1. PHY115 – Sault College – Bazlurslide 1 Measurement and The Metric System

2. PHY115 – Sault College – Bazlurslide 2 Standards of Measure One cubit ?

3. PHY115 – Sault College – Bazlurslide 3 Standards of Measure

4. PHY115 – Sault College – Bazlurslide 4 Standards of Measure When two people work together, they should both use the same standards of measure.

5. PHY115 – Sault College – Bazlurslide 5 Standards of Measure http://news.bbc.co.uk/1/shared/spl/hi/sci_nat/03 /race_to_mars/timeline/html/1999.stmhttp://news.bbc.co.uk/1/shared/spl/hi/sci_nat/03 /race_to_mars/timeline/html/1999.stm September 1999 Another Nasa space craft, Mars Climate Orbiter, is lost as it arrives at the Red Planet. A mix-up over units for a key space craft operation is blamed - one team used English units while the other used metric.

6. PHY115 – Sault College – Bazlurslide 6 SI Units

7. PHY115 – Sault College – Bazlurslide 7 Derived SI Units http://physics.nist.gov/cuu/Units/SIdiagram.html

8. PHY115 – Sault College – Bazlurslide 8 Prefixes for SI Units http://en.wikipedia.org/wiki/SI http://en.wikipedia.org/wiki/SI_prefix

9. PHY115 – Sault College – Bazlurslide 9 Prefixes for SI Units http://en.wikipedia.org/wiki/SI_prefix

10. PHY115 – Sault College – Bazlurslide 10 Metric System During the 1790s, a decimal system based on our number system, the metric system, was being developed in France. –Easy to use –Easy to remember –Uses prefixes, that made the basic units larger or smaller by multiples or fractions of 10 For example: 1km = 1000 m = 10,000 dm = 100,000 cm 1 mi = 1760 yd = 5280 ft = 63,360 in The only country left behind is the USA.

11. PHY115 – Sault College – Bazlurslide 11 Imperial and U.S. customary systems of measurement http://en.wikipedia.org/wiki/Comparison_of_the_Imperial_and_US_customary_systems Both the Imperial (UK and Canada) and U.S. customary systems of measurement derive from earlier English systems.ImperialU.S. customarysystems of measurementEnglish systems Comparison of Imperial and U.S. volume measures 1 liquid U.S. gallon = 3.785 411 784 litres ≈ 0.833 Imperial gallon 1 Imperial gallon = 4.546 09 litres ≈ 1.201 liquid U.S. gallons On January 1, 1983, the metric systems and SI units were introduced in Canada.January 11983

12. PHY115 – Sault College – Bazlurslide 12 Systems of Measurement United States Customary System (USCS) Formally called British System Used in the US and Burma –Length: foot –Weight/force: pound –Time: second Systeme International (SI) Also called the Metric or International System Used everywhere else in the world!

13. PHY115 – Sault College – Bazlurslide 13 Systeme International (SI) QuantityUnitSymbol Lengthmeterm Masskilogramkg Timeseconds ForcenewtonN EnergyjouleJ CurrentampereA TemperaturekelvinK

14. PHY115 – Sault College – Bazlurslide 14 SI Conversions Major advantage – the decimal system – all digits are related to one another – multiples of 10! 1 kilometer = 1000 meters = 100,000 cm 1 meter = 100 cm = 0.001 kilometer

15. PHY115 – Sault College – Bazlurslide 15 Scientific Notation Scientists often use very large or very small numbers that can not be conveniently written as fractions or decimal fractions. For example, the thickness of an oil film on water is about 0.0000001 m In scientific notation it is 1 x 10 -7 m 0.1 = 1 x 10 -1 0.001 = 1 x 10 -3 10,000 = 1 x 10 4

16. PHY115 – Sault College – Bazlurslide 16 Scientific Notation 0.1 = 1 x 10 -1 0.001 = 1 x 10 -3 10,000 = 1 x 10 4 Any number can be written as a product of a number between 1 and 10 and a power of 10. In general, M x 10 n ; Where M, is the a number between 1 and 10 and n, is the exponent or power of 10.

17. PHY115 – Sault College – Bazlurslide 17 Decimal to Scientific Notation 578 = 5.78 x 10 2 0.025 = 002.5 x 10 -2 3.5 = 3.5 x 10 0 Place a decimal point after the first nonzero digit reading from left to right. Place a caret (^) at the position of the original decimal point. The exponent of 10 is the number of places from the caret to the decimal point. If the decimal point is to the right of the caret, the exponent of 10 is a negative number. ^ ^ ^

18. PHY115 – Sault College – Bazlurslide 18 Scientific Notation to Decimal 5.78 x 10 2 = 578 2.5 x 10 -2 = 0.025 3.5 x 10 0 = 3.5 Multiply the decimal part by the power of 10. –Move the decimal point to the right by the exponent - If the exponent is a positive number –Move the decimal point to the left by the exponent - If the exponent is a negative number Add zeros as needed.

19. PHY115 – Sault College – Bazlurslide 19 Metric Length The basic SI unit of length is the metre (m). Originally 1m = distance from the equator to either pole/10,000,000 “The metre is the length of path traveled by light in a vacuum during a time interval of 1/299,792,458 s –Km –m –cm

20. PHY115 – Sault College – Bazlurslide 20 Conversion Factor A conversion factor is an expression used to change from one unit to another. Expressed as a fraction whose numerator and denominator are equal quantities in two different units. The information necessary for forming a conversion factor is usually found in their conversion table as follows: 1 m = 100 cm So, the conversion factors are: 1 m and 100 cm 100 cm 1 m

21. PHY115 – Sault College – Bazlurslide 21 Conversion using Conversion Factor So, convert 5m to cm: 5 m x 100 cm = 500 cm 1 m Where the unit of the denominator should be the same as the original unit, so they cancels out. So, convert 7 cm to m: 7 cm x 1 m = 0.07 m 100 cm

22. PHY115 – Sault College – Bazlurslide 22 Conversion Factors as unit values A conversion factor is an expression used to change from one unit to another. 1 m = 100 cm So, the conversion factors are: 1 m and 100 cm 100 cm 1 m These conversion factors can be read as: per cm (or, 1 cm = m) per m (or, 1 m = 100 cm) 1 m 100 100 cm 1 1 10 0

23. PHY115 – Sault College – Bazlurslide 23 Conversion using units value Or, it can be converted as follows: 5 m = 5 x 1 m = 5 x 100 cm = 500 cm Similarly, 7 cm = 7 x 1 cm = 7 x 1 m = 0.07 m 100 1 m = 100 cm 100 cm = 1 m Therefore, 1 cm = (1/100) m

24. PHY115 – Sault College – Bazlurslide 24 Metric-English Conversion To change from an English unit to a metric unit or from a metric unit to an English unit, we use a conversion factor, from the relation 1 in = 2.54 cm. So, the conversion factors are: 1 in and 2.54 cm 2.54 cm1 in

25. PHY115 – Sault College – Bazlurslide 25 Area The area of a plane surface is the number of square units that it contains. To measure the surface area of an object, you must first decide on a standard unit of area. Standard units of area are based on the square of standard lengths, for example 1 square m.

26. PHY115 – Sault College – Bazlurslide 26 Area Find the area of a rectangle 5 m long and 3 m wide. By simply counting the number of squares, we find the area of the rectangle is 15 m 2. Or, by using the formula A = l x w = 5 m x 3 m = (5 x 3) (m x m) = 15 m 2

27. PHY115 – Sault College – Bazlurslide 27 Volume The volume of a figure is the number of cubic units that it contains. Standard units of volume are based the cube of standard lengths, such as cubic meter, cubic cm, cubic in.

28. PHY115 – Sault College – Bazlurslide 28 Volume Find the volume of a rectangular prism 6 cm long, 4 cm wide, and 5 cm high. To find the volume of the rectangular solid, count the number of cubes in the bottom layer and then multiply by the number of layers. Or, V = l w h = 6 x 4 x 5 cm x cm x cm = 120 cm 3

29. PHY115 – Sault College – Bazlurslide 29 Mass The mass of an object is the quantity of material making up the object. One unit of mass in the metric system is the gram (g). The gram is defined as the mass of 1 cm 3 of water at its maximum density (at 4  C). Since the gram is so small, kg is the basic unit of mass in SI (Système international d'unités).

30. PHY115 – Sault College – Bazlurslide 30 Weight The weight of an object is a measure of the gravitational force or pull acting on an object. The weight unit in the metric system is the newton (N). An apple weighs about one newton. A newton is the amount of force required to accelerate a mass of one kilogram by one meter per second squared.acceleratemasskilogrammeter per second squared 1 N = 1 kg·m/s² The pound (lb), a unit of force, is one of the basic English system units. It is defined as the pull of the earth on a cylinder of a platinum-iridium alloy that is stored in a vault at the U.S. Bureau of Standards. 1 N = 0.225 lb 1 lb = 4.45 N

31. PHY115 – Sault College – Bazlurslide 31 kg with weight When the weight of an object is given in kilograms, the property intended is almost always mass. Occasionally the gravitational force on an object is given in "kilograms", but the unit used is not a true kilogram: it is the deprecated kilogram-force (kgf), also known as the kilopond (kp).kilopond An object of mass 1 kg at the surface of the Earth will be subjected to a gravitational force of approximately 9.80665 newtons (the SI unit of force).Earthnewtons http://en.wikipedia.org/wiki/Kilogram http://en.wikipedia.org/wiki/Newton

32. PHY115 – Sault College – Bazlurslide 32 Time The basic unit of time is second (s) in both system. It was defined as 1/86400 of a mean solar day. Now the standard second is defined more precisely in terms of frequency of radiation emitted by cesium atoms when they pass between two particular states; that is, the time required for 9,192,631,770 periods of this radiation.

33. PHY115 – Sault College – Bazlurslide 33 Electrical Units The ampere (A) is the basic unit and is measure of the amount of electric current. Derived units are: Columb (C) – is a measure of the amount of electrical charge Volt (V) – is a measure of electric potential Watt (W) - is a measure of power

34. PHY115 – Sault College – Bazlurslide 34 Accuracy vs. Precision Accuracy: A measure of how close an experimental result is to the true value. Precision: A measure of how exactly the result is determined. It is also a measure of how reproducible the result is. –Absolute precision: indicates the uncertainty in the same units as the observation –Relative precision: indicates the uncertainty in terms of a fraction of the value of the result

35. PHY115 – Sault College – Bazlurslide 35 Accuracy Physicists are interested in how closely a measurement agrees with the true value. This is an indication of the quality of the measuring instrument. Accuracy is a means of describing how closely a measurement agrees with the actual size of a quantity being measured.

36. PHY115 – Sault College – Bazlurslide 36 Error The difference between an observed value and the true value is called the error. The size of the error is an indication of the accuracy. Thus, the smaller the error, the greater the accuracy. The percentage error determined by subtracting the true value from the measured value, dividing this by the true value, and multiplying by 100.

37. PHY115 – Sault College – Bazlurslide 37 Error

38. PHY115 – Sault College – Bazlurslide 38 Significant Digits The accuracy of a measurement is indicated by the number of significant digits. Significant digits are those digits in the numerical value of which we are reasonably sure. More significant digits in a measurement the accurate it is:

39. PHY115 – Sault College – Bazlurslide 39 Significant Digits More significant digits in a measurement the accurate it is: E.g., the true value of a bar is 2.50 m Measured value is 2.6 m with 3 significant digits. The percentage error is (2.6-2.50)*100/2.50 = 4% E.g., the true value of a bar is 2.50 m Measured value is 2.55 m with 3 significant digits. The percentage error is (2.55-2.50)*100/2.50 = 0.2% Which one is more accurate? The one which has more significant digits

40. PHY115 – Sault College – Bazlurslide 40 Rules for Determining “Significant Digits” All non zero digits are significant All zeros between significant non zero digits are significant. 450.09  5 significant digits A zero in a number (> 1) which is specially tagged, such as by a bar above it, is significant. 250,000  3 significant digits Zeros at the right in whole number. 5600  2 significant digits All zeros to the right of a significant digits and a decimal point. 5120.010  7 significant digits Zeros at the left in measurements less than 1 are not significant. 0.00672  5 significant digits

41. PHY115 – Sault College – Bazlurslide 41 Determine the “Accuracy” and “Precision” 3463 m4 S.D.s1m 3005 km 36000  8800 V 1349000 km 0.00632 kg 0.0060 g 14.20 A 30.00 cm 100.060 g6 SDs0.001 g 0.00004 m 2.4765 m

42. PHY115 – Sault College – Bazlurslide 42 Precision Being precise means being sharply defined. The precision of a measuring instrument depends on its degree of fineness and the size of the unit being used. Using an instrument with a more finely divided scale allows us to take a more precise measurement.

43. PHY115 – Sault College – Bazlurslide 43 Precision The precision of a measuring refers to the smallest unit with which a measurement is made, that is, the position of the last significant digit. In most cases it is the number of decimal places. e.g., The precision of the measurement 385,000 km is 1000 km. (the position of the last significant digit is in the thousands place.) The precision of the measurement 0.025m is 0.001m. (the position of the last significant digit is in the thousandths place.)

44. PHY115 – Sault College – Bazlurslide 44 How precise do we need? Physicists are interested in how closely a measurement agrees with the true value. That is, to achieve a smaller error or more accuracy. For bigger quantities, we do not need to be precise to be accurate.

45. PHY115 – Sault College – Bazlurslide 45 How precise do we need? For bigger quantities, we do not need to be precise to be accurate. E.g., the true value of a bar is 25 m Measured value is 26 m with 2 significant digits. The percentage error is (26-25)*100/25 = 4% E.g., the true value of a bar is 2.5 m Measured value is 2.6 m with 2 significant digits. The percentage error is (2.6-2.5)*100/2.5 = 4% Which one is more precise? The one which has the precision of 0.1m Which one is more accurate? Both are same accurate as both have 2 significant digits

46. PHY115 – Sault College – Bazlurslide 46 Accuracy or Relative Precision An accurate measurement is also known as a relatively precise measurement. Accuracy or Relative Precision refers to the number of significant digits in a measurement. A measurement with higher number of significant digits closely agrees with the true value.

47. PHY115 – Sault College – Bazlurslide 47 Estimate Any measurement that falls between the smallest divisions on the measuring instrument is an estimate. We should always try to read any instrument by estimating tenths of the smallest division.

48. PHY115 – Sault College – Bazlurslide 48 Accuracy or Relative Precision In any measurement, the number of significant figures are critical. The number of significant figures is the number of digits believed to be correct by the person doing the measuring. It includes one estimated digit. A rule of thumb: read a measurement to 1/10 or 0.1 of the smallest division. This means that the error in reading (called the reading error) is 1/10 or 0.1 of the smallest division on the ruler or other instrument. If you are less sure of yourself, you can read to 1/5 or 0.2 of the smallest division. http://www.astro.washington.edu/labs/clearinghouse/labs/Scimeth/mr-sigfg.html

49. PHY115 – Sault College – Bazlurslide 49 Estimate to 1/10 th of a cm

50. PHY115 – Sault College – Bazlurslide 50 Estimate to 1/10 th of a mm

51. PHY115 – Sault College – Bazlurslide 51 6 or 6.0 cm?

52. PHY115 – Sault College – Bazlurslide 52 Measurement An object measured with a ruler calibrated in millimeters. One end of the object is at the zero mark of the ruler. The other end lines up exactly with the 5.2 cm mark. What reading should be recorded for the length of the object? Why?

53. PHY115 – Sault College – Bazlurslide 53 Precision Which of the following measured quantities is most precise? Why? 126 cm 2.54 cm 12.65 cm 48.1 mm 0.081 mm

54. PHY115 – Sault College – Bazlurslide 54 Exact vs. Approximate numbers An exact number is a number that has been determined as a result of counting or by some definition. E.g., 41 students are enrolled in this class 1in = 2.54 cm Nearly all data of a technical nature involve approximate numbers. That is numbers determined as a result of some measurement process, as with a ruler. No measurement can be found exactly.

55. PHY115 – Sault College – Bazlurslide 55 Calculations with Measurements The sum or difference of measurements can be no more precise than the least precise measurement. 42.28 mm Using a micrometer 54 mm Using a ruler, Precision of the ruler is 1 mm But actually it can be anywhere between 53.50 to 54.50 mm This means that the tenths and hundredths digits in the sum 96.28 mm are really meaningless, the sum should be 96 mm with a precision of 1 mm

56. PHY115 – Sault College – Bazlurslide 56 Calculations with Measurements The sum or difference of measurements can be no more precise than the least precise measurement. Round the results to the same precision as the least precise measurement. 42.28 mm Using a micrometer 54 mm Using a ruler, Precision of the ruler is 1 mm But actually it can be anywhere between 53.50 to 54.50 mm This means that the tenths and hundredths digits in the sum 96.28 mm are really meaningless, the sum should be 96 mm with a precision of 1 mm

57. PHY115 – Sault College – Bazlurslide 57 Calculations with Measurements The product or quotient of measurements can be no more accurate than the least accurate measurement. Round the results to the same number of significant digits as the measurement with the least number of significant digits. http://www.astro.washington.edu/labs/clearingh ouse/labs/Scimeth/mr-sigfg.htmlhttp://www.astro.washington.edu/labs/clearingh ouse/labs/Scimeth/mr-sigfg.html Length of a rectangle is 54.7 m Width of a rectangle is 21.5 m Area is 1176.05 m 2 Area should be rounded to 1180 m 2 To express with same accuracy

58. PHY115 – Sault College – Bazlurslide 58 Rounding Numbers To round a number to a particular place value: If the digit in the next place to the right is less than 5, drop that digit and all other following digits. Replace any whole number places dropped with zeros. If the digit in the next place to the right is 5 or greater, add 1 to the digit in the place to which you are rounding. Drop all other following digits. Replace any whole number places dropped with zeros

59. PHY115 – Sault College – Bazlurslide 59 Special case, Rounding Numbers If the digit in the next place to the right is exactly 5, add 1 to the digit in the place to which you are rounding if the previous digit is an odd number other wise just drop the digit. Replace any whole number places dropped with zeros. This rule means that if the digit to be dropped is 5 followed only by zeroes, the result is always rounded to the even digit. The rationale is to avoid bias in rounding: half of the time we round up, half the time we round down.

60. PHY115 – Sault College – Bazlurslide 60 Examples of Rounding http://www.astro.washington.edu/labs/clearingh ouse/labs/Scimeth/mr-sigfg.htmlhttp://www.astro.washington.edu/labs/clearingh ouse/labs/Scimeth/mr-sigfg.html

61. PHY115 – Sault College – Bazlurslide 61 Add the Measurements 1250 cm, 1562 mm, 2.963 m, 9.71 m Convert all measurements to the same units. In this case m will be the best choice of units. 1250 cm = 12.5 m 1562 mm = 1.562 m 12.5 m 1.562 m 2.963 m 9.71 m 26.735 m Round to ?  Should we round before adding?

62. PHY115 – Sault College – Bazlurslide 62 Calculations with Measurements A rectangular has dimensions of 15.6 m by 11.4 m. What is the area of the rectangle? A = L x W = 15.6 m x 11.4 m = 177.84 m 2 = ? m 2

63. PHY115 – Sault College – Bazlurslide 63 Calculations with Measurements A rectangular plot of land has an area of 78000 m 2. one side has a length of 654 m. What is the length of the second side? A = L x W  W= A/L = 78000 m 2 / 654 m = 119.266 m = ? m

64. PHY115 – Sault College – Bazlurslide 64 Calculations with Measurements Subtract the measurements: 2567 g – 1.60 kg Express your answer in g. Convert all measurements to the same units. 1.60 kg = 1600 g 2567 g 1600 g 970 g Round to ?  Should we round before subtracting?

65. PHY115 – Sault College – Bazlurslide 65 Calculations with Measurements and Exact numbers To round the result of a calculation use the precesion and the accuracy of the measured number not the exact number.

66. PHY115 – Sault College – Bazlurslide 66 Calculations with Measurements and Exact numbers 2 equal rectangular plots of land has an area of 75 m 2. What is the area of one plot?  Area of one plot= Total Area / 2 = 75 m 2 / 2 = 37.5 m 2 = ? m 2

67. PHY115 – Sault College – Bazlurslide 67 So far… Accuracy and precision Exact number and Approximate number Estimate Rounding USCS (United States Customary System) Systeme International (SI) or Metric system Quantities, units and symbols of the SI system Prefixes of SI system Major advantage of the SI system (multiples of 10)!