Measurement

Physical Quantities• Measurable characteristics that describe

object’s size, position, speed, energy, etc• All measured quantities have a dimension

(length, time, mass, etc)• Units used must be expressed with the

measurement• Quantities are of two types: scalars, with no

directional component, and vectors, which must include a direction

Units of Measurement

• Metric system used for all scientific measurements

• MKS based on meter, kilogram, second; also called SI system

• CGS based on centimeter, gram, second sometimes used for smaller quantities

• We will use the MKS system excluslively

MKS Fundamental UnitsQuantity Symbol Unit Abbrev.

length l meter m mass m kilogram kg time t second s

current I ampere A temp. T kelvin K

amount n mole mol luminous intensity

I candela cd

Standard Units

• Original standard meter was made of platinum, stored in Paris

• Now, meter is based on wavelength of a certain light emission

• Second, once based on part of a day, now based on atomic vibrations

• Only the kilogram is based on physical standard, stored in Paris

Derived Units

• Volume (liter) is derived from the meter

• Other units are combinations of fundamental units

• Newton, volt, joule, meters per second all derived units

Metric Prefixes

• Very large and very small numbers are common in physics, often expressed as powers of ten using scientific notation

• Multiples or fractional parts of any unit can be expressed using metric prefixes combined with a base unit

Metric Prefixes

• giga- G x 109

• mega- M x 106

• kilo-k x 103

• centi- c x 10-2

• milli- m x 10-3

• micro- x 10-6

• nano- n x 10-9

• pico- p x 10-12

Scientific Notation

• Used to simplify operations involving very large or very small numbers

• Use with numbers larger than 9,999 or smaller than 0.001

• Consists of a coefficient multiplied by ten raised to an exponent

Scientific Notation

• All significant digits and only significant digits are placed in coefficient, with one digit to left of decimal

• Exponent of 10 is found by how many places decimal must be moved from original number

• Movement to left is positive, to right is negative

Significant Figures

• Number of significant figures that can be reported depends on precision of measuring instrument

• When calculations are made, significance of answer depends on least significant measurement

• Counting numbers and fundamental constants are not considered in sig. figs., only measured numbers

Rules for Significant Figures• Non-zero numbers are always significant• Zeros between other nonzero digits are

significant• Zeros in front of nonzero digits are not

significant• Zeros at the end of a decimal number are

significant• Zeros at the end of a whole number are not

significant unless they have been measured and are indicated by a line over the zero

Calculations With Significant Figures

• Multiplication and division: answer must be rounded off to the same number of digits as the least significant measurement used to obtain the answer

• Addition and subtraction: answer must be rounded off to the same number of decimal places as the measurement with the smallest number of decimal places

Rules for Rounding

• When the digit(s) following the last significant figure is <5, round down

• When the digit(s) following the last significant figure is >5, round up

• When the digit(s) following the last significant figure is exactly = 5, round down if the last sig fig is even, round up if the last sig fig is odd

Accuracy

• Accuracy: how close a measurement is to the actual or accepted value

• Absolute error is the difference between a measurement and the accepted value

• Percent error is the absolute error divided by the accepted value (times 100)

Precision

• The degree of exactness of a measurement

• A measure of how many digits can be read from an instrument

• How well a series of measurements agrees with each other

• Precision is often estimated as one-half of the smallest division of the instrument

Types of Error

• Experimental error is not a mistake• Error is a measure of uncertainty of the

measurement• Systematic or systemic error: instruments

not properly calibrated or adjusted or used incorrectly; example: parallax

• Random error: unknown or unpredictable variation in experimental conditions

Graphing Data

• Helps show relationships between measured quantities

• Independent variable: controlled by experimenter, usually plotted on horizontal axis

• Dependent variable: depends on what is done to independent variable, usually plotted on vertical axis

• If time is one of the variables, it is often plotted on the horizontal axis

Important Graph Types

• Direct proportion or direct relationship between variables, linear graph; y=mx+b

• Inverse proportion or relationship: if one variable increases, other must decrease, hyperbola graph; xy=k

• Quadratic or squared relationship, parabola graph; y = ax2 + bx + c

Reading the Graph

• Graph is often used to estimate value not measured in experiment

• Interpolation: estimation between measured points

• Extrapolation: estimation beyond range of measured values

Elements of a Good Graph

• Graph must be large enough to be easily read and neat—use a straightedge for all lines

• Decide which variable goes on each axis

• Examine data to find the range; set up scales on axes that are consistent and easy to read

• Decide if the origin is a valid data point—if so, include it in the data set.

Elements of a Good Graph

• Axes must be labeled with units

• Plot the points—make them easily visible

• Determine the relationship shown by the data

• Draw best fit line or curve—don’t connect the points

• Graph must have descriptive title

Physics Equations

• Equations are used to write relationships between variables shown by experimental data and graphs

• Letters and often Greek letters are used to represent quantities

• Don’t confuse the symbol used in the equations with the abbreviation for the unit

Dimensional Analysis

• Dimensions can be treated as algebraic quantities

• Quantities can be added or subtracted only if they have the same units

• When multiplying or dividing quantities, units must work out to be the proper unit of the answer

• A good way to check your work

Orders of Magnitude

• Round a number to the nearest power of 10 to find its order of magnitude

• Useful in estimating quantities or checking answers for reasonableness

Problem Solving

• Read problem carefully, write down given information and what is asked for with proper symbols. Draw a sketch

• Find an equation that relates given quantities and unknown. May be a 2 step problem needing 2 equations

• Solve basic equation for the unknown in terms of given quantities

Problem Solving

• Substitute numbers into equation including units and significant digits

• Check dimensions (units) to make sure they match the desired answer

• Do the math, rounding to correct sig figs

• Check to see if answer is reasonable

Vocabulary

• accuracy• precision• fundamental units• derived units• significant digits• absolute error• percent error• scalar

• interpolation• extrapolation• direct proportion• inverse proportion• hyperbola• independent variable• dependent variable• vector

Vocabulary

• personal error• systematic error• random error• parallax