metric review metric base units meter (m) length mass volume time gram (g) liter (l) second (s)...

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.

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

Metric Prefixes

Kids Have Dropped (over) Dead Converting Metrics!

____ ____ ____ ____ ____ ____ ____

k h da d c m

Kilo Hecto Deka Base Deci Centi Milli

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/

http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/

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

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

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

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

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

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

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

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)

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 1/273.16 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.


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:

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:

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

►___ cm has the least # of sig figs, so answer must have only that many

►6.762 cm2___ cm2

2.1

6.8

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.

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 (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


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


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


Excel

Length (±0.05 cm)

14.88

14.84

15.02

14.57

14.76

14.66

C:%5CUsers%5CWStoll3%5CDocuments%5CSchool%5CPhysics%5CIntro%5CResources%5CRandom%20Error.xlsx

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


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.= %505.004.0

02.0

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

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 or

► 2. Adding or subtracting multiple measurements

► 3. Multiplying or dividing multiple measurements

► 4. Measured value raised to a power

For


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


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


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


► 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


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


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

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

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

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

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

12

12

xx

yy

run

riseSlope

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. , , , etc.


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


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


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

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:

►

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!

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



Power Law & Logs– Gravitational Force

From the graph ► is ► Algebraically simplifying► k or ► where k=GMm ►


► 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



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

Best Fit Line


Min & Max Slopes


Min & Max Y-Intercepts


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

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