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Measurement & Calculations

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Biblical Reference He measured its wall and it was 144 cubits thick, by man's measurement, which the angel was using. Revelation 21:17

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Accuracy vs. Precision For a single measurement: Accuracy - An indication of how close a measurement is to the accepted value Precision - An indication of the degree of exactness of a measurement For multiple measurements: Accuracy - An indication of how close the average measurement is to the accepted value Precision - An indication of the agreement among a number of similar measurements

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Accuracy – measurement of closeness to the true value of a number

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Precision – measure of how close a series of measurements are to one another

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602,000,000,000,000,000,000,000 copper atoms –6.02 x copper atoms grams –3.27 x grams Scientific Notation: Only 1 digit to the left of the decimal place.

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When you make a measurement there is some estimation Recorded numbers in a measurement or calculation are called significant figures What is the length of the black line in cm? How many significant figures are in the measurement? Significant Figures:

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You can't always keep all the digits a calculator produces. You can only keep the significant ones, the ones that are not beyond the accuracy of the measuring device. These digits are called significant digits or figures. Length cm Number of Significant Figures = 2

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1.Non zeros count (123.34) – 5 Sig Figs 2. Leading zeros do NOT count (0.004) – 1 Sig Fig 3. Captive zeros count (2004, 2.004, 20.04) – 4 Sig Figs 4. Trailing zeros count IF there is a decimal point (20., 20.00, 0.200) – 2, 4, and 3 Sig Figs Significant Figure Rules :

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A calculated answer cannot be more precise than the least precise measurement from which it was calculated Rounding –If the last number is less than 5, round down –If the last number is greater than 5, round up –If the last number is a 5, round so that the rounded number is even. Significant Figures in Calculations:

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Rounding with Addition and Subtraction –Round to the least number of decimal places Example: = 11.0 Rounding with Multiplication and Division –Round to the same number of significant figures as the measurement with the least number of significant figures Example: 540. g / 62 ml = 8.7 g/ml Mathematical Operations:

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Accepted value – correct value based on reliable references Experimental value – value measured in lab Determining Error: Error = Accepted Value – Experimental value Percent Error = |Error| x 100 Accepted Value *|Error| is the absolute value of the error

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SI units Measurements are fundamental to the experimental sciences Science uses the International System of Measurements (SI) MKS and CGS

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Measuring with SI Units - MKS QuantitySI Base UnitSymbol Lengthmeterm Masskilogramkg TemperatureKelvinK Timeseconds Amount of substancemolemol Luminous intensitycandelacd Electric currentampereA

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Metric Prefixes PrefixMeaningFactor mega (M)1 million times larger than preceding10 6 kilo (k)1000 times larger than preceding10 3 deci (d)10 times smaller than preceding10 -1 centi (c)100 times smaller than preceding10 -2 milli (m)1000 times smaller than preceding10 -3 micro ( ) 1 million times smaller than preceding10 -6 nano (n)1 billion times smaller than preceding10 -9 pico (p)1 trillion times smaller than preceding10 -12

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Metric Units of Length UnitRelationshipExample Kilometer (km)1 km = 10 3 mFive city blocks Meter (m)Base unitHeight of doorknob Decimeter (dm)10 1 dm = 1 mLarge orange Centimeter (cm)10 2 cm = 1 mShirt button Millimeter (mm)10 3 mm = 1 mThickness of dime Micrometer ( m) 10 6 mm = 1 mDiameter of bacteria Nanometer (nm)10 9 nm = 1 mThickness of RNA

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Metric Units of Mass & Volume UnitRelationshipExample Liter (L)Base unitQuart of milk Milliliter (mL)10 3 mL = 1 L20 water drops Cubic centimeter (cm 3 )1 cm 3 = 1 mLSugar cube Microliter ( L)10 6 L = 1 L Single salt crystal UnitRelationshipExample Kilogram (kg)1 kg = 10 3 gSmall textbook Gram (g) 1 g = kgDollar bill Milligram (mg)10 3 mg = 1 gSugar cube Microgram ( g)10 6 g = 1 g Single salt crystal

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Standard Kilogram Platinum-Iridium Cylinder Height = 3.9 cm Diameter = 3.9 cm

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Metric Units of Temperature The Celsius scale sets the freezing point of water at 0°C and the boiling point at 100°C. The Kelvin scale sets 0 at absolute zero. The units of Kelvin and Celsius are equivalent K = °C °C = K –

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Metric Units of Energy 1 Joule (J)= calories (cal) 1 calorie (cal) = Joules (J) One calorie is the amount of heat that raises the temperature of 1 g of pure water by 1 ° C

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Derived Units

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A conversion factor is a ratio that, when multiplied by the item you are converting, cancels out the units you do not want and leaves you with the units you want. Dimensional Analysis is a technique where you use the dimensions/units to check if a relationship is correct.

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1 dollar –4 quarters –10 dimes –20 nickels –100 pennies Everyday Conversion Factors

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Metric Conversion Factors 1 meter decimeters centimeters10 2 1,000 millimeters10 3 1,000,000 micrometers10 6 1,000,000,000 nanometers10 9

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Conversion Factor Steps Step 1 – Determine what units you are given and what units you need. Step 2 – Determine what conversion factors you need to use. Step 3 – Arrange the conversion factors so that the units you do not want cancel out. Step 4 – Make sure your last unit is the unit need.

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Conversion Examples 500 cm 1 m =5 m 100 cm 6 cm 10 mm =60 mm 1 cm 7.3 cm 1 m 10 6 m = 7.3 10 2 cm 10 2 cm1 m 8 h 60 min 60 sec =28,800 s 1 h1 min

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Example Problem K2 is the world’s second tallest peak at 8000 m. What is the height of K2 in feet? With sig figs

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Dimensional Analysis Question The period (T) of oscillation of a simple pendulum depends upon the acceleration of gravity(g) and the length (L) of the pendulum. Which expression below represents the relationship between T, g and L?

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Dimensional Analysis Question

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Coordinate Systems - Cartesian

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Coordinate Systems - Polar

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Density Density = Mass / Volume A cm 3 sample of platinum has a density of 21.4 g/cm 3. What is its mass?

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Comparative Densities

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