What is Physics? “Physics is a tortured assembly of contrary qualities: of skepticism and rationality, of freedom and revolution, of passion and aesthetics, and of soaring imagination and trained common sense.” - Leon M Lederman (Nobel Prize for Physics, 1988)

International System of Units (Metric System) (Newell, 2014, p. 36)

Metric Review Metric Base Units meter (m) Length  Mass  Volume  Time  gram (g) Liter (L) second (s) Note: In physics the kilogram (kg) is used as the fundamental unit for mass not the gram.

Easy as Ten Prefix Abbreviation Multiply By Conversion Kilo____k_x k_ = 1000 _ Hecto____h_x 1001 h_ = 100 _ Deka____da_x 101 da_ = 10 _ Base: x 1 Deci____d_x 1/1010 d_ = 1 _ Centi____c_x 1/ c_ = 1 _ Milli____m_x 1/ m_ = 1 _ Length = meter Volume = Liter Mass = gram

Metric Prefixes Kids Have Dropped (over) Dead Converting Metrics! ____ ____ ____ ____ ____ ____ ____ k h da d c m k h da d c m KiloHectoDekaBaseDeciCentiMilli

Largersmaller 1 kilo (k) =___________ base 1 mega (M)=___________ base 1 giga (G)=___________ base 1 base= ___________ deci (d) 1 base= ___________ centi (c) 1 base =___________ milli (m) 1 base=___________ micro (μ) 1 base=___________ nano (n) Notice that the 1 always goes with the larger unit!! There are always Lots of small units in a single large one! ,000,000 1,000,000, ,000,000 1,000,000,000

Scales Object Length (m) Distance to the edge of the observable universe Diameter of the Milky Way galaxy Distance to the nearest star10 16 Diameter of the solar system10 13 Distance to the sun10 11 Radius of the earth10 7 Size of a cell10 -5 Size of a hydrogen atom Size of a nucleus Size of a proton Planck length Object Mass (kg) The Universe10 53 The Milky Way galaxy10 41 The Sun10 30 The Earth10 24 Boeing 747 (empty)10 5 An apple.25 A raindrop10 -6 A bacterium Mass of smallest virus A hydrogen atom An electron Order of magnitude  The difference between exponents. Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005

Order of Magnitude ► Give an order of magnitude estimate for the mass (kg) of  An egg  The earth  The difference between the mass of an egg and the earth. ► The ratio to the nearest order of magnitude is

Examples: 1000m = _______ dm Base  d (1 step right)  _______ dm 1400mm = ________m m  base (3 steps left)  _____ m 154 cm = _______km 1.456hm = ________cm 1. Find the prefix of the given quantity. 2. Move toward the desired quantity counting the steps you move. 3. Which way did you move? Move the decimal point the same # of spaces in that direction. Sliding Decimal Scale M.. k h da base d c m.. μ.. n.. p ,560 10,

Factor Label Method a. Write the quantity and units equivalent b. Multiply the known by an unit or conversion factor that include the units you are looking for. Set this up so the quantities cancel. c. Let the UNITS be your guide! d. Give an answer with correct units! Example: ? pennies  2.46 dollars a. 1 dollar = 100 pennies Set up a ratio to express this  b dollars x = ______ pennies 246

FLM - Examples 1. Ms. Frisbee has 18 eggs. How many dozens does she have? a. Conversion Factor: 1 dozen = 12 eggs b. 18 eggs x = _____ dozen 2. How many grams are in 340 mg? 340 mg = ______ g 3. How many seconds are in 3.5 hours? 3.5 hours = ______ s ,600

Power of Ten ► Scientific Notation ► Large numbers can be written as the product of a number and raised to a power of ten. ► 10 n = 10 x 10 x 10 x 10… (n times) ► 10 -n = 1/(10 x 10 x 10 x 10… ) (n times) ► Examples: ► = x 10 7 ► x =

Scientific Notation 1. move decimal point until only one non-zero digit remains on left (ex becomes 6.0 and.0025 becomes 2.5) 2. count the number of places the decimal moved 3. For every place the decimal moved right, subtract one from the exponent 4. For every place the decimal moved left, add one to the exponent LARS  Left Add, Right Subtract!

Review of Scientific Notation Standard7,200, places to the left places to the right Scientific Notation 7.2 x x 10 -5

Fundamental vs. Derived Units: Fundamental Units ► Basic quantities that can be measured directly ► Examples: length, time, mass, etc… Derived Units ► Calculated quantities from fundamental units ► Examples: speed, acceleration, area, etc… Volume can be measured in liters (fundamental units), or calculated by multiplying length x width x height to give derived units in meters 3

IB Fundamental Units ► Length – meter (m)  Defined as the distance travelled by light in a vacuum in a time of 1/299,792,458 seconds ► Mass – kilogram (kg)  Standard is a certified quantity of a platinum-iridium alloy stored at the Bureau International des Poides et Measures (France) ► Time – second (s)  Defined as the duration 9,192,631,770 full oscillations of the electromagnetic radiation emitted in a transition between the two hyperfine energy levels in the ground state of a cesium-133 (Cs) atom

IB Fundamental Units ► Temperature – Kelvin (K)  Defined as 1/ of the thermodynamic temperature of the triple point of water. ► Molecules – mole (mol)  One mole contains as many molecules as there are atoms in 12 g of carbon 12. (6.02 x molecules – Avogadro’s number) ► Current – Ampere (A)  Defined as the current which when flowing in two parallel conductors 1m apart, produces a force of 2 x N on a length of 1m of the conductors. ► Light Intensity – candela (cd)  The intensity of a source of frequency 5.40 x Hz emitting 1/685 W per steradian. Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005

Present SI Base Units Seven direct measurements (Newell, 2014, p. 37)

New SI Base Quantities (2018) ► Seven fundamental constants (Newell, 2014, p. 37)

Precision ► describes the reproducibility of a measurement. ► If Chris and his lab partner both recorded the acceleration due to gravity as 12 m/s 2 and so did the teacher, then this measurement is reproducible, so it is also precise. ► When measurements are precise and not accurate, faulty instruments are usually to blame.

Significant Digits ► These “valid” digits in a measurement are called significant digits ► The more significant digits you have, the more precise your measurement.

Significant Digits: 1. not zero 2. zero between two non-zero digits 3. zero to far right of decimal 4. Zeros used as placeholders are NOT significant ► 1.23 ► ► or ► or Digits in a measurement are significant when:

Significant Digits – Math Rules Addition or Subtraction ► find the sum ► round answer to the largest least precise measurement in the problem ► NOTE: this will be to the smallest number of decimal places! Example: ► 18.2m m = __m ► 18.2 is measured only to a tenth of a meter, so answer must be only this precise ► =  24.7m

Math Rules - Continued Multiplication & Division ► complete the calculation ► find the factor with the least # of sig. digits ► round answer to that # of sig. digs. Example: ► 3.22cm X 2.1cm = _ cm 2 ► ___ cm has the least # of sig figs, so answer must have only that many ► cm 2  ___ cm

Remember: Significant digits are an indication of how PRECISE your measurement is, and you can only be a sure as your least sure measurements. In other words…you can’t multiply 2.1 x 2.3 and give an answer that looks like NOTE: On the IB test if sig. digits are not used a max of 1 pt will be deducted from your test. ► The same policy applies to your Physics Labs.

Measurements When you read any scale: ► record the measurement by reading the smallest division on the scale ► then “approximate” or estimate to the tenth of the smallest division.

A. B.

Accuracy ► the closeness of a measurement to a best or accepted value. ► For example, the constant for the acceleration due to gravity is 9.8 m/s 2 this is the accepted value. If Chris measured this value to be 12 m/s 2 and Tiffany measured this value to be 15 m/s 2 ► Chris would have the more accurate reading because it is closer to the accepted value.

Precision vs Accuracy ► Notice that it is possible for measurements to be precise, but not accurate. When this happens, instrument error is often to blame.

Errors Source: Kirk, 2007, p. 3

Errors ► Systematic Errors – error that arises for all measurements taken.  incorrectly calibrated instrument (not zeroed) ► Reading Errors – impreciseness of measurement due to limitations of reading the instrument. ► ► Digital scale   Safe to estimate the reading error (uncertainty) as the smallest division (Ex. Digital stopwatch – smallest division is.01 s so the uncertainty is ±0.01 s) ► ► Analog scale   Safe to estimate the uncertainty as half the smallest scale division (Ex. Ruler - smallest division is.001 m so the reading error is ± m) x Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005

Errors ► Random Errors – shown by fluctuations both high and low in the data.  Reduced by averaging repeated measurements (¯)  Error calculated with the standard deviation. whereMeasurement is whereMeasurement is  Estimating random error ► Calculate the average ► Find the highest deviation in the data above and below the average. ► The largest of these deviations becomes the uncertainty. x Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005

Estimating Uncertainty ► Suppose a ruler was used to make the following measurements with the observer noting the reading error to be ±0.05 cm. ► Calculate the average, standard deviation, uncertainty. ► Estimate the uncertainty Length (±0.05 cm) Deviation Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005 Excel Length (±0.05 cm)

Estimating Uncertainty ► Average ( ► Average (¯) = cm ► ► Standard deviation = ► ► Since the random error is larger than the reading error it must be included. ► ► Thus, the measurement is ± 0.16 cm.   Note: IB rounds uncertainty to one significant digit and you match the SD of measurement to the uncertainty ±0.2 cm ► ► Estimation of uncertainty   Largest deviations above/below 0.23 &   Estimated uncertainty ± 0.23 w/ IB rounding 14.8 ± 0.2 cm Length (±0.05 cm) Deviation x Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005

Errors in Measurements ► ► Best estimate ± uncertainty (x best ± Δx) standard error notation ► ► Rule for Stating Uncertainties – experimental uncertainties should almost always be rounded to one significant digit. ► ► Rule for Stating Answers – The last significant figure in any stated answer should be of the same order of magnitude as the uncertainty (same decimal position) ► ► Number of decimals places reflect the precision of the measuring instrument ► ► For clarity in graphing we need to convert all data into standard form (scientific notation). ► ► If calculations are made the uncertainties are propagated. ► ►

Relative and Absolute Uncertainty ► Absolute uncertainty is the uncertainty of the measurement.  Ex ±0.02 s  Ex.=

Absolute and Relative (%) Error: ► Useful when comparing to an established value. ► Absolute Error: E a =  O – A  Where  O = observed value  A = accepted value ► Relative or % Error: or E a /A x 100

Sample Problem: ► In a lab experiment, a student obtained the following values for the acceleration due to gravity by timing a swinging pendulum: m/s m/s m/s m/s 2 The accepted value for g at the location of the lab is m/s 2. ► Give the absolute error for each value. ► Find the relative error for each value.

Rules for the Propagation of Error

► To state our answer we now choose the number half-way between these two extremes and for the uncertainty we take half of the difference between them. or S = 4000mm² ± 130mm² or S = 4000mm² ± 130mm² 3. If two (or more) measured quantities are multiplied or divided then their relative uncertainties are added. ► Relative uncertainties: x is 1/50 or 0.02mm and y is 1/80 or mm. So, the relative uncertainty in the final result should be ( ) = ► Checking, the relative uncertainty in final result for S is 130/4000 =

Rules for the Propagation of Error

As previously we now state the final result as 4. If a measured quantity is raised to a power then the relative uncertainty is multiplied by that power. ► Relative uncertainty in r is 0.05/25 = ► Relative uncertainty in V is 393/65451 = ► x 3 = so, again the theory is verified V = 65451mm 3 ± 393mm 3

Summary

Propagation Step by Step ► For more complicated calculations, we break them down into a sequence of steps each involving one of these operations  Sums and differences  Products and quotients  Computation of a function of one variable (x n ) We then apply the propagation rule for each step and total the uncertainty.

Error Propagation

Graphs for Physics ► Graphs are one method of finding out how one quantity is related to another. ► We find the relationship by keeping all quantities constant EXCEPT the two in question. ► One quantity is varied and the other quantity is measured.

Independent Variable ► ► The quantity that is deliberately varied ► ► also called the manipulated variable ► ► Plotted on the x-axis of the graph

Dependent Variable ► ► The quantity that changes due to the variation in the independent variable ► ► also called the responding variable ► ► plotted on the y-axis of a graph

Variable Identification Read each of the following statements. Underline each independent (manipulated) variable and circle each dependent (responding) variable. 1. Beans were soaked in water for different lengths of time and their gain in mass was recorded. 2.A ball is dropped from several distances above the floor and the height it bounces up is then measured.

Graph Requirements 1. A title (dependent vs independent or y vs x) 2. Label the y-axis (vertical) with the dependant variable and corresponding units  Distance (m) 3. Label the x-axis (horizontal) with the independent variable and corresponding units  Time (s) 4. Start both x- and y-axis at zero, increasing by equal intervals (ex. x-axis can increase by 1 second, y-axis can increase by 5 meters – mark axes like a ruler!) ► Data should be plotted over full graph

Graph Requirements 5. Draw a best fit line (straight or curved) through the data points. The line may not hit all of the data points, but shows the general shape of the graph. ► DO NOT CONNECT THE DOTS! 6. If the graph is a straight line, calculate the slope of the line ► Choose points on the line and as far apart as possible to calculate the slope 7. Describe the relationship/proportionality of the 2 variables in the graph

Linear Relationship ► y changes directly with x ► Best Fit – Straight Line ► Linear Equation: y ≈ x ory=mx+b  m = slope = rise/run  b = y intercept ► Positive slope  variables are directly proportional ► Negative slope  variables are still directly proportional (negatively)

Power Relationship ) Power Relationship (exponential) The dependent variable varies with the power of the independent variable ► ► Best Fit  parabola ► ► Equation: y ≈ x 2 ory = kx 2

Inverse Relationship One variable relies on the inverse of the other. ► Best Fit  hyperbola ► Equation: Y ≈ 1/x or y=k(1/x)

Square Root Relationship The dependent variable varies with the square root of the independent variable ► Equation: y ≈ x 1/n (n>1) or y =kx 1/n

Interpolation: ► Points between Find the money the student earned after 3 hours? After 7 hours?

Extrapolation: ► Points beyond What will the temperature be after heating for 70 minutes? For 100 minutes?

Proportionality – Linearizing Relationships

Example: The gravitational force F that acts on an object at a distance r away from the center of a planet is given by ► M is the mass of a planet ( 6.0 x kg) ► m is the mass of an object (100 kg) ► G is a gravitational constant (6.67 x ) What type of relationship does the graph shape resemble? Inverse or Y ≈ 1/x (F ≈ 1/r) If we plotted F vs. 1/r what would we expect our graph to look like? ► Plotting F vs. r

Proportionality – Linearizing Relationships Is the graph linear? What relationship does the shape resemble? Exponential or y ≈ x 2 (F ≈ (1/r) 2 ) So the next step is to plot F vs. (1/r) 2 ► Plotting F vs. 1/r

Proportionality – Linearizing Relationships Is the graph linear? What relationship does the shape resemble? Exponential or y ≈ x 2 (F ≈ (1/r) 2 ) So the next step is to plot F vs. (1/r) 2 Is the graph linear? To verify add a best fit line. So our relationship is (F ≈ (1/r) 2 ) The slope (m) of our best fit line 4.27x ≈ (6.67 x ) ( 6.0 x kg) (100 kg) or GMm so our relationship is y=mx or ► Plotting F vs. (1/r) 2

Logarithms in Relationships

Log Examples Ln(X) Ln(Y) Slope= p This technique works for all logarithms no matter what the base is!

Power Law & Logs - Pendulum ► From the graph how could we identify the relationship? ► Taking the natural log (ln) of both plotted variables and plot them ► Show from the original relationship why this is the result Source: Kirk, 2007, p. 7

Power Law & Logs– Gravitational Force Source: Kirk, 2007, p. 7

► Comparing this to the equation of a straight line ► y=ln(R), m= -λ and x = t ► Graphing ln (R) vs. t in a log-linear plot Exponentials and Logs - Radioactivity ► Many physics’ relationships are exponential. ► Radioactivity is defined as where R o and λ are constants. Taking the log of both sides Source: Kirk, 2007, p. 7

Error Bars ► Lines plotted to represent the uncertainty in the measurements. ► If we plot both vertical and horizontal bars we have what might be called "error rectangles” ► The best-fit line could be any line which passes through all of the rectangles. x was measured to ±0·5s y was measured to ±0·3m Error Bars

Best Fit Line Source: Kirk, 2007, p. 3

Min & Max Slopes Source: Kirk, 2007, p. 9

Min & Max Y-Intercepts Source: Kirk, 2007, p. 9

Sources ► Kirk, T. (2007) Physics for the IB diploma: Standard and higher level. (2nd ed.). Oxford, UK: Oxford University Press ► Newell, D. B. (2014). A more fundamental International System of Units. Physics Today, 67(7), ► Taylor, J. R. (1997) An introduction to error analysis: The study of uncertainties in physical measurements. (2nd ed.). Sausalito, CA: University Science Books ► Tsokos, K. A. (2009) Physics for the IB diploma: Standard and higher level. (5th ed.). Cambridge, UK: Cambridge University Press