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