A Physics Toolkit

1. Chapter A Physics Toolkit 1

2. Chapter A Physics Toolkit 1 In this chapter you will: • Use mathematical tools to measure and predict. • Apply accuracy and precision when measuring. • Display and evaluate data graphically.

3. Section Mathematics and Physics 1.1 In this section you will: • Be able to answer the question, what is physics? • Review the algebra required for this class. • Learn the GUESS problem solving strategy. • Use and do conversions with SI units. • Evaluate answers using dimensional analysis. • Perform arithmetic operations using scientific notation. • Learn about and use significant figures.

4. Section Mathematics and Physics 1.1a What is Physics? • Physics is a branch of science that involves the study of the physical world: energy, matter, and how they are related. • Learning physics will help you to understand the physical world. • Physics is considered the basis for all other sciences: - Biology, Chemistry, Astronomy, Geology, etc. • Physics is the fundamental science.

5. Section Mathematics and Physics 1.1 What does one do as a physicist? • Many research physicists work in environments where they perform basic research in industry, research universities, and astronomical observations. • Physicists who find new ways to use physics are often employed by engineering, business, law, and consulting firms. • Physicists are also extremely valuable in areas such as computer science, medicine, communications, and publishing. • Finally, many physicists who love to see young people get excited about physics become teachers.

6. Section Mathematics and Physics 1.1 What jobs do non-physicists hold that use physics every day? • Every job has some relation to physics! • Athletes - The laws of motion to lift, throw, push, hit, tackle, run, drag, jump, and crawl. - The more an athlete and coach understand and use their knowledge of physics in their sport, the better the athlete will become.

7. Section Mathematics and Physics 1.1 • Automotive mechanics • ◦ areas such as optics, electricity and magnetism, thermodynamics, and mechanics play greater roles as vehicles become more and more complex. • CAT scan, computed tomography, and magnetic resonance imaging (MRI) • ◦ technicians in hospitals and medical clinics who use such technology must have an understanding of what X- rays and magnetic resonance imaging are, how they behave, and how such high-tech instruments are to be used.

8. Section Mathematics and Physics 1.1 Mathematics in Physics • Mathematics is the language of science. • In physics, equations are important tools for modeling observations and for making predictions.

9. Section Mathematics and Physics 1.1 Order Of Operations Simply: 2 + 5 x √ Correct: 2 + 5 x X Incorrect: ( 2 + 5 ) x = 7 x

10. Section Mathematics and Physics 1.1 Order Of Operations : PEMA P arenthesesand Brackets Exponents M ultiplicationand Division (from left to right) A dditionand Subtraction (from left to right)

11. Section Mathematics and Physics 1.1 Order Of Operations : Example 1 Step 1: parentheses Step 2: exponent Step 3: multiplication Step 4: addition Step 5: solution

12. Section 1.1 Order Of Operations : Example 2 Step 1: Step 2: distribute 8 into (x + 1) Step 3: remove 1st parenthesis Step 4: combine like terms Step 5: within parenthesis, left to right, first comes division Step 6: then multiplication Step 7: simplify exponent Step 8: solution

13. Section Mathematics and Physics 1.1 Algebra Review with Physics • Variables are used to represent concepts. ex: d for displacement, t for time, p for momentum • Units also are abbreviated. ex: m for meters, s2 for seconds squared • Do not confuse variables with units. • Subscripts are used to give more information about a variable. ex: vave : average velocity vi : initial velocity

14. Section Mathematics and Physics 1.1 Problem Solving Strategy • Given: Write down given or known quantities. Draw a picture. • Unknown: Write down the unknown variable. • Equation: Find an applicable equation. Isolate the unknown variable. • Substitutenumbers with units into the equation. Solve. • Sense: Are the units correct? Does the answer make sense?

15. Section Mathematics and Physics 1.1 Example: Electric Current The voltage across a circuit equals the current multiplied by the resistance in the circuit. That is, V (volts) = I (amperes) × R (ohms). What is the resistance of a light bulb that has a 0.75 amperes current when plugged into a 120-volt outlet? Step 1: Given - Write the given quantities with units. Step 2: Unknown - Identify unknown variable. Given: I = 0.75 amperes V = 120 volts Unknown: R = ?

16. Section Mathematics and Physics 1.1 Electric Current Step 3: Equation. Rewrite the equation so that the unknown value is alone on the left. V = IR IR = V Reflexive property of equality. Divide both sides by I. Step 4: Substitute 120 volts for V, 0.75 amperes for I. Solve. 120 volts = R 0.75 amperes R = 160  Resistance will be measured in ohms.

17. Section Mathematics and Physics 1.1 Electric Current • Are the units correct? 1 volt = 1 ampere-ohm, so the answer in volts/ampere is in ohms, as expected. • Does the answer make sense? 120 is divided by a number a little less than 1, so the answer should be a little more than 120. Step 5: Sense

18. Section Mathematics and Physics 1.1 SI Units • Units are CRITICAL in physics. • It is the units that give meaning to the numbers. It is helpful to use units that everyone understands. • Scientific institutions have been created to define and regulate measures. • The worldwide scientific community and most countries currently use an adaptation of the metric system to state measurements.

19. Section Mathematics and Physics 1.1 SI Units • The Système International d’Unités, or SI, uses seven base quantities, also called the fundamental units.

20. Section Mathematics and Physics 1.1 SI Units • The base quantities were originally defined in terms of direct measurements. Other units, called derived units, are created by combining the base units in various ways. ex: speed is measured in meters per second (m/s) • The SI system is regulated by the International Bureau of Weights and Measures in Sèvres, France. • This bureau and the National Institute of Science and Technology (NIST) in Gaithersburg, Maryland, keep the standards of length, time, and mass against which our metersticks, clocks, and balances are calibrated.

21. Section Mathematics and Physics 1.1 SI Units • Measuring standards for kilogram and meter are shown below.

22. Section Mathematics and Physics 1.1 SI Units • You probably learned in math class that it is much easier to convert meters to kilometers than feet to miles. • The ease of switching between units is another feature of the metric system. • To convert between SI units, multiply or divide by the appropriate power of 10.

23. Section Mathematics and Physics 1.1 • Prefixes are used to change SI units by powers of 10, as shown in the table below.

24. Section Mathematics and Physics 1.1 Scientific Notation • A number in the form a x 10n is written in scientific notation where 1 ≤ a < 10, and n is an integer. (An integer is a whole number, not a fraction, that can be positive, negative, or zero.) • When moving the decimal point to the right, you reduce the exponent when using scientific notation. Right – REDUCE When moving the decimal point to the left, you make the exponent larger when using scientific notation. LEFT – LARGER • Common powers of ten include 100 = 1, 101 = 10, 102 = 100, etc.

25. Section Mathematics and Physics 1.1 Scientific Notation: Example 1 Write 7,530,000 in scientific notation. The value for a is 7.53 (The decimal point is to the right of the first non-zero digit.) So the form will be 7.53 x 10n. 7,530,000. = 7.53 x 106 (Move the decimal point 6 places to the left; exponent gets larger.)

26. Section Mathematics and Physics 1.1 Scientific Notation: Example 2 Write 0.000000285 in scientific notation. The value for a is 2.85 (The decimal point is to the right of the first non-zero digit.) So the form will be 2.85 x 10n. 0.000000285 = 2.85 x 10-7 (Move the decimal point to the right 7 places; exponent gets smaller.)

27. Section Mathematics and Physics 1.1 Dimensional Analysis • You often will need to use different versions of a formula, or use a string of formulas, to solve a physics problem. • To check that you have set up a problem correctly, write the equation or set of equations you plan to use with the appropriate units. • Before performing calculations, check that the answer will be in the expected units. • For example, if you are finding a speed and you see that your answer will be measured in s/m or m/s2, you know you have made an error in setting up the problem. • This method of treating the units as algebraic quantities, which can be cancelled, is called dimensional analysis.

28. Section Mathematics and Physics 1.1 Dimensional Analysis example: Calculate the distance a car travels when it is moving at a velocity of 20 meters per second for 10 seconds. Use the formula: distance = velocity x time ? Use dimensional analysis: meters = meters x seconds second Treat the units as if they were algebraic quantities. ? meters = meters x seconds second Seconds in the numerator cancel seconds in the denominator. The formula is therefore dimensionally correct. meters = meters 

29. Section Mathematics and Physics 1.1 Dimensional Analysis • Dimensional analysis also is used in choosing conversion factors. This is also known as the factor-label method. • A conversion factor is a multiplier equal to 1. For example, because 1 kg = 1000 g, you can construct the following conversion factors:

30. Section Mathematics and Physics 1.1 Factor-Label Method • Choose a conversion factor that will make the units cancel, leaving the answer in the correct units. • For example, to convert 1.34 kg of iron ore to grams, do as shown below: 1.34 kg 1000 g = 1,340 g 1 kg

31. Section Mathematics and Physics 1.1 8760 hours/year Factor-Label Method: Example 2 Convert 1 year to hours. 1 year 365 days 24 hours = 8,750 hours 1 year 1 day

32. Section Mathematics and Physics 1.1 8760 hours/year Warmup Problem Pressure is defined as force divided by area of contact ( P = F / A). What pressure must you apply to your pen in order to create a force of 0.25 N on a piece of paper, if the tip of the pen has a surface area of 3 mm2 touching the paper?

33. Section Mathematics and Physics 1.1 Rules for Rounding Off 1. In a series of calculations, carry the extra digits through to the final answer, then round off. Rounding only occurs ONCE in a calculation! 2. If the digit to be removed is:  <5, the preceding stays the same. example: 1.33 rounds to 1.3  5 or greater, the preceding digit increases by 1. example: 1.36 rounds to 1.4. Example: Round 62.5347 to four significant figures. Look at the fifth figure. It is a 4, a number less than 5. Therefore, simply drop every figure after the fourth, and the number becomes 62.53

34. Section Mathematics and Physics 1.1 Significant Digits (Significant Figures or Sig Figs) • Definition: All the valid digits in a measurement, the number of which indicates the measurement’s precision (degree of exactness). • Do not count sig figs for non-measurement quantities such as counting (4 washers) or exact conversion factors (24 hrs. in 1 day). • Use the Atlantic & Pacific Rule to determine the sig figs. PACIFIC OCEAN ATLANTIC OCEAN

35. Section Mathematics and Physics The Atlantic /Pacific Rule for Sig Figs If the… 1.1 • Decimal is Present • Count all digits from the Pacific side from the first non-zero digit. • Decimal is Absent • Count from the Atlantic side from the first non-zero digit. • Trailing zeros are indeterminate; they may or may not be significant. Use scientific notation to remove the ambiguity.

36. Section Mathematics and Physics 1.1 How many sig figs are there? Count all the digits from the first non-zero digit. PACIFIC OCEAN Decimal point present ATLANTIC OCEAN Decimal point absent 3 sig figs (and 1 indeterminate) 5 sig figs 705.00 g 2130 m 523.0 g 706, 000 g 0.0098070 mm 9,010, 000 km 4 sig figs 3 sig figs (and 3 indeterminate) 5 sig figs 3 sig figs (and 4 indeterminate)

37. Section Mathematics and Physics 1.1 Significant Digits • Using the Atlantic rule, we can’t be sure if trailing zeros are significant or not. To specify the exact number of sig figs, use scientific notation. • Exponents do not count towards significant digits. Example: 9,010, 000 km (3 sig figs and 4 indeterminate). To write this number indicating: 3 significant digits: 4 significant digits: 5 significant digits: 9.01 x 106 km 9.010 x 106 km 9.0100 x 106 km

38. Section Mathematics and Physics 1.1 Operations with Significant Digits • When you perform any arithmetic operation, it is important to remember that the result never can be more precise than the least-precise measurement. • Add / Subtract: Round to the least number of DECIMAL places as determined by the original calculation. (All numbers must be the same power of 10). Example: 23.1 4.77 125.39 + 3.581 156.841 Round to 156.8 (one decimal place)

39. Section Mathematics and Physics 1.1 Significant Digits – Addition / Subtraction Example: Add 5.861 dL + 2.614 L + 3.5 mL Convert to the same units and power of 10. Add in column form. 5.861 dL = 5.861 x 10-1 L = 0.5861 L 2.614 L 3.5 mL = 3.5 x 10-3 L = 0.0035 L 3.2036 L Round to the least amount of decimal places 3.204 L  fewest decimal places

40. Section Mathematics and Physics 1.1 Significant Digits – Multiplication / Division • Multiplication / Division: Round to the fewest number of SIGNIFICANT FIGURES. Example: (3.64928 x 105) (7.65314 x 107) (5.2 x 10-3) (5.7254 x 105) least precise measurement = (3.64928 x 105) x (7.65314 x 107) ÷ (5.2 x 10-3) ÷ (5.7254 x 105) = 9.3808 x 109 = 9.4 x 109 because the least precise measurement has 2 sig figs.

41. Section Mathematics and Physics 1.1 Significant Digits – Combination Operations • When doing a calculation that requires a combination of addition/subtraction and multiplication/division, use the multiplication rule. Example: slope = 70.0 m – 10.0 m = 3.3 m/s 29 sec – 11 sec 29 sec and 11 sec only have two significant digits each, so the answer should only have two significant digits

42. Section Mathematics and Physics 1.1 Multistep Calculations Do not round to 580 N2 and 1300 N2 Do not round to 1800 N2 Final answer, so it should be rounded to two sig figs.

43. Section Section Check 1.1 Question 1 The potential energy, PE, of a body of mass, m, raised to a height, h, is expressed mathematically as PE = mgh,where g is the gravitational constant. If m is measured in kg, g in m/s2, h in m, and PE in joules, then what is 1 joule described in base unit? • 1 kg·m/s • 1 kg·m/s2 • 1 kg·m2/s • 1 kg·m2/s2

44. Section Section Check 1.1 Answer 1 Answer:D Reason:

45. Section Section Check 1.1 Question 2 A car is moving at a speed of 90 km/h. What is the speed of the car in m/s? (Hint: Use Dimensional Analysis) • 2.5×101 m/s • 1.5×103 m/s • 2.5 m/s • 1.5×102 m/s

46. Section Section Check 1.1 Answer 2 Answer:A Reason:

47. Section Section Check 1.1 Question 3 Which of the following representations is correct when you solve 0.03 kg + 3.333 kg? • 3 kg • 3.4 kg • 3.36 kg • 3.363 kg

48. Section Section Check 1.1 Answer 3 Answer:C Reason:When you add or subtract, round to the least number of decimal places.