Fundamentals In Physics PDF

SH1685 FUNDAMENTALS IN PHYSICS PHYSICAL QUANTITIES a. Length is a quantity that shows dimension. Relative words are how long, how far, how thick, and how high. b. Mass is a quantity that denotes how heavy an object is. Defined as a quantity or aggregate of ma...

SH1685 FUNDAMENTALS IN PHYSICS PHYSICAL QUANTITIES a. Length is a quantity that shows dimension. Relative words are how long, how far, how thick, and how high. b. Mass is a quantity that denotes how heavy an object is. Defined as a quantity or aggregate of matter that has direct proportion to size. c. Electric Current is a quantity defined as the amount of electricity passing through a medium. Can also be defined as the rate of flow of electricity moving continuously in a certain direction. d. Amount of Substance is a quantity that tells how much matter makes up that object. Connected with mass. e. Luminous Intensity is a quantity defined as the amount of brightness exerted by something. It is also the amount of light given off by an object. f. Time is a quantity that shows change in a certain quantifiable limit. It is a dimension explained that it moves along or changes with the perspective of the observer. UNITS List of Figures Length Energy 1 km = 0.62137 mi 1 J = 1 kg-m2/s2 1 mi = 5280 ft 1 J = 0.2390 cal 1 mi = 1.6093 km 1J=1C×1V 1 m = 1.0936 yd 1 cal = 4.184 J 1 in = 2.54 cm (exactly) 1 ev = 1.602 × 10–19 J 1 ft = 12 in 1 cm = 0.39370 in Pressure 1 Å = 10–10 m 1 Pa = 1 N/m2 1 Pa = 1 kg /m-s2 Mass 1 atm = 101,325 Pa 1 kg = 2.2046 lb 1 atm = 760 torr 1 lb = 453.59 g 1 atm = 14.70 lb/in2 1 lb = 16 oz 1 bar = 105Pa 1 amu = 1.66053873 × 10–24 g Volume Temperature 1 L = 10–3 m3 0 K = –273.15 °C 1 L = 1 dm3 0 K = –459.67 °F 1 L = 103 cm3 K = °C + 273.15 1 L = 1.0567 qt °C = 59 (°F – 32°) 1 gal = 4 qt 1 gal = 3.7854 L °F = ( 95 °C) + 32° 1 cm3 = 1 mL 1 in3 = 16.4 cm3 01 Handout 1 *Property of STI Page 1 of 10  SH1685 ENGLISH SYSTEM VS METRIC SYSTEM MEASUREMENT FACTS U.S. CUSTOMARY SYSTEM METRIC SYSTEM LENGTH 1 foot (ft) = 12 inches (in) 10 millimeters (mm) = 1 centimeter (cm) 1 yard (yd) = 3 ft = 36 in 10 cm = 1 decimeter (dm) = 100 mm 1 chain = 22 yd 10 dm = 1 meter (m) = 1000 mm 1 furlong = 10 chains 10 m = 1 decameter (dam) 1 mile (mi) = 8 furlongs = 1760 yd = 5280 ft 10 dam = 1 hectometer (hm) = 100 m 1 nautical mile (nm) = 1.15078 mi 10 hm = 1 kilometer (km) = 1000 m 1 in = 2.54 cm WEIGHT 1 mass-pound (lbm) = 16 ounces (oz) 10 milligrams (mg) = 1 centigram (cg) 1 ton = 2000 lbm 10 cg = 1 decigram (dg) = 100 mg 10 dg = 1 gram (g) = 1000 mg 10 g = 1 decagram (dag) 10 dag = 1 hectogram (hg) = 100 g 10 hg = 1 kilogram (kg) = 1000 g 1 oz = 28.3495 g VOLUME 1 cup = 8 fluid ounces (fl oz) 10 milliliters (mL) = 1 centiliter (cL) 1 pint (pt) = 2 cups = 16 ounces (oz) 10 cL = 1 deciliter (dL) = 100 mL 1 quart (qt) = 2 pt = 4 cups 10 dL = 1 liter (L) = 1000 mL 1 gallon (gal) = 4 qt 10 L = 1 decaliter (daL) 1 cubic yard (y3) = 27 cubic feet (ft3) 10 daL = 1 hectoliter (hL) = 100 L 1 cubic foot (ft3) = 1728 cubic inches (in3) 10 hL = 1 kiloliter (kL) = 1000 L 1 mL = 1 cubic centimeter (cm3 or cc) 1 cubic meter (m3) = 1000 cubic decimeters (dm3) = 1000 cc 1 fl oz = 29.574 mL AREA 1 square yard (yd2) = 9 square feet (ft2) = 1290 1 square meter (m2) = 100 square decimeters square inches (in2) (dm2) = 10,000 square centimeters (cm2) 1 ft2 = 144 in2 1 dm2 = 100 cm2 1 acre = 43,560 ft2 1 are (a) = 100 m2 1 square mile (mi2) = 640 acres 1 hectare (ha) = 100 ares 1 square kilometer (km2) = 100 ha = 1,000,000 m2 1 in2 = 6.4516 cm2 TEMPERATURE Fahrenheit (℉ ) Celsius (℃ ) a. The English or Imperial System is a unit system that was created in Great Britain that unified various standards of measurement across the entire country. This system was developed by the creative minds of early natural philosophers and apothecaries or pharmacists. b. The International or the Metric System is a unit system designed to simplify the way things are measured, due to the inconsistency of the units used in the English System way before they were unified. Even Great Britain and the US adopted this system. Prefix Symbol 𝟏𝟎𝒏 Decimal Short scale Long Scale Yotta Y 1024 1,000,000,000,000,000,000,000,000 septillion quadrillion Zetta z 1021 1,000,000,000,000,000,000,000 sextillion trilliard Exa E 1018 1,000,000,000,000,000,000 quintillion trillion Peta P 1015 1,000,000,000,000,000 quadrillion billiard Tera T 1012 1,000,000,000,000 trillion billion Giga G 109 1,000,000,000 billion milliard Mega M 106 1,000,000 million Kilo K 103 1,000 thousand 01 Handout 1 *Property of STI Page 2 of 10  SH1685 Hecto h 102 100 hundred Deca da 101 10 ten 100 1 one deci d 10−1 0.1 tenth centi c 10−2 0.01 hundredth milli m 10−3 0.001 thousandth micro μ 10−6 0.000 001 millionth nano n 10−9 0.000 000 001 billionth milliardth pico p 10−12 0.000 000 000 001 trillionth billionth femto f 10−15 0.000 000 000 000 001 quadrillionth billiardth atto a 10−18 0.000 000 000 000 000 001 quintillionth trillionth zepto z 10−21 0.000 000 000 000 000 000 001 sextillionth trilliardth yocto y 10−24 0.000 000 000 000 000 000 000 001septillionth quadrillionth Note: 1. Long scale reading of the Metric prefixes is usually ignored. 2. Some of these unit prefixes are used rarely, like the deci-, deca-, and hecto-. RULES AND GUIDELINES IN WRITING SCIENTIFIC NOTATION Scientific Notation is a way scientists compress extremely large or small numbers into manageable values widely understood by many. Although there are numbers that do not need such configurations, such as 125 or 0.12, scientific notation enables scientists to read and organize values that can help in their research. Writing numbers in scientific notation has two (2) advantages: 1. It saves space, especially when handling extremely large amounts of data using complicated formulae. 2. It allows for faster unit conversions. Guidelines on how to write in Scientific Notation. 1. Coefficient must be greater than or equal to one (1), but less than 10, for there must be only one (1) nonzero digit. Example: 1.23 × 1023 This is CORRECT. 12.3 × 1022 This is WRONG. 24 0.123 × 10 This is also WRONG. 2. The base is always 10. 3. Exponents in the scientific notation represent the number of digits the decimal point has crossed, and are either positive or negative only. a. The exponent is POSITIVE if the decimal point moves to the left. This means that the number has a large value (see Example a). b. The exponent is NEGATIVE if the decimal point moves to the right. This means that the number has a small value (see Example b). c. Although fractional exponents represent radical values, there is no radical notation in the scientific notation. Example: a. 1.23 × 1023 is 123,000,000,000,000,000,000,000 when expanded. 01 Handout 1 *Property of STI Page 3 of 10  SH1685 b. 1.23 × 10−23 is 0.0000000000000000000000123 when expanded. Operations involving Scientific Notations are straightforward. This method utilizes the basic four (4) operations of Mathematics when handling decimal and exponent values. LINEAR FITTING OF DATA Definition of Terms a. Variables refer to a set of data gathered and observed over the experiment. There are several kinds of variables, each one influencing the other. b. Axes are the projections of the Cartesian plane that denote and track changes within the observable period of the experiment or research. i. The X-axis is the horizontal part of the Cartesian plane that usually represents the independent variable, ii. The Y-axis is the vertical part of the Cartesian plane that represents the dependent variable. c. A Curve is a linear representation of the trend that the variables show. d. An Asymptote is a trend line that lies exactly close to one of the axes, but never touches it. e. A Cusp is a singular point on a curve wherein a moving point starts to move backward. f. An Inflection Point is a point in a curve wherein the curvature changes. g. The Maximum and Minimum are points with the largest and smallest values, respectively, along the curve. Kinds of Variables: 01 Handout 1 *Property of STI Page 4 of 10  SH1685 1. Plotted on the Abscissa: a. Independent – variation does not depend on the outcome of the other. b. Explanatory – variable that influences the outcome of the other; can have more than one explanatory variables. 2. Plotted on the Ordinate: a. Dependent – usually plotted on the ordinate; variation changes accordingly as the value of the independent variable changes. b. Response – variable that answers the question in the study, and is influenced by explanatory variables. 3. Other Variables: a. Categorical – represented by various symbols on the same coordinate system. b. Lurking – hidden variables. TYPES OF GRAPHS 1. Time Series – graphs used to represent changes in the y-axis as time passes (which is represented by the x-axis). 2. Scatter Plot – graphs used to represent data variance and shows the relationship between one variable to another. 3. Histogram – graph used to represent the frequency of a given variable spread over certain intervals. UNCERTAINTIES AND ERRORS  Accuracy refers to the degree of closeness and agreement between the acquired value and the true value.  Precision refers to the degree of closeness of the results repeatedly gathered to the true value.  Uncertainty pertains to the errors in a given value which the observer might suspect to obtain due to: 1. Limitations of the measuring device 2. The skills of the one making the measurement 3. Irregularities in the object 4. Other situational factors Formula: ∆𝑎′ % 𝑢𝑛𝑐 = ( ) × 100 𝑎 It is patterned after the given value. Ways to represent uncertainties:  Absolute uncertainties are uncertainties marked by having the smallest significant figure of the given value. Examples: 13.21 m ± 0.01 0.002 g ± 0.001 1.2 s ± 0.1 12 V ± 1  Fractional uncertainties are simply uncertainty values divided by the given value. 01 Handout 1 *Property of STI Page 5 of 10  SH1685 Examples: 1.2 s ± 0.1 Fractional uncertainty: 0.1 / 1.2 = 0.0625  Percentage uncertainties are fractional uncertainties of a piece of data converted into percentile values. Examples: 1.2 s ± 0.1 Percentage uncertainty: 0.1 / 1.2 x 100 = 6.25 % Uncertainty exists as bars over the value, like a mini Cartesian plane (with the obtained value as the origin) called Error bars. Figure 1 Source: http://ibguides.com/physics/notes/measurement-and-uncertainties The given values and the error bars that determine the uncertainty of the obtained value ERRORS Error refers to the miscalculation of data or a lapse of judgment during the experiment or research that yielded dubious results. Low credibility is the result of a high amount of error. Types of Errors 1. Random – errors due to variations in the environment and/or with measuring techniques  data values are scattered alongside the true value  error values have varied magnitude and direction  usually affects the reading of data 2. Systematic – errors caused by faulty devices and/or incorrect handling of such instruments  data values are deviating away from the true value  error values have consistent magnitude and direction  error occurs during data gathering 01 Handout 1 *Property of STI Page 6 of 10  SH1685 Error Relationships:  Higher random errors mean lower precision.  Higher systematic errors mean lower accuracy. Figure 2 Source: http://www.xtremepapers.com/revision/a-level/physics/measurement.php The relationships of errors from their respective true values SLOPE OF THE CURVE Sample Problem: Woody and Tom, backpacks ready, set foot for a trek. They traveled for 3500 meters (m) in 3000 seconds (sec), and then took a break when they have traveled for 3600 sec. When they took the road again, this time they have traveled for 8000 m in 4800 sec. How far did they travel in 3600 sec? If we are to quantify the given data, we can safely assume that time is independent (influences the other) from the distance. So, if we are to tabulate the data, the result is as follows Distance Traveled (m) Time (sec) 3500 3000 8000 4800 And, applying the values to Algebra, we get: Distance (y) Time (x) 3500 3000 8000 4800 01 Handout 1 *Property of STI Page 7 of 10  SH1685 Plotting the initial data, we get the chart as seen on the right. Adding the data with the missing value, we get: Distance (y) Time (x) 3500 3000 y 3600 8000 4800 As shown, we cannot simply define the missing value (y) because we haven’t found the relationship between the values, yet. To do so, we need to find first the slope of the curve. The slope of the curve (m) defines the relationship of all the given set of points in a graph. In this linear regression graph, we can see here that the relationship is a simple relationship. From the two (2) given points, we can say that a change exists. To determine the slope using two (2) points, we use the formula: 𝜕𝑦 𝑚= 𝜕𝑥 where: 𝑚 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑙𝑜𝑝𝑒 𝜕𝑦 𝑖𝑠 𝑡ℎ𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 (𝑦 − 𝑦0 ) 𝜕𝑥 𝑖𝑠 𝑡ℎ𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 (𝑥 − 𝑥0 ) So, using the obtained values, 𝜕𝑦 𝑚= 𝜕𝑥 (𝑦 − 𝑦0 ) 𝑚= (𝑥 − 𝑥0 ) (8000 − 3500) 𝑚= (4800 − 3000) 4500 𝑚= 1800 𝑚 = 2.5 01 Handout 1 *Property of STI Page 8 of 10  SH1685 So, the relationship or the slope between the two (2) values is equal to 2.5. Using this, we can now determine the value of the missing variable, y. Given the equation, 𝜕𝑦 𝑚= 𝜕𝑥 where: 𝑚 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑙𝑜𝑝𝑒 𝜕𝑦 𝑖𝑠 𝑡ℎ𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 𝑓𝑟𝑜𝑚 𝑡𝑤𝑜 𝑔𝑖𝑣𝑒𝑛 𝑝𝑜𝑖𝑛𝑡𝑠 (𝑦 − 𝑦0 ) 𝜕𝑥 𝑖𝑠 𝑡ℎ𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 𝑓𝑟𝑜𝑚 𝑡𝑤𝑜 𝑔𝑖𝑣𝑒𝑛 𝑝𝑜𝑖𝑛𝑡𝑠 (𝑥 − 𝑥0 ) We can now derive a new equation to find the missing value. 𝜕𝑦 𝑚= 𝜕𝑥 (𝑦 − 𝑦0 ) 𝑚= (𝑥 − 𝑥0 ) 𝑚(𝑥 − 𝑥0 ) = 𝑦 − 𝑦0 where: 𝑚 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑙𝑜𝑝𝑒 𝑦 𝑖𝑠 𝑡ℎ𝑒 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑝𝑜𝑖𝑛𝑡 𝑎𝑙𝑜𝑛𝑔 𝑦 𝑥 𝑖𝑠 𝑡ℎ𝑒 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑝𝑜𝑖𝑛𝑡 𝑎𝑙𝑜𝑛𝑔 𝑥 𝑦0 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑎𝑙𝑜𝑛𝑔 𝑦 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑝𝑜𝑖𝑛𝑡 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑝𝑜𝑖𝑛𝑡 𝑥0 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑎𝑙𝑜𝑛𝑔 𝑥 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑝𝑜𝑖𝑛𝑡 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑙 𝑝𝑜𝑖𝑛𝑡 Now, we can now look for 𝑦0. (2.5)[(4800 − 3600)] = 8000 − 𝑦0 3000 = 8000 − 𝑦0 𝑦0 = 8000 − 3000 𝑦0 = 5000 Completing the table, we get these values. Distance (y) Time (x) 3500 3000 5000 3600 8000 4800 In conclusion, Woody and Tim has traveled 5000 m, or 5 km, in 3600 seconds, or 1 hour. So, what do we call the formula that we used? We used the two-point slope formula. The linear graph that we have plotted has a standard formula, y = mx + b. We have used it to determine the answer to the previous question. Previously, we used the formula 𝑚(𝑥 − 𝑥0 ) = 𝑦 − 𝑦0. So, to determine the variables in y = mx + b, we need to derive it from 𝑚(𝑥 − 𝑥0 ) = 𝑦 − 𝑦0. 𝑚(𝑥 − 𝑥0 ) = 𝑦 − 𝑦0 𝑚𝑥 − 𝑚𝑥0 = 𝑦 − 𝑦0 𝑚𝑥 + 𝑦0 − 𝑚𝑥0 = 𝑦 𝑦 = 𝑚𝑥 + (𝑦0 − 𝑚𝑥0 ) 01 Handout 1 *Property of STI Page 9 of 10  SH1685 𝑦 = 𝑚𝑥 + 𝑏 The 𝑦 = 𝑚𝑥 + 𝑏 is the formula for every straight line. The variables here are as follows:  𝑚 is the slope  𝑦 is the given point along y  𝑥 is the given point along x  𝑏 is the point known as the y-intercept, a value computed from the difference between the initial y-value (𝑦0), and the product between the slope (𝑚) and initial x-value (x0) From these variables, we can then determine the statistically best line that will fit in with all the given points with the least distance from the line in the graph. Even if they are spread apart, finding this best-fitting line will determine their relationships. The general spreading of the points from the fitted line is what we call the standard deviation. The formulae for each deviation are as follows: 1. Standard deviation of the Slope – deviation/spread of the points from the slope itself 𝑛 𝑆𝑚 = 𝑆𝑦 √ 2 𝑛 ∑ 𝑥𝑖 − (∑ 𝑥𝑖 )2 2. Standard deviation of the y-intercept / b – deviation of the slope from the intercept ∑ 𝑥𝑖2 𝑆𝑏 = 𝑆𝑦 √ 𝑛 ∑ 𝑥𝑖2 − (∑ 𝑥𝑖 )2 References: Belleza, R., Gadong, E., & Sharma, M. (2016). General physics 1. Quezon City: Vibal Publishing House, Inc. Boundless. (2016). Boundless physics. Retrieved on October 13, 2016, from Boundless Physics: https://www.boundless.com/physics/textbooks/boundless-physics-textbook/the-basics- of-physics-1/units- 32/prefixes-and-other-systems-of-units-199-6022/ Brown, Theodore L., et al. (2004). Chemistry: The central science (9th ed.). Upper Saddle River, New Jersey: Pearson Education, Inc. IBGuides. (2012). Measurements and uncertainties. Retrieved on October 14, 2016, from IBGuides: http://ibguides.com/physics/notes/measurement-and-uncertainties XtremePapers. (2012). Measurement. Retrieved on October 13, 2016, from XtremePapers: http://www.xtremepapers.com/revision/a-level/physics/measurement.php 01 Handout 1 *Property of STI Page 10 of 10