Measurement Physical Quantities Measurable characteristics that describe objects

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 Units

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 • mega- M • kilo- k x 109 x 106 x 103 • • • centimillimicronanopico- c x 10 -2 m x 10 -3 m x 10 -6 n x 10 -9 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 = ax 2 + 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 vector

Vocabulary • • personal error systematic error random error parallax