1. Measurement Fundamental Quantities in Physics Units & Conversion
Chapter 1
Measurement
Fundamental Quantities in Physics
Units & Conversion

2. Three KEYS for Chapter 1
Fundamental quantities in physics (length, mass, time)
Units (meters, kilograms, seconds...)
Dimensional Analysis
Force = kg meter/sec2
Power = Force x Velocity = kg m2/sec3

3. Three KEYS for Chapter 1
Fundamental quantities in physics (length, mass, time)
Units (meters, kilograms, seconds...)
Dimensional Analysis
Significant figures in calculations
6.696 x 104 miles/hour
67,000 miles hour 3

4. Three KEYS for Chapter 1
Fundamental quantities in physics (length, mass, time)
Units (meters, kilograms, seconds...)
Dimensional Analysis
Significant figures in calculations
Estimation (order of magnitude ~10#) 4

5. Standards and units
Length, mass, and time = three fundamental quantities (“dimensions”) of physics.
The SI (Système International) is the most widely used system of units.
Meeting ISO standards are mandatory for some industries. Why?
In SI units, length is measured in meters, mass in kilograms, and time in seconds.

6. Converting Units A conversion factor is A ratio of units equal to 1
Used to convert between units
Units obey same algebraic rules as variables & numbers

7. Converting Units km

8. Converting Units km 1000 m = 1 km
Multiplying by 1 doesn’t change the overall value, just the units.

9. Converting Units km

10. Converting Units km

11. Converting Units km cm

12. Converting Units km cm

13. Unit consistency and conversions
An equation must be dimensionally consistent. Terms to be added or equated must always have the same units. (Be sure you’re adding “apples to apples.”)
OK: 5 meters/sec x 10 hours =~ 2 x 102 km
(distance/time) x (time) = distance

14. Unit consistency and conversions
An equation must be dimensionally consistent. Terms to be added or equated must always have the same units. (Be sure you’re adding “apples to apples.”)
OK: 5 meters/sec x 10 hours =~ 2 x 102 km
5 meters/sec x 10 hour x (3600 sec/hour)
= 180,000 meters = 180 km = ~ 2 x 102 km 14

15. Unit consistency and conversions
An equation must be dimensionally consistent. Terms to be added or equated must always have the same units. (Be sure you’re adding “apples to apples.”)
OK: 5 meters/sec x 10 hours =~ 2 x 102 km
NOT: 5 meters/sec x 10 kg = 50 Joules
(velocity) x (mass) = (energy) 15

16. Larger & smaller units for fundamental quantities.
Unit prefixes
Larger & smaller units for fundamental quantities.
Learn these – and prefixes like Mega, Tera, Pico, etc.!
Skip Ahead to Slide 24 – Sig Fig Example

17. Measurement & Uncertainty
No measurement is exact; there is always some uncertainty due to limited instrument accuracy and difficulty reading results.
Figure 1-2. Caption: Measuring the width of a board with a centimeter ruler. The uncertainty is about ±1 mm.

18. Measurement & Uncertainty
The precision – and also uncertainty - of a measured quantity is indicated by its number of significant figures.
Ex: 8.7 centimeters
2 sig figs
Specific rules for significant figures exist
In online homework, sig figs matter!
In exams, sig figs matter!!

19. Significant Figures
Number of significant figures = number of “reliably known digits” in a number.
Often possible to tell # of significant figures by the way the number is written:
23.21 cm = four significant figures.
0.062 cm = two significant figures (initial zeroes don’t count).

20. Significant Figures Significant figures are not decimal places
has 5 decimal places, but just significant figures
Generally, round to the least number of significant figures of the given data
25 x 18 → 2 significant figures;
25 x → still 2
Round up for 5+ (13.5 → 14, but 13.4 → 13)

21. Significant Figures Numbers ending in zero are ambiguous.
In general, trailing zeros are NOT significant
In other words, 3000 may have 4 significant figures
but usually 3000 will have only 1 significant figure!
Numbers ending in zero are ambiguous.
Does the last zero mean uncertainty to a factor of 10, or just 1?

22. Significant Figures Numbers ending in zero are ambiguous
Is 20 cm precise to 10 cm, or 1? We need rules!
20 cm = one significant figure (trailing zeroes don’t count w/o decimal point)
20. cm = two significant figures (trailing zeroes DO count w/ decimal point)
20.0 cm = three significant figures

23. Rules for Significant Figures
When multiplying or dividing numbers, or using functions, result has as many sig figs as term with fewest (the least precise).
ex: 11.3 cm x 6.8 cm = 77 cm.
When adding or subtracting, answer is no more precise than least precise number used.
ex: = 3, not 3.213!

24. Significant Figures
Calculators will not give right # of sig figs; usually give too many but sometimes give too few (especially if there are trailing zeroes after a decimal point).
top image: result of 2.0/3.0
bottom image: result of 2.5 x 3.2
Figure 1-3. Caption: These two calculators show the wrong number of significant figures. In (a), 2.0 was divided by 3.0. The correct final result would be In (b), 2.5 was multiplied by 3.2.The correct result is 8.0.

25. Scientific Notation Scientific notation commonly used
Uses powers of 10 to write large & small numbers

26. Scientific Notation
Scientific notation allows the number of significant figures to be clearly shown.
Ex: cannot easily tell how many significant figures in “36,900”.
Clearly x has three and x 104 has four!
Remember trailing zeroes DO count with a decimal point (always in Scientific Notation!)

27. Measurement & Uncertainty
No measurement is exact; there is always some uncertainty due to limited instrument accuracy and difficulty reading results.
Photo illustrates this – it would be difficult to measure the width of this board more accurately than ± 1 mm.
Figure 1-2. Caption: Measuring the width of a board with a centimeter ruler. The uncertainty is about ±1 mm.

28. Uncertainty and significant figures
Every measurement has uncertainty
Ex: 8.7 cm (2 sig figs)
“8” is (fairly) certain
8.6? 8.8?
8.71? 8.69?
Good practice – include uncertainty with every measurement!
8.7  0.1 meters 28

29. Uncertainty and significant figures
Uncertainty should match measurement in the least precise digit:
8.7  0.1 centimeters
8.70  0.10 centimeters
8.709  centimeters
8  1 centimeters
Not…
8.7 +/ cm 29

30. Relative Uncertainty
Relative uncertainty: a percentage, the ratio of uncertainty to measured value, multiplied by 100.
ex. Measure a phone to be 8.8 ± 0.1 cm What is the relative uncertainty in this measurement?

31. Uncertainty and significant figures
Physics involves approximations; these can affect the precision of a measurement. 31

32. Uncertainty and significant figures
As this train mishap illustrates, even a small percent error can have spectacular results! 32

33. Conceptual Example: Significant figures
Using a protractor, you measure an angle to be 30°. (a) How many significant figures should you quote in this measurement?
Figure 1-4. Caption: Example 1–1. A protractor used to measure an angle.
Response: (a) If you look at a protractor, you will see that the precision with which you can measure an angle is about one degree (certainly not 0.1°). So you can quote two significant figures, namely, 30° (not 30.0°). (b) If you enter cos 30° in your calculator, you will get a number like However, the angle you entered is known only to two significant figures, so its cosine is correctly given by 0.87; you must round your answer to two significant figures.

34. Conceptual Example: Significant figures
Using a protractor, you measure an angle to be 30°. (a) How many significant figures should you quote in this measurement? What uncertainty?
2 sig figs! (30. +/- 1 degrees or 3.0 x 101 +/- 1 degrees)
Figure 1-4. Caption: Example 1–1. A protractor used to measure an angle.
Response: (a) If you look at a protractor, you will see that the precision with which you can measure an angle is about one degree (certainly not 0.1°). So you can quote two significant figures, namely, 30° (not 30.0°). (b) If you enter cos 30° in your calculator, you will get a number like However, the angle you entered is known only to two significant figures, so its cosine is correctly given by 0.87; you must round your answer to two significant figures.

35. Conceptual Example: Significant figures
Using a protractor, you measure an angle to be 30°. (b) What result would a calculator give for the cosine of this result? What should you report?
Figure 1-4. Caption: Example 1–1. A protractor used to measure an angle.
Response: (a) If you look at a protractor, you will see that the precision with which you can measure an angle is about one degree (certainly not 0.1°). So you can quote two significant figures, namely, 30° (not 30.0°). (b) If you enter cos 30° in your calculator, you will get a number like However, the angle you entered is known only to two significant figures, so its cosine is correctly given by 0.87; you must round your answer to two significant figures.

36. Conceptual Example: Significant figures
Using a protractor, you measure an angle to be 30°. (b) What result would a calculator give for the cosine of this result? What should you report?
, but to two sig figs, 0.87!
Figure 1-4. Caption: Example 1–1. A protractor used to measure an angle.
Response: (a) If you look at a protractor, you will see that the precision with which you can measure an angle is about one degree (certainly not 0.1°). So you can quote two significant figures, namely, 30° (not 30.0°). (b) If you enter cos 30° in your calculator, you will get a number like However, the angle you entered is known only to two significant figures, so its cosine is correctly given by 0.87; you must round your answer to two significant figures.

37. 1-3 Accuracy vs. Precision
Accuracy is how close a measurement comes to the true value.
ex. Acceleration of Earth’s gravity = 9.81 m/sec2 Your experiment produces 10 ± 1 m/sec2
You were accurate! How accurate? Measured by ERROR.
|Actual – Measured|/Actual x 100%
| 9.81 – 10 | / 9.81 x 100% = 1.9% Error

38. Accuracy vs. Precision
Accuracy is how close a measurement comes to the true value
established by % error
Precision is a measure of repeatability of the measurement using the same instrument.
established by uncertainty in a measurement
reflected by the # of significant figures

39. Accuracy vs. Precision Example
Example: You measure the acceleration of Earth’s gravitational force in the lab, which is accepted to be 9.81 m/sec2
Your experiment produces m/sec2
Were you accurate? Were you precise?

40. Accuracy vs. Precision
Accuracy is how close a measurement comes to the true value. (established by % error)
ex. Your experiment produces m/sec2 for the acceleration of gravity (9.81 m/sec2)
Accuracy: (9.81 – 8.334)/9.81 x 100% = 15% error
Is this good enough? Only you (or your boss/customer) know for sure! 

41. Accuracy vs. Precision
Precision is the repeatability of the measurement using the same instrument.
ex. Your experiment produces m/sec2 for the acceleration of gravity (9.81 m/sec2)
Precision indicated by 4 sig figs
Seems (subjectively) very precise – and precisely wrong!

42. Better Technique: Include uncertainty
Accuracy vs. Precision
Better Technique: Include uncertainty
Your experiment produces m/sec2 +/ m/sec2
Your relative uncertainty is
.077/8.334 x 100% = ~1%
But your error was ~ 15%
NOT a good result!

43. Better Technique: Include uncertainty
Accuracy vs. Precision
Better Technique: Include uncertainty
Your experiment produces m/sec2 +/- 1.2 m/sec2
Your relative uncertainty is
1.2 / 8.3 x 100% = ~15%
Your error was still ~ 15%
Much more reasonable a result!

44. established by uncertainty in a measurement
Accuracy vs. Precision
Precision is a measure of repeatability of the measurement using the same instrument.
established by uncertainty in a measurement
reflected by the # of significant figures
improved by repeated measurements!
Statistically, if each measurement is independent
make n measurements (and n> 10)
Improve precision by √(n-1)
Make 10 measurements, % uncertainty ~ 1/3

45. 1-6 Order of Magnitude: Rapid Estimating
Quick way to estimate calculated quantity:
round off all numbers in a calculation to one significant figure and then calculate.
result should be right order of magnitude
expressed by rounding off to nearest power of 10
104 meters
108 light years

46. Order of Magnitude: Rapid Estimating
Example: Volume of a lake
Estimate how much water there is in a particular lake, which is roughly circular, about 1 km across, and you guess it has an average depth of about 10 m.
Figure 1-7. Caption: Example 1–5. (a) How much water is in this lake? (Photo is of one of the Rae Lakes in the Sierra Nevada of California.) (b) Model of the lake as a cylinder. [We could go one step further and estimate the mass or weight of this lake. We will see later that water has a density 1000 kg/m3, of so this lake has a mass of about (103 kg/m3)(107 m3) ≈ 1010 kg, which is about 10 billion kg or 10 million metric tons. (A metric ton is 1000 kg, about 2200 lbs, slightly larger than a British ton, 2000 lbs.)]
Answer: The volume of the lake is about 107 m3.

47. Order of Magnitude: Rapid Estimating
Example: Volume of a lake
Volume = Area x depth = (p x r2) x depth
= ~ 3 x 500 x 500 x 10
= ~75 x 105
= ~ 100 x 105
= ~ 107 cubic meters
Figure 1-7. Caption: Example 1–5. (a) How much water is in this lake? (Photo is of one of the Rae Lakes in the Sierra Nevada of California.) (b) Model of the lake as a cylinder. [We could go one step further and estimate the mass or weight of this lake. We will see later that water has a density 1000 kg/m3, of so this lake has a mass of about (103 kg/m3)(107 m3) ≈ 1010 kg, which is about 10 billion kg or 10 million metric tons. (A metric ton is 1000 kg, about 2200 lbs, slightly larger than a British ton, 2000 lbs.)]
Answer: The volume of the lake is about 107 m3.

48. Order of Magnitude: Rapid Estimating
Example: Volume of a lake
Volume = (p x r2) x depth
= 7,853, cu. m But…. Round to power of 10 for Order of Mag:
So ~ 107 cubic meters
Figure 1-7. Caption: Example 1–5. (a) How much water is in this lake? (Photo is of one of the Rae Lakes in the Sierra Nevada of California.) (b) Model of the lake as a cylinder. [We could go one step further and estimate the mass or weight of this lake. We will see later that water has a density 1000 kg/m3, of so this lake has a mass of about (103 kg/m3)(107 m3) ≈ 1010 kg, which is about 10 billion kg or 10 million metric tons. (A metric ton is 1000 kg, about 2200 lbs, slightly larger than a British ton, 2000 lbs.)]
Answer: The volume of the lake is about 107 m3.

49. 1-6 Order of Magnitude: Rapid Estimating
Example: Thickness of a page.
Estimate the thickness of a page of your textbook. (Hint: you don’t need one of these!)
Figure 1-8. Caption: Example 1–6. Micrometer used for measuring small thicknesses.
Answer: Measure the thickness of 100 pages. You should find that the thickness of a single page is about 6 x 10-2 mm.

50. Solving problems in physics
The online system offers a HUGE array of additional resources to help you visualize how to solve problems 50

51. Solving problems in physics
The online system offers a HUGE array of additional resources to help you visualize how to solve problems 51

52. Solving problems in physics
The textbook sample problems are IMPORTANT 52

53. Solving problems in physics – Step by Step!
Step 1: Identify KEY IDEAS, relevant concepts, variables, what is known, what is needed, what is missing. 53

54. Solving problems in physics
Step 2: Set up the Problem – MAKE a SKETCH, label it, act it out, model it, decide what equations might apply. What units should the answer have? What value? 54

55. Solving problems in physics
Step 3: Execute the Solution, and EVALUATE your answer! Are the units right? Is it the right order of magnitude? Does it make SENSE? 55

56. Solving problems in physics
Good problems to gauge your learning
“Test your Understanding” Questions throughout the book
Conceptual “Clicker” questions linked online
“Two dot” problems in the chapter
Good problems to review before exams
Checkpoints along the way
ODD problems with answers in the back
Exam reviews published online 56