Transcript Physics Introduction Notes

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
Length  meter (m)
Mass  gram (g)
Volume  Liter (L)
Time  second (s)
Note: In physics the kilogram (kg) is used as the fundamental
unit for mass not the gram.
Easy as Ten
Prefix
Abbreviation
Kilo____ k_
Hecto____ h_
Deka____ da_
Base: Length = meter
Volume = Liter
Mass = gram
Deci____ d_
Centi____ c_
Milli____ m_
x 1000
x 100
x 10
Conversion
1 k_ = 1000 _
1 h_ = 100 _
1 da_ = 10 _
x
x
x
x
10 d_ = 1 _
100 c_ = 1 _
1000 m_ = 1 _
Multiply By
1
1/10
1/100
1/1000
Metric Prefixes
Kids Have Dropped (over) Dead Converting Metrics!
Kilo
____ Hecto
____ Deka
____
k
h
da
Base
____ Deci
____
d
Centi
____
Milli
____
c
m
Larger
1 kilo (k)
1 mega (M)
1 giga (G)
1 base
1 base
1 base
1 base
1 base
=
=
=
=
=
=
=
=
smaller
1000
___________
base
___________
base
1,000,000
___________
1,000,000,000 base
___________
deci (d)
10
___________
centi (c)
100
1000
___________
milli (m)
___________
1,000,000 micro (μ)
___________ nano (n)
1,000,000,000
Notice that the 1 always goes with the larger unit!!
There are always Lots of small units in a single large one!
http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/
Scales
Object
Length
(m)
Distance to the edge of the
observable universe
1026
Diameter of the Milky Way
galaxy
1021
Distance to the nearest star
Object
Mass
(kg)
The Universe
1053
The Milky Way galaxy
1041
The Sun
1030
1016
The Earth
1024
Diameter of the solar system
1013
Boeing 747 (empty)
105
Distance to the sun
1011
An apple
.25
Radius of the earth
107
10-6
Size of a cell
10-5
A raindrop
Size of a hydrogen atom
10-10
A bacterium
10-15
Size of a nucleus
10-15
Mass of smallest virus
10-21
Size of a proton
10-17
A hydrogen atom
10-27
Planck length
10-35
An electron
10-30
Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005
Order of magnitude  The difference between exponents.
Order of Magnitude
► Give
an order of magnitude estimate for the
mass (kg) of
10-1
 An egg
 The earth
1024
 The difference between the mass of an egg and
the earth.
1025
► The
diameter of hydrogen atom
ratio diameter of hydrogen nucleus
order of magnitude is
105
to the nearest
Sliding Decimal Scale
1.
2.
3.
Find the prefix of the
given quantity.
Move toward the
desired quantity
counting the steps you
move.
Which way did you
move? Move the
decimal point the same
# of spaces in that
direction.
Examples:
1000m = _______ dm
Base  d (1 step right)
10,000 dm
1000.0  _______
1400mm = ________m
m  base (3 steps left)
1.4 m
1400.  _____
0.00154
154 cm = _______km
14,560
1.456hm = ________cm
M . . k h da base d c m . . μ . . n . . p
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  1dollar
100 pennies
1
b. 2.46 dollars x 100 pennies
1dollar
246 pennies
= ______
FLM - Examples
1. Ms. Frisbee has 18 eggs. How many dozens
does she have?
1dozen
a. Conversion Factor: 1 dozen = 12 eggs
1
12eggs
1.5 dozen
b. 18 eggs x 1dozen = _____
12eggs
2. How many grams are in 340 mg?
0.34 g
340 mg = ______
3. How many seconds are in 3.5 hours?
3.5 hours = 12,600
______ s
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
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!
Review of Scientific Notation
Standard
Scientific Notation
7,200,000.
7.2 x 106
6 places to the left
0.000045
5 places to the right
4.5 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 meters3
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)
► Time
– second (s)
 Standard is a certified quantity of a platinum-iridium
alloy stored at the Bureau International des Poides
et Measures (France)
 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 1/273.16 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
Present SI Base Units
Seven direct measurements
New SI Base Quantities (2018)
► Seven
fundamental constants
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.
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:
Digits in a measurement are significant when:
1.
2.
3.
4.
not zero
zero between two
non-zero digits
zero to far right of
decimal
Zeros used as
placeholders are
NOT significant
► 1.23
► 43.089
► 13.00
or
13.50
► 00.34 or
0.0045
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
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
2.1 cm has the least #
► ___
of sig figs, so answer
must have only that
many
6.8 cm2
► 6.762 cm2___
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.
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.
B.
A.
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.
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
x
(Ex. Ruler - smallest division is .001 m so the reading error is
±0.0005 m)
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 (¯)
x
 Error calculated with the standard deviation.
e
x1 2  x2 2  ...  x N 2 where xi  xi  x Measurement is x  e
N 1
 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.
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
Excel
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
Estimating Uncertainty
►
►
►
►
Average (¯)
x = 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 rounding
14.8 ± 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
Source: Tsokas, T.A. Physics for the IB Diploma, Cambridge University Press 2005
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.
Relative and Absolute Uncertainty
► Absolute
uncertainty is the uncertainty
of the measurement.
 Ex. 0.04 ±0.02 s
 Ex.=
0 .02
 0 .5  50 %
0 .04
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:
Observed  Accepted
%  Error 
 100
Accepted
or Ea/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:
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.
Rules for the Propagation of Error
► 1.
Multiply or divide by a constant
Δ𝑞 = 𝐵 Δ𝑥 or Δ𝑞 =
Δ𝑥
𝐵
► 2.
Adding or subtracting multiple measurements
Δ𝑞 ≤ Δ𝑥 + Δ𝑦
► 3. Multiplying or dividing multiple measurements
Δ𝑞
𝑞
► 4.
≤
Δ𝑥
𝑥
+
Δ𝑦
𝑦
Measured value raised to a power
For 𝑞 =
𝑥𝑛

Δ𝑞
𝑞
=
Δ𝑥
𝑛
𝑥
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.
Δ𝑥
𝐵
Δ𝑞 = 𝐵 Δ𝑥 or Δ𝑞 =
► 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 9.0 mm  0.09mm
100
► Average thickness of one page:
9 .0
0 .1
► Result:
mm 
mm  0.09mm  0.001mm
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 .
Δ𝑞 ≤ Δ𝑥 + Δ𝑦
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
Rules for the Propagation of Error
3.
If two (or more) measured quantities (x & y) are
multiplied or divided then their relative
Δ𝑥
Δ𝑦
uncertainties ( & ) are added.
𝑥
𝑦
Δ𝑞 Δ𝑥 Δ𝑦
≤
+
𝑞
𝑥
𝑦
►
►
►
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
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
Rules for the Propagation of Error
4.
If a measured quantity (x) is raised to a power (n)
Δ𝑥
then the relative uncertainty ( ) is multiplied by
𝑥
that power.
For 𝑞 = 𝑥
►
►
►
►
𝑛

Δ𝑞
𝑞
=
Δ𝑥
𝑛
𝑥
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
Rules for the Propagation of Error
As previously we now state the final result as
V = 65451mm3 ± 393mm3
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
Summary
► 1.
Multiply or divide by a constant
Δ𝑞 = 𝐵 Δ𝑥 or Δ𝑞 =
Δ𝑥
𝐵
► 2.
Adding or subtracting multiple measurements
Δ𝑞 ≤ Δ𝑥 + Δ𝑦
► 3. Multiplying or dividing multiple measurements
Δ𝑞
𝑞
► 4.
≤
Δ𝑥
𝑥
+
Δ𝑦
𝑦
Measured value raised to a power
For 𝑞 =
𝑥𝑛

Δ𝑞
𝑞
=
Δ𝑥
𝑛
𝑥
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.
Error Propagation
A pendulum can be used to
► If our measurements were:
measure the acceleration of
► l=92.95 ± 0.1 cm
gravity (g) by the relationship
► T=1.936 ± 0.004 s
4𝜋 2 𝑙
Calculate g
𝑔=
𝑇2
4𝜋 2 × 92.95𝑐𝑚
Where l is the length of the
𝑔𝑏𝑒𝑠𝑡 =
1.936𝑠 2
pendulum an T is the period.
= 979.0 𝑐𝑚 𝑠 2
Here g is the product or quotient of
Relative uncertainties
three factors, 4π2, l, T2
Δ𝑙
.1
2
 4π has no uncertainty
=
= .001 = 0.1%
𝑙
92.95
Δ𝑇
 T2 has a relative uncertainty of 2
Δ𝑇
.004
𝑇
=
= .002 = 0.2%
𝑇
1.936
Using the product rule
►
Δ𝑔
𝑔
=
Δ𝑙
𝑙
+
Δ𝑇
2
𝑇

Δ𝑔
𝑔

0.5%
Δ𝑔 = 0.005 × 979 𝑐𝑚 𝑠 2 = 5 𝑐𝑚 𝑠 2
g = 979 ± 5𝑐𝑚 𝑠 2

=
Δ𝑙
𝑙
+2
Δ𝑇
𝑇
= 0.1 + 2 × 0.2 =
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
rise y 2  y1
the line
Slope 

run
►
x 2  x1
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
or y=mx+b
 m = slope = rise/run
 b = y intercept
► Positive
slope variables are directly proportional
► Negative slope  variables are inversely
proportional
Quadratic Relationship (exponential)
The dependent variable varies with the square
of the independent variable
► Best Fit parabola
► Equation: y ≈ x2 or y = kx2
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 ≈ x1/n (n>1) or y =kx1/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
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.

1
2
3
𝑥 ,𝑥 ,
𝑥
𝑥, etc.
Proportionality – Linearizing
Relationships
Example: The gravitational force ► Plotting F vs. r
F that acts on an object at a
distance r away from the center
𝐺𝑀𝑚
of a planet is given by 𝐹 = 2
𝑟
M is the mass of a planet ( 6.0 x 10 kg)
► m is the mass of an object (100 kg)
► G is a gravitational constant (6.67 x 10-11
𝑚3
2)
24
►
𝑘𝑔∙𝑠
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?
Proportionality – Linearizing
Relationships
Is the graph linear?
► Plotting F vs. 1/r
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
Proportionality – Linearizing
Relationships
Is the graph linear?
► Plotting F vs. (1/r)2
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
𝑚
4.27x 10 ≈ (6.67 x 10 𝑘𝑔∙𝑠 ) ( 6.0 x 10 kg) (100 kg)
or GMm so our relationship is
y=mx or
𝐺𝑀𝑚
𝐹= 2
𝑟
16
-11
3
2
24
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)=b
if p=eq then ln(p)=q
Rules of Logs:
► 𝑙𝑛 𝐶 ∙ 𝐷 = 𝑙𝑛 𝐶 + 𝑙𝑛 𝐷
► 𝑙𝑛 𝐶 𝐷 = 𝑙𝑛 𝐶 − 𝑙𝑛 𝐷
► 𝑙𝑛 𝑐 𝑛 = 𝑛 ∙ 𝑙𝑛 𝑐
► 𝑙𝑛 1 𝑐 = −𝑙𝑛 𝑐
Log Examples
Ln(Y)
Example: Find the relationship
between x and y In the
equation y=kxp, k and p are
constants.
► 𝑙𝑛 𝑦 = 𝑙𝑛 𝑘𝑥 𝑝
► 𝑙𝑛 𝑦 = 𝑙𝑛 𝑘 + 𝑝 ∙ 𝑙𝑛 𝑥
► ln(k) is a constant so, it can
Slope= p
be ignored to find the
relationship. The axis can be
made ln(y) and ln(x), making
p the slope.
This technique works for all logarithms no matter what the base is!
Ln(X)
Power Law & Logs - Pendulum
►
The period of a pendulum is ► From the graph how could we
defined by a relationship of
identify the relationship?
the following form
𝑝
►
𝑇 = 𝑘𝑙
►
Plotting it
►
where k and p are constants
Taking the natural log (ln) of
both plotted variables and plot
them
Source: Kirk, 2007, p.
7
►
Source: Kirk, 2007, p. 7
Show from the original
relationship why this is the
result
Power Law & Logs– Gravitational
Force
From the graph
► 𝑦 = 𝑚𝑥 + 𝑏 is
►
𝑙𝑜𝑔 𝐹 = −2 𝑙𝑜𝑔 𝑟 + 𝑙𝑜𝑔 𝑘
► Algebraically simplifying
►𝐹
=
𝑟 −2 k
or 𝐹 =
► where k=GMm
► 𝐹
Source: Kirk, 2007, p.
7
=
𝐺𝑀𝑚
𝑟2
𝑘
𝑟2
Exponentials and Logs - Radioactivity
Comparing thisSource:
toKirk,
the
2007, p.
7
► Many physics’ relationships
equation of a straight
line
are exponential.
𝑦 = 𝑚𝑥 + 𝑏
► Radioactivity is defined as ► y=ln(R), m= -λ and x = t
𝑅 = 𝑅𝑂 𝑒 −𝜆𝑡
► Graphing ln (R) vs. t in a
log-linear plot
where Ro and λ are constants.
Taking the log of both sides
►
►
𝑙𝑛 𝑅 = 𝑙𝑛 𝑅𝑂 𝑒 −𝜆𝑡
►
𝑙𝑛 𝑅 = 𝑙𝑛 𝑅𝑂 + 𝑙𝑛 𝑒 −𝜆𝑡
𝑙𝑛 𝑅 = 𝑙𝑛 𝑅𝑂 + −𝜆𝑡 ∙
𝑙𝑛 𝑒
► 𝑙𝑛 𝑅 = 𝑙𝑛 𝑅𝑂 + −𝜆𝑡
►
Source: Kirk, 2007, p.
Error Bars
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
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), 35-41.
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