Lecture 1. > Introduction. > Measurement, Trigonometry

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1 Lecture 1 > Introduction > Measurement, Trigonometry Giancoli, Physics 6/e Serway & Vuille, College Physics 8/e Serway & Jewett, Physics for Scientists & Engineers, 6/e Tippens, Physics 7/e Young, Freedman & Ford, University Physics 14/e Villacorta--DLSUM-SCITECS-L Term01 1

2 Nature of Science > Aim of science: ideas from data, observations; collecting facts, making theories > Observing nature, performing experiments + Aristotle, Galileo on the moving cart + Observation & experimentation + Formulating a theory + Testing the theory + Accepting the theory, range of the theory > Theories: attempts at explaining observed phenomena + Observations lead to theories about how they came about + Theories are tested by further experiments and observations Villacorta--DLSUM-SCITECS-L Term01 2

3 Introduction to Physics > Physics: expt'l science, sci method + Observe nature + Find patterns > Physical theories: patterns observed in nature; Laws, principles: well established & widely used physical theories Aristotle BC Galileo Galilei > Galileo on falling objects + Experiment ~ Light, heavy ~ Rate of falling ~ Weight-independent + Experimental process, tests > Falsifiability + Testing theories through experiments + New experiments: refine or discard + How to disprove a theory: find an observation inconstent w/ it + Range of validity: ignoring air resistance 3

4 Introduction to Physics contd > Physics: the study of matter and energy + Most basic of the sciences + Classical: motion, fluids, heat, sound, light, electricity, & magnetism + Modern: relativity, atomic structure, condensed matter, nuclear physics, elementary particles, & cosmology and astrophysics + Physics is related to many fields: mathematics, biology, chemistry, etc > Picturing & explaining nature + Model: using simplified, known ideas & applying it to other phenomena + Theory: more expansive, testable results + Law: natural behavior + Principle: specific behavior + Prescription (setting limits) vs. description (describe as observed) Real object Point particle Simplified Young, Freedman, c01 4

5 Measurement and Uncertainty > Uncertainty from the inaccuracy of measuring devices Vernier caliper: ½ of 1/10 of 1/10 of a centimeter or cm Ruler: ½ of 1/10 of a centimeter or 0.05 cm > Accuracy (closeness to the true value) vs Precision (measurement repeatabilty) 5

6 Measurement and Uncertainty contd > Significant figures: number of reliable digits + Since measuring devices can never be completely accurate, the measurements taken using them cannot be 100% reliable. + Non-zeroes are always significant cm 3 significant figures kg 4 significant figures + Zeroes to the right of the decimal point are significant km 3 significant figures m 5 significant figures + Zeroes to the right of a significant digit and to the left of a decimal point are NOT significant lb 2 significant figures 7090 m 2 3 significant figures + Zeroes are significant when these are between significant digits in 4 significant figures s > 5 significant figures 6

7 Measurement and Uncertainty contd > When adding measures, the final answer must have the same decimal places to the right of the decimal point as the addend with the least number of decimal places. > When multiplying measures, the final answer must have the same number of significant figures as the factor with the least number of significant figures. Ex. What is the total mass of three objects: 14.0 kg, kg, and kg? Ans. Simple addition 14.0 kg kg kg = kg Since 14.0 kg has one decimal place w/c is the least among those given, then the final answer should be 38.1 kg, w/c also has one decimal place. Ex. What is the area of a triangle that is 12.0 cm tall and has a base cm wide? Ans. Use the formula for the traingle area: A = (½) b h = (½) (35.25 cm) (12.0 cm) = cm 2 Since the height measure only has three significant figures (sf), then the final answer should be 212 cm 2, w/c also has three sf. 7

8 Measurement and Uncertainty contd > Scientific Notation, Powers of Ten: a compact way of writing very small, large measures that keep the significant figures > Ex. 2,300,000 kg = 2.3 x 10 6 kg (two s.f. before & after conversion) > Ex m = 7.03 x 10 4 m (three s.f. before & after conversion) > Percentage Error: a measure of how far an experimental value is from an accepted, theoretical value Percentage Error (%E)= Ex. Value Th. Value 100% Th. Value > Percentage Difference: a measure of how close two measurements are to each other (Ex. The same quantity measured in different ways) Meas. 1 Meas. 2 Percentage Difference (%D)= 100% Average Meas. 8

9 Units & Standards > Physical quantity: number describing a physical phenomenon Ex. height, weight, length, etc > Standard Units + Time: second = duration of 9,192,631,770 periods of cesium atom radiation Operational definition: fundamental description of a physical quantity Ex. length: measure using a ruler time: measure using a stopwatch > Unit: the measurement standard of a quantity + Ex. centimeters, kilometers, grams + A number is not enough to describe a quantity: the unit must be included + Standard: a reproducible measure for a unit Young, Freedman, c01 9

10 Units & Standards contd + Length: meter = distance traveled by light in 1 / 299,792,458 seconds + Mass: kilogram = mass of a platinum-iridium alloy in Paris, France Early defn (French Academy of Sciences, 1797): Dist from N Pole to Equator 1 m= 10,000,000 FR Orig in Saint-Cloud US Prototype in Maryland Young, Freedman, c01 10

11 Units & Standards contd > Systems of Units + International System (SI): metric system, cgs (centimeter-gram-second), mks (meter-kilogram-second) length: meters (m) 1 kilometer (km) = 1000 m = 10 3 m mass: kilograms (kg) 1 metric ton/ tonne (t) = 1000 kg time: second (s) 1 minute (min) = 60 s 1 hour (h) = 60 min = 3600 s area: 1 hectare (ha) = (100 m) 2 = 10 4 m 2 + Imperial Units: foot-pound-second length: 1 inch (in) = 2.54 cm [defn] 1 foot (ft) = 12 in 1 yard (yd) = 3 ft 1 mile (mi) = 5280 ft = km mass: 1 pound (lb) = kg 1 stone (st) = 14 lb 1 ton (t) = 2240 lb area: 1 acre = 66 ft x 660 ft = 4840 yd 2 = 4046 m 2 volume: 1 fluid ounce (fl oz) = ml 1 pint (pt) = 20 fl oz 1 quart (qt) = 2 pt 1 gallon (gal) = 4 qt volume: 1 liter (L) = (10 cm) 3 = m

12 Units & Standards contd > Quantities can be: + Base: defined using the standard units. 7 Base Qts: 1. Length 5. Electric charge 2. Mass 6. Amount of substance 3. Time 7. Luminous intensity 4. Temperature + Derived: combination of base quantities: Ex. speed, volume, force, etc. Serway & Vuille, Ch01 12

13 Sample Problem 1. (a) The recommended daily allowance (RDA) of the trace metal magnesium is 410 mg/day for males. Express this quantity in µg/day. (b) For adults, the RDA of the amino acid lysine is 12 mg per kg of body weight. How many grams per day should a 75-kg adult receive? (c) A typical multivitamin tablet can contain 2.0 mg of vitamin B 2 (riboflavin), and the RDA is g/day. How many such tablets should a person take each day to get the proper amount of this vitamin, if he gets none from other sources? (d) The RDA for the trace element selenium is g/day. Express this dose in mg/day. [Young, Freedman, c01] 13

14 Measure Estimates > Order of magnitude + Quick estimate of measure + Rough estimate + Ex. area of an empty lot, floor space of a room, fluid ounces of medicine, etc. > Dimensional analysis + Equations must have the same units on both sides + Dimensions refer to the type of base quantities making up a unit + Ex. L = length, T = time, M = mass, etc. distance = (velocity) (time) or d = v t dimension: [distance] = L {length} [velocity] = L/T {length over time} [time] = T {time} L = ( L/T ) (T) = L, showing that the eq is consistent dimensionally 14

15 Trigonometric Functions > The three trigonometric functions + Sine + Cosine + Tangent Pythagorean Theorem Pythagoras c BC c 2 = a 2 + b 2 SOH CAH - TOA 90º 15

16 Sample Problem 1. A high fountain of water is located at the center of a circular pool as shown... Not wishing to get his feet wet, a student walks around the pool and measures its circumference to be 15.0 m. Next, the student stands at the edge of the pool and uses a protractor to gauge the angle of elevation at the bottom of the fountain to be How high is the fountain? [Serway, Vuille, c01] 16

17 Law of Cosines > Relates the side of one triangle to the other sides and the angle opposite it > Applicable for different kinds of triangles c 2 =a 2 +b 2 2 a b cos γ a 2 =b 2 +c 2 2 b c cosα b 2 =c 2 +a 2 2 c a cosβ β a Ex. Two sides of a triangle have measures 3.0 cm and 4.0 cm and form a 45º angle. What is the length of the third side? c γ Ans. Use the cosine law with a = 3.0 cm b = 4.0 cm and γ = 45º c 2 = (3.0cm) 2 + (4.0cm) 2 2 (3.0cm)(4.0cm)cos 45º = cm 2 => c = cm The third side is 2.8 cm long. α Note: If γ = 90º, then the cosine law for c 2 is just the Pythagorean formula. b 17

18 Law of Sines > Gives the relationship between the different ratios of the sides of a triangle and the angle opposite said side > Applicable for different kinds of triangles a sinα = b sinβ = c sin γ β a Ex. Find the angle opposite 3.0-cm side and the angle opposite the 4.0-cm side in the triangle in the previous item. Ans. Use the sine law: sin α = (a/c) sin γ and sin β = (b/c) sin γ. c γ sin α = (3.0cm/2.834cm) sin 45º = => α =48.46º sin β = (4.0cm/2.834cm) sin 45º = => β =86.41º Thus, the angle opposite the 3.0-cm side is 48º while that of the 4.0-cm side is 86º. α b Note: The sine law may lead to 2 different triangles under certain conditions. 18

19 Sample Problem 1. You and a friend are out hiking across a large flat plain and decide to determine the height of a distant mountain peak, and also the horizontal distance from you to the peak... In order to do this, you stand in one spot and determine that the sightline to the top of the peak is inclined at 7.5º above the horizontal. You also make note of the heading to the peak at that point: 13º east of north. You stand at the original position, and your friend hikes due west for 1.5 km. He then sights the peak and determines that its sightline has a heading of 15º east of north. How far is the mountain from your position, and how high is its summit above your position? [Tipler, c01] 19

20 Sample Problem contd 1. 20

21 Vectors > Vectors, having both magnitude and direction, can be modeled using a finite ray. + The length of the ray representing the vector magnitude. + The arrowhead representing the vector direction. > Basic vectors can be drawn in one, two, or three dimensions: + On a number line, for 1-D + On a Cartesian plane, for 2-D > Vectors can also be written symbolically or in mathematical terms. 21

22 Vectors contd. > Adding two or more vectors can be done in a number of ways. Unlike scalars that only require adding numbers using simple algebra, vectors require further steps due to its direction. > The addition can be done either: + graphically by triangulation or head-to-tail, or + computationally by components. > Other instances such as the negative of a vector and multiplication by a scalar are also common ways of combining vectors. 22

23 Decomposing Vectors > Aside from adding vectors by graphical means, one can also combine vectors by decomposing them into their x- and y-components. > These components come from the idea that if adding many vectors yields another vector (head-to-tail), then any vector can be decomposed into a number of vectors. > In practice, one decomposes a vector into two perpendicular components for the 2-D case. + Horizontal, vertical components + x-, y-components 23

24 Decomposing Vectors contd. > Suppose we add the following vectors: L = 5.00 u, 30.0º above the +x-axis M = 7.00 u, 45.0º below the +x-axis > Components: L x = u; L y = u M x = u; M y = u R x = u; R y = u > Resultant: R = 9.60 u, 14.8º below the +x-axis L x = L cosθ L ; L y = L sin θ L M x = M cosθ M ; M y = M sin θ M R x = L x + M x R y = L y + M y R= R x 2 +R y 2 θ R =arctan R y R x 24