measurement & calculations. biblical reference he measured its wall and it was 144 cubits thick,...
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Measurement & Calculations
Biblical Reference
He measured its wall and it was 144 cubits thick, by man's measurement, which the angel was using.
Revelation 21:17
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
• Accuracy – measurement of closeness to the true value of a number
• Precision – measure of how close a series of measurements are to one another
• 602,000,000,000,000,000,000,000 copper atoms– 6.02 x 1023 copper atoms
• 0.000000000000000000000327 grams– 3.27 x 10-22 grams
Scientific Notation:
Only 1 digit to the left of the decimal place.
• 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:
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 - 0.43 cm
Number of Significant Figures = 2
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 :
• 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:
Rounding with Addition and Subtraction– Round to the least number of decimal places
Example: 6.7 + 4.321 = 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:
• 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
SI units
• Measurements are fundamental to the experimental sciences
• Science uses the International System of Measurements (SI)
• MKS and CGS
Measuring with SI Units - MKS
Quantity SI Base Unit SymbolLength meter mMass kilogram kgTemperature Kelvin KTime second sAmount of substance mole molLuminous intensity candela cdElectric current ampere A
Metric Prefixes
Prefix Meaning Factormega (M) 1 million times larger than preceding 106
kilo (k) 1000 times larger than preceding 103
deci (d) 10 times smaller than preceding 10-1
centi (c) 100 times smaller than preceding 10-2
milli (m) 1000 times smaller than preceding 10-3
micro (m) 1 million times smaller than preceding 10-6
nano (n) 1 billion times smaller than preceding 10-9
pico (p) 1 trillion times smaller than preceding 10-12
Metric Units of Length
Unit Relationship Example
Kilometer (km) 1 km = 103 m Five city blocks
Meter (m) Base unit Height of doorknob
Decimeter (dm) 101 dm = 1 m Large orange
Centimeter (cm) 102 cm = 1 m Shirt button
Millimeter (mm) 103 mm = 1 m Thickness of dime
Micrometer (mm) 106 mm = 1 m Diameter of bacteria
Nanometer (nm) 109 nm = 1 m Thickness of RNA
Metric Units of Mass & Volume
Unit Relationship ExampleLiter (L) Base unit Quart of milkMilliliter (mL) 103 mL = 1 L 20 water dropsCubic centimeter (cm3) 1 cm3 = 1 mL Sugar cubeMicroliter (mL) 106 mL = 1 L Single salt crystal
Unit Relationship ExampleKilogram (kg) 1 kg = 103 g Small textbookGram (g) 1 g = 10-3 kg Dollar billMilligram (mg) 103 mg = 1 g Sugar cubeMicrogram (mg) 106 mg = 1 g Single salt crystal
Standard Kilogram
Platinum-Iridium Cylinder
Height = 3.9 cm
Diameter = 3.9 cm
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 + 273.15
°C = K – 273.15
Metric Units of Energy• 1 Joule (J)= 0.2390 calories (cal)
• 1 calorie (cal) = 4.184 Joules (J)
• One calorie is the amount of heat that raises the temperature of 1 g of pure water by 1 ° C
Derived Units
Area WidthLength mm Area][ 2m
Volume HeightWidthLength mmm [Volume]
DensityVolume
Mass
3 [Density] m
kg
3m
VelocityTime
ntDisplaceme
s
m [Velocity]
onAccelerati Time
Velocity
ss
m ion][Accelerat
2s
m
Momentum VelocityMass s
mkg [Momentum]
Derived Units
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.
• 1 dollar– 4 quarters– 10 dimes– 20 nickels– 100 pennies
Everyday Conversion Factors
Metric Conversion Factors
• 1 meter 100
• 10 decimeters 101
• 100 centimeters 102
• 1,000 millimeters 103
• 1,000,000 micrometers 106
• 1,000,000 micrometers 106
• 1,000,000,000 nanometers 109
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.
Conversion Examples
500 cm ´ 1 m= 5 m
100 cm
6 cm ´ 10 mm= 60 mm
1 cm
7.3 ´ 10-2 cm ´1 m
´106 mm
= 7.3 ´ 102 cm102 cm 1 m
8 h ´ 60 min ´ 60 sec= 28,800 s
1 h 1 min
Example Problem
K2 is the world’s second tallest peak at 8000 m. What is the height of K2 in feet?
719.2624612
1
54.2
11008000
in
ft
cm
in
m
cmm
ft000,30 With sig figs
Dimensional Analysis QuestionThe 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?
gL
2T )C( Lg
2T )B( gL2T )A(
mL s
mg sT
2
gL
2T )C(
Lg
2T )B(
gL2T )A( ms
m
2
sm
ms
m
2
s1
2s
mm
s
mL s
mg sT
2
Dimensional Analysis Question
Coordinate Systems - Cartesian
Coordinate Systems - Polar
Density
Density = Mass / Volume
A 0.750-cm3 sample of platinum has a density of 21.4 g/cm3. What is its mass?
V
m
gcm
gcmVm 0.16)4.21)(750.0(
33
Comparative Densities