Introduction to Mechanics Dimensional Analysis and Unit Conversion

Transcription

1 Introduction to Mechanics Dimensional Analysis and Unit Conversion Lana Sheridan De Anza College Jan 10, 2018

2 Last time physics vocabulary definitions of base units dimensional analysis

3 Overview symbols for scaling units scientific notation measurement uncertainty and significant figures precision and accuracy unit conversions (non-si units)

4 Scale of Units Scale Prefix zetta peta teragigamegakilohectodeka decicentimillimicronanopicofemto- Symbol Z P T G M k h da d c m µ n p f

5 Unit Scaling Examples What is 8 cm in meters?

6 Unit Scaling Examples What is 8 cm in meters? (8 cm) ( 1 m 100 cm) = 0.08 m.

7 Unit Scaling Examples What is 8 cm in meters? (8 cm) ( 1 m 100 cm) = 0.08 m. What is 508 µs in seconds?

8 Unit Scaling Examples What is 8 cm in meters? (8 cm) ( 1 m 100 cm) = 0.08 m. What is 508 µs in seconds? ( (508 µs) 1 s 1,000,000 µs ) = s.

9 Unit Scaling Examples What is 8 cm in meters? (8 cm) ( 1 m 100 cm) = 0.08 m. What is 508 µs in seconds? ( (508 µs) 1 s 1,000,000 µs ) = s. What is 3 g/cm 3 in kg/m 3?

10 Unit Scaling Examples What is 8 cm in meters? (8 cm) ( 1 m 100 cm) = 0.08 m. What is 508 µs in seconds? ( (508 µs) 1 s 1,000,000 µs ) = s. What is 3 g/cm 3 in kg/m 3? (3 g/cm 3 ) ( ) 1 kg 1000 g ( ) 100 cm 3 1 m = 3000 kg/m 3.

11 Scientific Notation An alternate way to write numbers is in scientific notation. This is especially useful when numbers are very large or very small.

12 Scientific Notation An alternate way to write numbers is in scientific notation. This is especially useful when numbers are very large or very small. For example, the speed of light in a vacuum is roughly: 300,000,000 m/s In scientific notation, we could write this as: m/s

13 Scientific Notation An alternate way to write numbers is in scientific notation. This is especially useful when numbers are very large or very small. For example, the speed of light in a vacuum is roughly: 300,000,000 m/s In scientific notation, we could write this as: m/s This is the same thing = 100, 000, 000 so, ,000,000 = 300,000,000 m/s

14 Scientific Notation: One digit only before decimal! One reason to use scientific notation is to clearly convey the number of significant figures in a value. When a number is in scientific notation, there is one digit, followed by a decimal point, followed by more digits, if there is more than one significant figure. Here there are two significant figures: m/s Here there are 4 significant figures: m/s one digit one digit before the decimal + 3 digits after the decimal = 4 s.f.s

15 Scientific Notation vs Unit Scaling Prefixes In scientific notation, m/s Alternatively, we could write this with a unit prefix: 300 Mm/s where 1 Mm is one mega-meter,

16 Scientific Notation vs Unit Scaling Prefixes In scientific notation, m/s Alternatively, we could write this with a unit prefix: 300 Mm/s where 1 Mm is one mega-meter, or use kilometers: 300, 000 km/s or use a prefix with scientific notation: km/s

17 Measurement Uncertainty and Significant Figures All measuring devices are only so accurate. For example, a ruler has millimeter (mm) marks, but not micrometer (µm) marks.

18 Measurement Uncertainty and Significant Figures All measuring devices are only so accurate. For example, a ruler has millimeter (mm) marks, but not micrometer (µm) marks. The arrow is 35 mm from the end of the ruler. The uncertainty in this ruler measurement is ±0.5 mm.

19 Measurement Uncertainty and Significant Figures All measuring devices are only so accurate. For example, a ruler has millimeter (mm) marks, but not micrometer (µm) marks. The arrow is 35 mm from the end of the ruler. The uncertainty in this ruler measurement is ±0.5 mm.

20 Measurement Uncertainty and Significant Figures The uncertainty in any ruler measurement is ±0.5 mm.

21 Measurement Uncertainty and Significant Figures The uncertainty in any ruler measurement is ±0.5 mm. Does it make sense to report a ruler measurement to six significant figures: cm? or mm?

22 Measurement Uncertainty and Significant Figures The uncertainty in any ruler measurement is ±0.5 mm. Does it make sense to report a ruler measurement to six significant figures: cm? or mm? No. Quote it as 35 mm or even 35.0 ± 0.5 mm

23 Measurement Uncertainty and Significant Figures The uncertainty in any ruler measurement is ±0.5 mm. Does it make sense to report a ruler measurement to six significant figures: cm? or mm? No. Quote it as 35 mm or even 35.0 ± 0.5 mm A simple rule: If inputs to a problem or experiment are given to 3 significant figures, give the output to 3 significant figures. If given to 2 significant figures, give answer to 2 significant figures, and so on.

24 Measurement Uncertainty and Significant Figures The uncertainty in any ruler measurement is ±0.5 mm. Does it make sense to report a ruler measurement to six significant figures: cm? or mm? No. Quote it as 35 mm or even 35.0 ± 0.5 mm A simple rule: If inputs to a problem or experiment are given to 3 significant figures, give the output to 3 significant figures. If given to 2 significant figures, give answer to 2 significant figures, and so on. How could this be written in scientific notation in units of meters?

25 Measurements: Precision and Accuracy Ideally, a measurement should be both precise and accurate. Precision A measurement is precise if it yields very similar results when repeated. Accuracy A measurement is accurate if it its result is very close to the true value.

26 Measurements: Precision and Accuracy 1 Graph created by Pekaje, based on PNG version by Anthony Cutler, Wikipedia.

27 Unit Conversion [L] represents any length unit, whereas [m] is specifically meters. There are other units of length such as feet, inches, miles, bu, li, parsecs, etc. It is sometimes necessary to change units. Example: what is 9 inches (in) in feet (ft)?

28 Unit Conversion [L] represents any length unit, whereas [m] is specifically meters. There are other units of length such as feet, inches, miles, bu, li, parsecs, etc. It is sometimes necessary to change units. Example: what is 9 inches (in) in feet (ft)? 3/4 of a foot, or 0.75 feet. 12 in = 1 ft. ( ) 1 foot (9 inches) = 9 12 inches 12 ft

29 Unit Conversion [L] represents any length unit, whereas [m] is specifically meters. There are other units of length such as feet, inches, miles, bu, li, parsecs, etc. It is sometimes necessary to change units. Example: what is 9 inches (in) in feet (ft)? 3/4 of a foot, or 0.75 feet. 12 in = 1 ft. ( ) (9 1 foot inches) = 3 12 inches 4 ft

30 Unit Conversion Examples To solve that problem, we multiplied the value we wished to convert by 1. ( ) 1 foot (9 inches) = 0.75 ft 12 inches }{{} 1 Any number times 1 remains unchanged. The value remains the same, but the units change, in this case, from inches to feet.

31 Unit Conversion Example: what is 9 inches (in) in feet (ft)? 3/4 of a foot, or 0.75 feet. 12 in = 1 ft. ( ) 1 foot (9 inches) 12 = 3 inches 4 ft

32 Unit Conversion Examples To solve that problem, we multiplied the value we wished to convert by 1. ( ) 1 foot (9 inches) = 0.75 ft 12 inches }{{} 1 Any number times 1 remains unchanged. The value remains the same, but the units change, in this case, from inches to feet.

33 Unit Conversion Examples The distance between two cities is 100 mi. What is the number of kilometers between the two cities? A smaller than 100 B larger than 100 C equal to 100

34 Unit Conversion Examples The distance between two cities is 100 mi. What is the number of kilometers between the two cities? A smaller than 100 B larger than 100 C equal to 100

35 Unit Conversion Examples It may be necessary to change units several times to get to the unit you need. Example: how many seconds are there in a day?

36 Unit Conversion Examples It may be necessary to change units several times to get to the unit you need. Example: how many seconds are there in a day? (1 day) ( ) ( ) ( ) 24 hr 60 min 60 s 1 day 1 hr 1 min

37 Unit Conversion Examples It may be necessary to change units several times to get to the unit you need. Example: how many seconds are there in a day? ( 24 (1 hr day) 1 day ) ( 60 min 1 hr = s = 86, 400 s ) ( 60 s 1 min )

38 Unit Conversion Examples What is 60.0 mi/hr in m/s? (mi is miles, hr is hours)

39 Unit Conversion Examples What is 60.0 mi/hr in m/s? (mi is miles, hr is hours) 1 mi = km

40 Unit Conversion Examples What is 60.0 mi/hr in m/s? (mi is miles, hr is hours) 1 mi = km ( km (60.0 mi/hr) 1 mi ) ( ) ( 1000 m 1 hr 1 km 60 min ) ( ) 1 min 60 s

41 Unit Conversion Examples What is 60.0 mi/hr in m/s? (mi is miles, hr is hours) 1 mi = km ( 60.0 mi hr ) ( km 1 mi ) ( 1000 m 1 km ) ( 1 hr 60 min ) ( ) 1 min 60 s

42 Unit Conversion Examples What is 60.0 mi/hr in m/s? (mi is miles, hr is hours) 1 mi = km ( 60.0 mi hr ) ( km 1 mi ) ( ) ( ) ( 1000 m 1 hr 1 ) min 1 km 60 min 60 s = m/s = 26.8 m/s

43 Summary symbols for scaling units scientific notation measurement uncertainty and significant figures precision and accuracy unit conversions with non-si units Quiz Start of class tomorrow (Thursday). Homework unit conversion worksheet, due tomorrow Thurs, Jan 11. Walker Physics: (this will not be collected) Ch 1, onward from page 14. Questions: Problems: 15, 23, 25, 41, 49