Mathematics for Physical Sciences.pdf

Mathematics for Physical Sciences Objective In this lesson, you will understand basic mathematical concepts important to the physical sciences and successfully carry out mathematical operations Systems of Measurement The SI system originated i...

Mathematics for Physical Sciences Objective In this lesson, you will understand basic mathematical concepts important to the physical sciences and successfully carry out mathematical operations Systems of Measurement The SI system originated in France and has become an ______________________ accepted system of measurements to facilitate international __________ and communication. Although the United States has adopted the SI system for global trade, the United States _________________ system (USCS) continues to be the commonly used system of measurement within the United States. SI Prefixes This table lists some prefixes commonly used in the SI system. You can use it to find multiples of units. kilo 103 hecto h 102 100 deka da 101 10 d 10-1 0.1 c 10-2 milli 10-3 0.001 You may have noticed that the SI prefixes denote the integer powers of ______. So, to convert between these prefixes, you need to either multiply or ____________ by a suitable conversion factor, which is a power of 10. Scientific Notation A number written in scientific notation has three parts: a _______________, which is always greater than or equal to 1 and less than _____; a base, which is always 10; and an _______________, or the power to which the base is raised. Let's write 1,200,000 milligrams in scientific notation. First, determine the coefficient. The decimal point moved six places to the left to get the 1,200,000 = _____ × 106 coefficient, so the power of 10 is 6. Now let's write 0.0000000061 kilometers in scientific notation. The 0.0000000061 = _____ × 10-9 decimal point moved nine places to the right, so -9 is the power of 10. The exponent is positive for numbers with an absolute value greater than _____. It's negative for numbers between ____ and ____, and it's zero for numbers between _____ and _____. Converting from Scientific Notation to Standard Notation The moon is about 3.8 × 105 kilometers away from us. Let's walk through the process to convert this number from scientific notation to standard notation. 1. Write down the coefficient. In this case, it is ______. 2. Use the exponent to determine how many _________ to write after the coefficient. 3. The exponent is ____, so move the decimal point ________ places to the right to get 380,000. Qualities of a Measurement In physics, measurements are not just numbers. They represent both quantity and ____________. Accurate and _____________ measurements give dependable results and conclusions. Accuracy Versus Precision Accurate measurements are as close to the correct ___________ as possible. On the other hand, precise measurements are close to _______ ___________ but might not be close to the correct value. Precise measurements indicate that the experiment is _______________, which means it will yield similar ____________ when performed by a different scientist or at a different location, provided that all other conditions are the _________. Significant Figures To address the fact that measurements may not be as ______________ as they appear, scientists have adopted rules for what are called __________________ figures. In any measurement, the significant figures in a number include the digits that are definitely __________, plus the _______ digit, which is approximate. Here are some of the rules that define significant figures: All ______________ digits (1, 2, 3, 4, 5, 6, 7, 8, and 9) are significant. All zeros ______________ two nonzero digits (sandwiched zeros) are significant. Zeros that are both to the __________ of the decimal point and at the ______ of the measurement are significant. If a decimal place is displayed in a number greater than or equal to _____, any zeros to the ________ of the decimal point are significant. Zeros that are merely used as placeholders are _______ __________________. According to these rules, 12.234 has ______ significant figures, while 0.030 has _____ significant figures (3 and the final 0). So, we would not claim the same _________ of accuracy for a calculation that uses the value 0.030 grams as we would for a calculation using 12.234 grams. Adding and Subtracting Significant Figures It's important to follow certain rules when adding or subtracting numbers. Determine which is the __________ accurate value. Here, ________ is the least accurate of the two numbers. When setting up calculations, write numbers one above the other, making sure the decimal points are ______________. Then add or subtract: 25.113 + 3.5__ 28.613 We must round off the answer to the _________ number of decimal places as the least accurate value. So, we would round off our answer to _________. Multiplying and Dividing Significant Figures As with adding and subtracting, the answer when multiplying and dividing can be only as accurate as the __________ accurate value in the calculation. However, when we multiply or divide measurements, we count __________________ ____________ instead of decimal places. We must round off the answer to the same number of significant figures as the value with the ___________ significant figures. Scalar and Vector Quantities ❖ Quantities that need only magnitude are called ____________ quantities. Distance, for example, is a scalar quantity. ❖ Quantities that need magnitude and direction are called ____________ quantities. Displacement is a vector quantity. Speed and Velocity Speed is an example of a scalar quantity. It tells you the ___________________ (say, 12 kilometers per hour) but not the _________________. Velocity is a ______________ quantity. It tells you both the magnitude (12 kilometers per hour) and the direction (north). We can calculate your average speed by dividing the total _________________ you covered by the total ____________ it took. distance traveled The formula can be written as average speed =. time of travel Average Speed Versus Average Velocity The average velocity of a body in motion is based on the distance between the _______________ point and the ______________ point of the motion. Average velocity is equal to the ____________________ (or change in position) divided by the total time taken. 𝑑 It is written as 𝑣 =. 𝑡 In figure A, the starting and ending points are 1.2 kilometers apart, so the displacement is _______ kilometers. If you took an hour to run this 10–kilometer path, your average speed would be ______ kilometers per hour. But your average velocity would be ______ kilometers per hour at a heading of about _______° (as indicated by the compass). In figure B, the starting and ending points are the same, so the displacement is _________. If you took an hour to run this 10–kilometer path, your average speed would still be ______ kilometers per hour, but your average velocity would be ____ kilometers per hour. So, although you completed the race in one hour, moving at various speeds, your average velocity was ________ because you ended up at the ________________ point. Summary How can significant figures help you compare the accuracy of different values?