Welcome

1. Welcome Course: AP Physics Room: 207 Teacher: Mrs. LaBarbera Email:diana.labarbera@valleycentralschools.org Post session: Tue. – Fri.

2. Objectives • Introduction of AP physics curriculum • Lab safety • Sign in lab safety attendance sheet • Chapter 1 - units, physical quantities, and vectors

3. Chapter 1- units, physical quantities, and vectors • Know the fundamental quantities and units of mechanics. • Be able to determine the number of significant figures in calculations. • Differentiate between vectors and scalars • Be able to add and subtract vectors graphically. • Be able to determine the components of vectors and to use them in calculations • Know the unit vector and be able to use them with components to describe vectors • Know the two ways of multiplying vectors.

4. 1.1 The Nature of Physics • Physics is an experimental science. • Theories are formed through observation and experiments. However, no theory is ever regarded as the final and ultimate truth. • All theories can be revised by new observations. • All theories have a range of validity.

5. 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

6. 1.2 Solving Physics Problems • Identify the relevant concepts – determine target variable and the given quantities. • Set up the problem – choose equations based on the known and unknown from Identify step. • Execute the solution – “do the math” • Evaluate your answer – “Does the answer make sense?” • I SEE

7. 1.3 Standards and Units • SI Fundamental Quantities And Units Of Mechanics All other units can be expressed by combinations of these fundamental (base) units. The combined base units is called derived units.

8. Derived units • Like derived dimensions, when we combine base 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)

9. 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

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

11. The British System • Length: 1 inch = 2.54 cm • Force: 1 pound = 4.448221615260 N

12. 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, • The dimension for Energy: E = ML2/T2 • The dimension for Impulse: J = ML/T

13. L L = = L T2 1.4 Unit consistency and conversions • 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). Dimensionally, this looks like At least, the equation is dimensionally correct; it may still be wrong on other grounds, of course.

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

15. Conversion Strategies: I SEE • Identify the target units and the known conversion factors • Setup the problem using the given units and conversion factors to determine the unknown. Note units can be multiplied or divided like numbers. • Execute: do the math • Evaluate: “Does the answer make sense?” Note: the units are a part of the measurement as important as the number. They must always be kept together. Example: we wish to convert 2 miles into meters. (given conversion factors:1 miles = 1760 yards, 1 yd = 0.9144 m)

16. Example 1.1 • The official world land speed record is 1228.0 km/h, set on 10/15/1997, by Andy Green in the jet engine car Thrust SSC. Express this speed in m/s.

17. Example 1.2 • The world’s largest cut diamond is the First Star of Africa. Its volume is 1.84 cubic inches. What is tis volume in cubic centimeers? In cubic meters?

18. 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!!

19. Example 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.

20. 1.5 Uncertainty and Significant Figures • 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. Example 1.3 The rest energy E of an object with rest mass m is given by Einstein’s equation E = mc2 Where c is the speed of the light in vacuum (c = 2.99792458 x 108 m/s). Find E for an object with m = 9.11 x 10-31 kg.

23. Test Your Understanding 1.5 • The density of a material is equal to its mass divided by its volume. What is the density (in kg/m3) of a rock of mass 1.80 kg and volume 6.0 x 10-4 m3 • 3 x 103 kg/m3 • 3.0 x 103 kg/m3 • 3.00x 103 kg/m3 • 3.000x 103 kg/m3 • Any of these

24. 1.6 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.

25. Example 1.4 • You are writing an adventure novel in which the hero escapes across the border with a billion dollars’ worth of gold in his suitcase. Is this possible? Would that amount of gold fit in a suitcase? Would it be too heavy to carry? (given 1 g of gold ≈ $10.00 and density of gold ≈ 1 g/cm3)

26. Test Your Understanding 1.6 • What is approximate number of teeth in all the mouths of everyone at VC?

27. 1.7 vectors and vector additions • There are two kinds of quantities… • Vectors have both magnitude and direction • displacement, velocity, acceleration • Scalars have magnitude only • distance, speed, time, mass

28. Vectors • Vectors show magnitude and direction, drawn as a ray. Equal and Inverse Vectors

29. 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

30. Vector addition • Vectors may be added graphically, “head to tail.” or “parallegram

31. Commutative properties of vector addition

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

33. Subtract vectors: adding a negative vector

34. -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

35. Example 1.5 • A cross-country skier skies 1.00 km north and then 2.00 km east on a horizontal snow field. How far and in what direction is she from the starting point?

36. Test Your Understanding 1.7 • Two displacement vectors, S and T, have magnitudes S = 3 m and T = 4 m. Which of the following could be the magnitude of the difference vector S -T? (there may be more than one correct answer) • 9 m • 7 m • 5 m • 1 m • 0 m • -1 m

37. 1.8 Components of vectors • 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 A*cos θ and its y component using A*sin θ (where θ is the angle the vector A sweeps out from 0°).

38. The sign of the component depends on the angle from 0o Y is positive X is negative Y is negative X is negative

39. Example 1.6. • a) what are the x and y components of vector D? the magnitude of the vector is D = 3.00 m and the angle α = 45o. • b) what are the x and y components of vector E? the magnitude of the vector is E = 4.50 m and the angle β = 37.0o.

40. Doing vector calculations using components • Vector addition strategies • 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-1 Ry/Rx. Finding the direction of a vector sum by looking at the individual components

41. A A Multiplying a vector by a scalar • Multiplying a vector by a positive scalar changes the magnitude (length) of the vector, but not its direction. 2A is twice as long as A D =2A Dx = 2Ax, Dy = 2Ay • Multiplying a vector by a negative scalar changes the magnitude (length) of the vector and reverse its direction. -3A is three times as long as A and points in the opposite direction. D = -3A Dx = -3Ax, Dy = -3Ay

42. Example 1.7 • Three players are brought to the center of a large, flat field, each is given a meter stick, a compass, a calculator, a shovel, and the following three displacements: • 72. 4 m 32.0o east of north • 57.3 m 36.0o south of west • 17.8 m straight south • The three displacements lead to the point where the keys to a new Porsche are buried. Two players start measuring immediately, but the winner first calculates where to go. What does she calculate?

43. Example 1.8 • After an airplane takes off, it travels 10.4 km west, 8.7 km north, and 2.1 km up. How far us it from the takeoff point?

44. Test Your Understanding 1.8 • Two vectors A and B both lie in the xy-plane. • Is it possible for A to have the same magnitude as B but different components? • Is it possible for A to have the same components as B but a different magnitude?

45. ^ ^ ^ ^ ^ ^ i i k j k j 1.9 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

46. 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

47. 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:

48. Adding Vectors By Component using unit vector representation 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

49. 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.

50. 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