48 A jogger runs 145 m in a direction 20.0° east of north (displacement vector ) and then 105 m in a direction 35.0° south of east (displacement vector ). Using components, determine the magnitude and direction of the resultant vector for these two displacements.

49 Homework Assignment for CH1 PHYS I @ HCC 49 5,7,16,17,25,30,37,40,46,50


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