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Measurement

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Measurement

- 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

- 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

- 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

- Volume (liter) is derived from the meter
- Other units are combinations of fundamental units
- Newton, volt, joule, meters per second all derived units

- 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

giga-Gx 109

mega-Mx 106

kilo-kx 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

- 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

- 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

- 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

- 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

- 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

- 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: 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)

- 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

- 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

- 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

- 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

- Graph is often used to estimate value not measured in experiment
- Interpolation: estimation between measured points
- Extrapolation: estimation beyond range of measured values

- 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.

- 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

- 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

- 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

- Round a number to the nearest power of 10 to find its order of magnitude
- Useful in estimating quantities or checking answers for reasonableness

- 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

- 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

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

personal error

systematic error

random error

parallax