Welcome! Please find your seat

1. Welcome! Please find your seat

2. Welcome Course: AP Physics Room: 207 Teacher: Mrs. LaBarbera Email:dlabarbera@vcmail.ouboces.org Post session: Tue. – Fri.

3. 9/4 Objectives (A day) • Introduction of AP physics • Lab safety • Sign in lab safety attendance sheet • Classical Mechanics • Coordinate Systems • Units of Measurement • Dimensional Analysis/Units conversion

4. Classical Mechanics • Mechanics is a study of motion and its causes. • We shall concern ourselves with the motion of a particle. This motion is described by giving its position as a function of time. Specific position & time → event Position (time) → velocity (time) → acceleration • Ideal particle • Classical physics concept • Point like object / no size • Has mass • Measurements of position, time and mass completely describe this ideal classical particle. • We can ignore the charge, spin of elementary particles.

5. Position • If a particle moves along a • straight line → 1-coordinate • curve/surface → 2-coordinate • Volume → 3-coordinate • General description requires a coordinate system with an origin. • Fixed reference point, origin • A set of axes or directions • Instruction on labeling a point relative to origin, the directions of axes and the unit of axes. • The unit vector

6. Rectangular coordinates - Cartesian Simplest system, easiest to visualize. To describe point P, we use three coordinates: (x, y, z)

7. Spherical coordinate • Nice system for motion on a spherical surface • need 3 numbers to completely specify location: (r, Φ, θ) • r: distance between point to origin • Φ: angle between line OP and z: latitude = π/2 - Φ. • θ: angle in xy plane with x – longitude.

8. Time • Time is absolute. The rate at which time elapse is independent of position and velocity.

9. Unit of measurement International system of units (SI) consists of 7 base units. All other units can be expressed by combinations of these base units. The combined base units is called derived units

10. Physical Dimensions • The dimension of a physical quantity specifies what sort of quantity it is—space, time, energy, etc. • We find that the dimensions of all physical quantities can be expressed as combinations of a few fundamental dimensions: length [L], mass [M], time [T]. • For example, • Energy: E = ML2/T2 • Speed: V = L/T

11. Derived units • Like derived dimensions, when we combine basic unit to describe a quantity, we call the combined unit a derived unit. • Example: • Volume = L3 (m3) • Velocity = length / time = LT-1 (m/s) • Density = mass / volume = ML3 (kg/m3)

12. SI prefixes • SI prefixes are prefixes (such as k, m, c, G) combined with SI base units to form new units that are larger or smaller than the base units by a multiple or sub-multiple of 10. • Example: km – where kis prefix, m is base unit for length. • 1 km = 103 m = 1000 m, where 103 is in scientific notation using powers of 10

13. SI uses prefixes for extremes prefixes for power of ten

14. Standards and units • Base units are set for length, time, and mass. • Unit prefixes size the unit to fit the situation.

15. 1760 yd 0.9144 m 2 miles x x 1 mile 1 yd Dimensional analysis / Unit conversions Note: the units are a part of the measurement as important as the number. They must always be kept together. Suppose we wish to convert 2 miles into meters. (1 miles = 1760 yards, 1 yd = 0.9144 m) = 3218 m

16. m 1000 m 1 hr x x = 22 s 1 km 3600 s 80 km hr example • Convert 80 km/hr to m/s. • Given: 1 km = 1000 m; 1 hr = 3600 s Units obey same rules as algebraic variables and numbers!!

17. Dimensional Analysis Suppose we want to convert 65 mph to ft/s or m/s. Dimensional Analysis is simply a technique you can use to convert from one unit to another. The main thing you have to remember is that the GIVEN UNIT MUST CANCEL OUT.

18. L L = = L T2 Dimensional analysis • We can check for error in an equation or expression by checking the dimensions. Quantities on the opposite sides of an equal sign must have the same dimensions. Quantities of different dimensions can be multiplied but not added together. • For example, a proposed equation of motion, relating distance traveled (x) to the acceleration (a) and elapsed time (t). T2 Dimensionally, this looks like At least, the equation is dimensionally correct; it may still be wrong on other grounds, of course.

19. Another example use dimensional analysis to check if the equation is correct. d = v / t L = (L ∕ T ) ∕ T [L] ≠ L ∕ T2

20. Significant Figures (Digits) • Instruments cannot perform measurements to arbitrary precision. A meter stick commonly has markings 1 millimeter (mm) apart, so distances shorter than that cannot be measured accurately with a meter stick. • We report only significant digits—those whose values we feel sure are accurately measured. There are two basic rules: • (i) the last significant digit is the first uncertain digit • (ii) when multiply/divide numbers, the result has no more significant digits than the least precise of the original numbers. The tests and exercises in the textbook assume there are 3 significant digits.

21. Scientific Notation and Significant Digits • Scientific notation is simply a way of writing very large or very small numbers in a compact way. • The uncertainty can be shown in scientific notation simply by the number of digits displayed in the mantissa 2 digits, the 5 is uncertain. 3 digits, the 0 is uncertain.

22. Percent error • Measurements made during laboratory work yield an experimental value • Accepted value are the measurements determined by scientists and published in the reference table. • The difference between and experimental value and the published accepted value is called the absolute error. • The percent error of a measurement can be calculated by (absolute error) experimental value – accepted value Percent error = X 100% accepted value

23. Estimates and orders of magnitude • Estimation of an answer is often done by rounding any data used in a calculation. • Comparison of an estimate to an actual calculation can “head off” errors in final results. • Refer to Example 1.4.

24. There are two kinds of quantities… • Vectors have both magnitude and direction • displacement, velocity, acceleration • Scalars have magnitude only • distance, speed, time, mass

25. A A θ θ y y1 Magnitude: R = √x12 +y12 Direction: θ = tan-1(y1/x1) p(x1, y1) x o x1 Two ways to represent vectors • Geometric approach • Vectors are symbolized graphically as arrows, in text by bold-face type or with a line/arrow on top. Magnitude: the size of the arrow Direction: degree from East Algebraic approach • Vectors are represent in a coordinate system, e.g. Cartesian x, y, z. The system must be an inertial coordinate system, which means it is non-accelerated. θ

26. Vectors—Figures 1.9–1.10 • Vectors show magnitude and direction, drawn as a ray. Equal and Inverse Vectors

27. Vector addition—Figures 1.11–1.12 • Vectors may be added graphically, “head to tail.” or “parallegram

28. Vector additional II—Figure 1.13

29. B A E R Resultant and equilibrant A + B = R R is called the resultant vector! E is called the equilibrantvector!

30. Subtract vectors: adding a negative vector

31. -v1 v2 v2 -v1 example • At time t = t1, and object’s velocity is given by the vector v1 a short time later, at t = t2, the object’s velocity is the vector v2. If the magnitude of v1 = the magnitude of v2, which one of the following vectors best illustrates the object’s average acceleration between t = t1 and t = t2 v2 v2 –v1 v1 v2 v1 A B C D E

32. Vector addition III—Figure 1.16 • Refer to Example 1.5.

33. Components of vectors—Figure 1.17 • Manipulating vectors graphically is insightful but difficult when striving for numeric accuracy. Vector components provide a numeric method of representation. • Any vector is built from an x component and a y component. • Any vector may be “decomposed” into its x component using V*cos θ and its y component using V*sin θ (where θ is the angle the vector V sweeps out from 0°).

34. Components of vectors II—Figure 1.18

35. Finding components—Figure 1.19 • Refer to worked Example 1.6.

36. Calculations using components—Figures 1.20–1.21 • To find the components, follow the steps on pages 17 and 18. • Refer to Problem-Solving Strategy 1.3.

37. Component Addition of Vectors • Resolve each vector into its x- and y-components. Ax = Acos Ay = Asin Bx = Bcos By = Bsin etc. • Add the x-components together to get Rx and the y-components to get Ry. Rx = Ax + BxRy = Ay + By • Calculate the magnitude of the resultant with the Pythagorean Theorem • Determine the angle with the equation  = tan-1Ry/Rx.

38. Calculations using components II—Figure 1.22 • See worked examples 1.7 and 1.8.

39. ^ ^ ^ ^ ^ ^ i i k j k j Unit vectors • A unit vector is a vector that has a magnitude of 1, with no units. Its only purpose is to point, or describe a direction in space. • Unit vector is denoted by “^” symbol. • For example: • represents a unit vector that points in the direction of the + x-axis • unit vector points in the + y-axis • unit vector points in the + z-axis y x z

40. Any vector can be represented in terms of unit vectors, i, j, k Vector A has components: Ax, Ay, Az A = Axi + Ayj + Azk • In two dimensions: A = Axi + Ayj

41. The magnitude of the vector is |A| = √Ax2 + Ay2 The magnitude of the vector is |A| = √Ax2 + Ay2 + Az2 Magnitude and direction of the vector • In two dimensions: The direction of the vector is θ = tan-1(Ay/Ax) • In three dimensions:

42. Adding Vectors By Component s = a + b Where a = axi + ayj & b = bxi + byj s = (ax + bx)i + (ay + by)j sx = ax + bx; sy = ay + by s = sxi + syj s2 = sx2 + sy2 tanf = sy / sx

43. A = + + A = a (3.0 + 4.0 ) ^ ^ ^ ^ ^ i j k i j example • Is the vector a unit vector? • Can a unit vector have any components with magnitude greater than unity? Can it have any negative components? • If , where a is a constant, determine the value of a that makes A a unit vector.

44. E =(4 - 5+ 8 ) m D =(6 + 3 - ) m • Find the magnitude of the displacement 2D - E 2D - E =(8 + 11 - 10 ) m ^ ^ ^ ^ ^ ^ ^ ^ ^ i i j k k j i k j Example – 1.9 Given the two displacement • Its magnitude = (√ 82 + 112 + 102 )m = 17 m

45. Multiplication of Vectors: Scalar or Vector Product of a Scalar and a Vector Vector Product of Two Vectors

46. Multiplication of Vector by Scalar produce a vector • Examples: • momentum p = mv • Net force F = ma • Result • A vector with the same direction, a different magnitude and perhaps different units.

47. A scalar Product • Scalar product or dot product, yields a result that is a scalar quantity, • Examples: • work W = F d • Result • A scalar with magnitude and no direction.

48. The scalar product—Figures 1.25–1.26 1.25 1.26 • Termed the “dot product.” the result is a number only. C = AxBx + AyBy + AzBz A ∙ B = B ∙ A (A + B)∙C = A∙C + B∙C

49. -A A A A A∙A = |A||A|cos0o = |A|2 • Scalar product of parallel vectors: • Scalar product of anti-parallel vectors: A∙(-A) = |A||A|cos180o = -|A|2

50. W = F∙d Application of scalar product • When a constant force F is applied to a body that undergoes a displacement d, the work done by the force is given by • The work done by the force is • positive if the angle between F and d is between 0 and 90o (example: lifting weight) • Negative if the angle between F and d is between 90o and 180o (example: stop a moving car) • Zero and F and d are perpendicular to each other (example: waiter holding a tray of food while walk around)