DENSITY GEL2007 Q) Which weighs more:- A kilogram of feathers or a kilogram of iron?
Metric Review Metric Base Units meter (m) Length Mass Volume Time gram (g) Liter (L) second (s)...
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Transcript of Metric Review Metric Base Units meter (m) Length Mass Volume Time gram (g) Liter (L) second (s)...
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Metric Review
Metric Base Unitsmeter (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.
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Easy as Ten
Prefix Abbreviation Multiply By ConversionKilo____ k_ x 1000 1 k_ = 1000 _Hecto____ h_ x 100 1 h_ =
100 _Deka____ da_ x 10 1 da_ = 10 _Base:
x 1Deci____ d_ x 1/10 10 d_ = 1 _Centi____ c_ x 1/100 100 c_ = 1 _Milli____ m_ x 1/1000 1000 m_ = 1
_
Length = meter
Volume = Liter
Mass = gram
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Metric Prefixes
Kids Have Dropped (over) Dead Converting Metrics!
____ ____ ____ ____ ____ ____ ____
k h da d c m
Kilo Hecto Deka Base Deci Centi Milli
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Larger smaller1 kilo (k) = ___________ base1 mega (M) = ___________ base1 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!
1000
1,000,000
1,000,000,000
10100
1000
1,000,000 1,000,000,000
http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/
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ScalesObject Length
(m)Distance to the edge of the
observable universe1026
Diameter of the Milky Way galaxy
1021
Distance to the nearest star 1016
Diameter of the solar system 1013
Distance to the sun 1011
Radius of the earth 107
Size of a cell 10-5
Size of a hydrogen atom 10-10
Size of a nucleus 10-15
Size of a proton 10-17
Planck length 10-35
Object Mass(kg)
The Universe 1053
The Milky Way galaxy 1041
The Sun 1030
The Earth 1024
Boeing 747 (empty) 105
An apple .25
A raindrop 10-6
A bacterium 10-15
Mass of smallest virus 10-21
A hydrogen atom 10-27
An electron 10-30
Order of magnitude The difference between exponents.
Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005
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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
nucleushydrogen ofdiameter
atomhydrogen ofdiameter
105
10-1
1024
1025
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Examples:1000m = _______ dm Base d (1 step right)1000. _______ dm
1400mm = ________m m base (3 steps left)1400. _____ 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
0
0.00154
14,560
10,000
1.4
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Factor Label Methoda. 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
dollarsa. 1 dollar = 100
penniesSet up a ratio to
express this
b. 2.46 dollars x
= ______ pennies
1100
1
pennies
dollar
dollar
pennies
1
100
246
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FLM - Examples
1. Ms. Frisbee has 18 eggs. How many dozens does she have?
a. Conversion Factor: 1 dozen = 12 eggsb. 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
112
1
eggs
dozen
eggs
dozen
12
1 1.5
0.34
12,600
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Power of Ten►Scientific Notation►Large numbers can be written as the
product of a number and raised to a power of ten.
►10n = 10 x 10 x 10 x 10… (n times)►10-n = 1/(10 x 10 x 10 x 10… ) (n times)►Examples:►25903000 = 2.5903 x 107
►6.022 x 1023= 602200000000000000000000
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Scientific Notation
1. move decimal point until only one non-zero digit remains on left (ex. 6000 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!
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Review of Scientific Notation
Standard
7,200,000.
6 places to the left
0.000045
5 places to the right
Scientific Notation
7.2 x 106
4.5 x 10-5
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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 meters3
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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
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IB Fundamental Units
► Temperature – Kelvin (K) Defined as 1/273.16 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 1023 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 10-7 N on a length of 1m of the conductors.
► Light Intensity – candela (cd) The intensity of a source of frequency 5.40 x 1014
Hz emitting 1/685 W per steradian.
Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005
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Precision
►describes the reproducibility of a measurement.
►If Chris and his lab partner both recorded the acceleration due to gravity as 12 m/s2 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.
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Significant Digits
►These “valid” digits in a measurement are called significant digits
►The more significant digits you have, the more precise your measurement.
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Significant Digits:
1. not zero2. zero between two
non-zero digits3. zero to far right of
decimal4. Zeros used as
placeholders are NOT significant
►1.23►43.089
►13.00 or 13.50
►00.34 or 0.0045
Digits in a measurement are significant when:
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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 + 6.48m
= __m►18.2 is
measured only to a tenth of a meter, so answer must be only this precise
►= 24.68 24.7m
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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 = _
cm2
►___ cm has the least # of sig figs, so answer must have only that many
►6.762 cm2___ cm2
2.1
6.8
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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 4.345682
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.
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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.
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A.B.
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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/s2 this is the accepted value. If Chris measured this value to be 12 m/s2 and Tiffany measured this value to be 15 m/s2
►Chris would have the more accurate reading because it is closer to the accepted value.
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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.
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Errors
Source: Kirk, 2007, p. 3
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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 ±0.0005 m)
To simplify we will use the smallest scale division.
x
Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005
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Errors ► Random Errors – shown by fluctuations both
high and low in the data. Reduced by averaging repeated measurements
(¯) Error calculated with the standard deviation.
where Measurement 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
1
... 222
21
N
xxxe N
xxx ii ex
Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005
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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
14.88 0.09
14.84 0.05
15.02 0.23
14.57 -0.22
14.76 -0.03
14.66 -0.13
Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005
Excel
Length (±0.05 cm)
14.88
14.84
15.02
14.57
14.76
14.66
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Estimating Uncertainty
► Average (¯) = 14.79 cm► Standard deviation = 0.1611► Since the random error is
larger than the reading error it must be included.
► Thus, the measurement is 14.79 ± 0.16 cm. Note: IB rounds uncertainty to
one significant digit and you match the SD of measurement to the uncertainty.
14.8 ±0.2 cm► Estimation of uncertainty
Largest deviations above/below 0.23 & -0.22
Estimated uncertainty 14.79 ± 0.23 w/ IB
rounding14.8 ± 0.2 cm
Length (±0.05 cm)
Deviation
14.88 0.09
14.84 0.05
15.02 0.23
14.57 -0.22
14.76 -0.03
14.66 -0.13
x
Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005
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Errors in Measurements► Best estimate ± uncertainty (xbest ± 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.
►
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Relative and Absolute Uncertainty
►Absolute uncertainty is the uncertainty of the measurement. Ex. 0.04 ±0.02 s
Ex.= %505.004.0
02.0
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Absolute and Relative (%) Error:
►Useful when comparing to an established value.
►Absolute Error: Ea = O – A Where O = observed value A = accepted value
►Relative or % Error: or Ea/A x
100100%
Accepted
AcceptedObservedError
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Sample Problem:►In a lab experiment, a student obtained
the following values for the acceleration due to gravity by timing a swinging pendulum:
9.796 m/s2
9.803 m/s2
9.825 m/s2
9.801 m/s2
The accepted value for g at the location of the lab is 9.801 m/s2.
►Give the absolute error for each value.►Find the relative error for each value.
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Rules for the Propagation of Error
► 1. Multiply or divide by a constant or
► 2. Adding or subtracting multiple measurements
► 3. Multiplying or dividing multiple measurements
► 4. Measured value raised to a power
For
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Rules for the Propagation of Error
1. If a measured quantity(x) is multiplied or divided by a constant (B) then the absolute uncertainty (δx) is multiplied or divided by the same constant. Therefore, the relative uncertainty stays the same.
or ► You need to find the average thickness of a
page of a book. You find 100 pages of the book have a total thickness of 9mm. Your measuring instrument has a precision of 0.1mm,
9.0 mm ± 0.1mm ► Average thickness of one page:► Result:
mmmm 09.0100
0.9
mmmmmmmm 001.009.0100
1.0
100
0.9
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Rules for the Propagation of Error
2. If two measured quantities (x & y) are added or subtracted then their absolute uncertainties (δx & δy) are added .
► To find a change in temperature, ΔT, we find an initial temperature, T1, a final temperature, T2, and then use ΔT = T2 - T1 with the precision of the measurement ±1°C.
► If T1 is 20°C and if T2 is 40°C then ΔT= 20°C.► Remember, 19°C < T1 < 21°C and 39°C < T2 <
41°C ► The smallest difference is (39 - 21) = 18°C
and the biggest difference is (41 - 19) = 22°C► This means that 18°C < ΔT < 22°C or
ΔT = 20°C ± 2°C
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Rules for the Propagation of Error
3. If two (or more) measured quantities (x & y) are multiplied or divided then their relative uncertainties ( & ) are added.
► To measure a surface area, S, we measure two dimensions, say, x and y, and then use S=xy.
► Using a ruler marked in mm, we measure x = 50mm ± 1mm and y = 80mm ± 1mm
► Therefore, the area could be anywhere between (49 × 79)mm² and (51 × 81)mm² or 3871mm² < S < 4131mm²
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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
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 0.0125mm. So, the relative uncertainty in the final result should be (0.02 + 0.0125) = 0.0325.
► Checking, the relative uncertainty in final result for S is 130/4000 = 0.0325
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Rules for the Propagation of Error
4. If a measured quantity (x) is raised to a power (n) then the relative uncertainty () is multiplied by that power.
For ► To find the volume of a sphere, we first find its radius,
r, (usually by measuring its diameter) and use the formula: V = (4/3)πr3
► Suppose that the diameter of a sphere is measured as 50 mm (using an instrument having a precision of ±0.1mm).
► So, the diameter = 50.0mm ± 0.1mm where the radius is r = 25.0mm ± 0.05mm (Rule 1).
► V could be between (4/3)π(24.95)3 and (4/3)π(25.05)3 or 65058mm3 < V < 65843mm3
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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 = 0.002► Relative uncertainty in V is 393/65451 = 0.006 ► 0.002 x 3 = 0.006 so, again the theory is verified
V = 65451mm3 ± 393mm3
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Summary
► 1. Multiply or divide by a constant or
► 2. Adding or subtracting multiple measurements
► 3. Multiplying or dividing multiple measurements
► 4. Measured value raised to a power
For
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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
(xn) We then apply the propagation rule for each step and total the uncertainty.
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Error Propagation► A pendulum can be used to
measure the acceleration of gravity (g) by the relationship
Where l is the length of the pendulum an T is the period.Here g is the product or quotient of three factors, 4π2, l, T2
4π2 has no uncertainty T2 has a relative uncertainty of
Using the product rule
► If our measurements were:► l=92.95 ± 0.1 cm► T=1.936 ± 0.004 sCalculate g
Relative uncertainties
g = 979 ± 5
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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.
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Independent Variable
►The quantity that is deliberately varied
►also called the manipulated variable
►Plotted on the x-axis of the graph
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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
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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.
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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
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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
12
12
xx
yy
run
riseSlope
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Linear Relationship
► y changes directly with x► Best Fit – Straight Line► Linear Equation: y ≈ x
or y=mx+b m = slope = rise/run b = y intercept
► Positive slope variables are directly proportional
► Negative slope variables are inversely proportional
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Quadratic Relationship (exponential)
The dependent variable varies with the square of the independent variable
►Best Fit parabola►Equation: y ≈ x2 or y = kx2
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Inverse Relationship
One variable relies on the inverse of the other.
►Best Fit hyperbola►Equation: Y ≈ 1/x or y=k(1/x)
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Square Root Relationship
The dependent variable varies with the square root of the independent variable
►Equation: y ≈ x1/n (n>1) or y =kx1/n
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Interpolation:
►Points between
Find the money the student earned after 3 hours?
After 7 hours?
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Extrapolation:
►Points beyond
What will the temperature be after heating for 70 minutes?
For 100 minutes?
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Proportionality – Linearizing Relationships
Often, judging whether a set of points is best fit by a line or curve is difficult to determine
A better technique is to change the proportions being graphed so the graph results in a direct (linear) relationship.
► Identify your variables and your constants.► The quantities you plot on the x and y axes must
be variables.► You can plot any mathematical combination of your
original reading on one axis – it is still a variable. , , , etc.
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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 10 24 kg)► m is the mass of an object (100 kg) ► G is a gravitational constant (6.67 x 10-11 )
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
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Proportionality – Linearizing Relationships
Is the graph linear?What relationship does the shape resemble?Exponential or y ≈ x2
(F ≈ (1/r)2 )So the next step is to plot F vs. (1/r)2
► Plotting F vs. 1/r
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Proportionality – Linearizing Relationships
Is the graph linear?What relationship does the shape resemble?Exponential or y ≈ x2 (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 1016 ≈ (6.67 x 10-11 ) ( 6.0 x 10 24 kg) (100 kg)
or GMm so our relationship is y=mx or
► Plotting F vs. (1/r)2
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Logarithms in Relationships
In dealing with variable exponents, Logarithms can mathematically be used to manipulate the graphs easily into linear relations.
► Example: a=10b then log(a)=bif p=eq then ln(p)=q
Rules of Logs:
►
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Log ExamplesExample: Find the
relationship between x and y In the equation y=kxp, k and p are constants.
► ln(k) is a constant so, it can be ignored to find the relationship. The axis can be made ln(y) and ln(x), making p the slope.
Ln(X)
Ln(Y
)
Slope= p
This technique works for all logarithms no matter what the base is!
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Power Law & Logs - Pendulum
► The period of a pendulum is defined by a relationship of the following form
► where k and p are constants
► Plotting it
► 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
Source: Kirk, 2007, p. 7
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Power Law & Logs– Gravitational Force
From the graph ► is ► Algebraically simplifying► k or ► where k=GMm ►
Source: Kirk, 2007, p. 7
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► 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 Ro and λ are constants.
Taking the log of both sides
Source: Kirk, 2007, p. 7
Source: Kirk, 2007, p. 7
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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
±0·5sy was measured to ±0·3m
Error Bars
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Best Fit Line
Source: Kirk, 2007, p. 3
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Min & Max Slopes
Source: Kirk, 2007, p. 9
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Min & Max Y-Intercepts
Source: Kirk, 2007, p. 9
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Sources
► Kirk, T. (2007) Physics for the IB diploma: Standard and higher level. (2nd ed.). Oxford, UK: Oxford University Press
► 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