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 x 109 mega- M x 106 kilo- k x 103 centi- c x 10-2 milli- m x 10-3 micro- m 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