Degree Course in PharmacyDegree Course in Pharmacy

Ph i FIS/01 dit 5Physics FIS/01    credits 5

Massimiliano Lucci

• Physics Applied to Pharmacy• Prof. Massimiliano Lucci• e‐mail: massimiliano.lucci@uniroma2.it

massimiliano.lucci@roma2.infn.it

• Introduction to physics; Kinematics of point‐like objects;j ;

• Dynamics of point‐like objects; Work, energy, potentials;

• Linear momentum, collisions; Extended bodies; Elasticproperties of materials; Properties of fluids; p p ; p ;Thermodynamics;

• Electricity and magnetism; Light and sound.

SUGGESTED TEXBOOKS

S J “Ph i f i iSerway, Jewett: “Physics for scientists and engineers” Hallidayand engineers  Halliday,

Joseph W. KaneMorton M Sternheim “Life ScienceMorton M. Sternheim Life Science 

Physics” John Wiley & Sons.y y“ Fisica Biomedica”EMSI.

Exams:

Test you on your basic understanding of the material and your quantitative problem solving skills.y q p g

There will be 2 or 3 midterm exams.

Chapter 1.Making measurements.  Using units.Th i i h i d l d h b i f• Theories in physics are developed on the basis of experimental observations, or are tested by comparing predictions with the results of experimentspredictions with the results of experiments.

• Being able to carry out experiments and understand• Being able to carry out experiments and understand their limitations is a critical part of physics or any experimental science.experimental science.

• In every experiment you make errors; understandingIn every experiment you make errors; understanding what to do with these errors is required if you want to compare experiments and theories.p p

Making measurements.

d h l f i

Using units.• In order to report the results of experiments, we need to agree on a system of units to be used.

• Only if all equipment are calibrated with respect to the same standard we can compare the results of different experiments.

• Although different units can be used to report different measurements, we need to know what units are used and how to do unit conversions.

• Using the wrong units can lead to expensive mistakes.

Making measurements.Using units.

• In this course we will use the SI System of units:

– Length: meter

– Time: second

– Mass: kg

• The SI units are related to the units you use in your daily life:

– Length: 1” = 2.54 cm = 0.0254 m

– Conversion factors can be found in the front cover of the book.

The base units.

O t illi th f th idi li f th th l

The unit of length: changes over time!• One ten‐millionth of the meridian line from the north pole 

to the equator that passes though Paris.

• Distance between 2 fine lines engraved near the ends of a Platinum‐Iridium bar kept at the International Bureau of Weights and Measures in ParisWeights and Measures in Paris.

• 1,650,763.73 Wavelengths of a particular orange‐red light1,650,763.73 Wavelengths of a particular orange red light emitted by Krypton‐86 in a gas discharge tube.

• Path length traveled by light in vacuum during a time interval of 1/299,792,458 of a second.

The base units.Their current definitions.

• TIME UNIT SECOND ( )• TIME  ‐ UNIT: SECOND (s)– One second is the time occupied by 919 263 170 vibrations of the light 

(of a specified wavelength) emitted by a Cesium‐133 atom.

• LENGTH  ‐ UNIT: METER (m)( )– Path length traveled by light in vacuum during a time interval of 

1/299 792 458 of a second.

• MASS  ‐ UNIT: KILOGRAM (kg)– One kilogram is the mass of a Platinum‐Iridium cylinder kept at the 

International Bureau of Weights and Measures in Paris.

The base SI unitsThe base SI units.

The current standard of the kg and the old standard of the m. 

SI (2)SI (2)

A è l’intensità di corrente elettrica costante che fluendo in due conduttoriA:  è l intensità di corrente elettrica costante che fluendo in due conduttori rettilinei e paralleli di sezione cilindrica trascurabile e posti ad un metro di distanza, determinano tra loro una forza di 2x10‐7 N per metro di conduttore. (1948) ( )

K: è l’unità di temperatura termodinamica pari a 1/273,16 del punto triplo dell’acqua. (1957)q ( )

cd: La candela è la intensità luminosa in una direzione che una sorgente monocromatica di 540x1012 Hz e la cui energia è (1/683) W/sr (Watt per g ( / ) / ( psteradiante)  (1979)

mol: La mole e la quantità di materia di un sistema che contiene tante qentità elementari quanti atomi sono contenuti in 0,012 kg di 12C (1971)

multiplies and fractionsmultiplies and fractions

NAME S b l f t NAME S b l f tNAME Symbol factor NAME Symbol factor

deca da 101 deci d 10-1

etto h 102 centi c 10-2

kilo k 103 milli m 10-3

mega M 106 micro µ 10-6

giga G 109 nano n 10-9

tera T 1012 pico p 10-12

peta P 1015 femto f 10-15peta P 10 femto f 10

exa E 1018 atto a 10-18

• Cultural issues

"non‐SI units accepted for use with the SI"p

• The units of time (minute, min; hour, h; day, d) in use besides the SI second, are specifically accepted for use according to table 6.

Th i ifi ll i l d d b h d d i• The year is specifically not included but has a recommended conversion factor.[37]

• The Celsius temperature scale; kelvins are rarely employed in everyday p ; y p y y yuse.

• Electric energy is often billed in kilowatt hours, instead of megajoules. Similarly battery charge is often measured as milliampere hours (mA h)Similarly, battery charge is often measured as milliampere hours (mA∙h), instead of coulombs.

• The nautical mile and knot (nautical mile per hour) used to measure travel distance and speed of ships and aircraft (1 International nautical mile = 1852 m or approximately 1 minute of latitude). In addition to these, Annex 5 of the Convention on International Civil Aviation permits the "temporary p p yuse" of the foot for altitude. 

• Astronomical distances measured in astronomical units, parsecs, and light‐years instead of for example petametres (a light year is about 9 461 Pmyears instead of, for example, petametres (a light‐year is about 9.461 Pm or about 9461000000000000 m).

• Cultural issues

"non‐SI units accepted for use with the SI"p

• Atomic scale units used in physics and chemistry, such as the ångström, electron volt, atomic mass unit and barn. 

S h i i f h i d (CGS) i• Some physicists prefer the centimetre‐gram‐second (CGS) units, or systems based on physical constants, such as Planck units, atomic units, or geometric units. 

• In the US, blood glucose measurements are recorded in milligrams per decilitre (mg/dL), which normalises to cg/L. In Canada, Australia, New Zealand Oceania and Europe the standard is millimole per litre (mmol/L)Zealand, Oceania, and Europe the standard is millimole per litre (mmol/L) or mM (millimolar). 

• Blood pressure is usually measured in mmHg (≈Torr). 

• Atmospheric pressure in government weather reports is measured in inHgin the USA,[38] and in the SI unit hPa in Australia,[39] UK[40] and most other countries.

Error Analysis.Some important facts.

• Why should we care?T f• Types of errors.

• The Gaussian distribution ‐ not all results can be described in terms of such distribution butbe described in terms of such distribution, but most of them can.

• Estimate the parameters of the Gaussian• Estimate the parameters of the Gaussian distribution (the mean and the width).

• Error propagationError propagation.• The weighted mean.

Error Analysis.Error Analysis.Is statistics relevant to you personally?

Global Warming

Analytical medical diagnostics

Effect of EM radiationEffect of EM radiation

Error Analysis.Type of Errors.

• Statistical errors: 

R l f d fl i i h f– Results from a random fluctuation in the process of measurement.  Often quantifiable in terms of “number of measurements or trials” Tends to makeof measurements or trials .  Tends to make measurements less precise.

• Systematic errors:Systematic errors:  

– Results from a bias in the observation due to observing conditions or apparatus or technique or analysis.  Tend pp q yto make measurements less accurate.

The Gaussian distribution: the most common error distribution.

The Gaussian Distribution:

its mean and its standard deviation.

1σ is roughly the half-width at half-maximum

2σ

of the distribution.

2σ

( ) 1− x−µ( )2

2σ 2

g x( )=

2πσe 2σ

Making measurements: increasing the number of measurements increases thenumber of measurements increases the 

accuracy.L h 10 1 Length = 10 m, σ = 1 m.

Probability of a single measurement falling within ±1σ of the mean is 0,683.

Probability of a single measurement falling within ±2σ of the mean is 0,954.

Probability of a single measurement falling within ±3σ of the mean is 0,997.

How to determine the mean µ and width σ of a distribution based on Nmeasurements?

How to determine the mean µ and width σ of a distribution based on Nmeasurements?

L N ?

x =x1 + x2 +L + xN−1 + xN

N=

1N

xii=1

N

∑ =?

µ

How to determine the mean µ and width σ of a distribution based on Nmeasurements?

The “standard deviation” is a measure of the error is a measure of the error in each of the N measurements:

N

xN

ii∑

=

−1

2)( µσ

Ni= 1σ

How to determine the mean µ and width σ of 

N

a distribution based on Nmeasurements?

xN

ii∑

=

−1

2)( µ

Ni== 1σ• The standard deviation is equal to

• But ….. µ is unknown. So we will usethe mean (which is your best estimatethe mean (which is your best estimateof µ). We also change thedenominator to increase the errorli htl d t i th

Nslightly due to using the mean.

• This is the form of the standarddeviation you use in practice: σ =

(xi − x )2

i=1∑

y p• Note: This quantity cannot be

determined from a singlemeasurement

N −1

measurement.

What matters?  The standard deviation or the error in the mean?

• The standard deviation is a measure of the error made ineach individual measurementeach individual measurement.

• Often you want to measure the mean and the error in themean.Whi h h ld h ll i di id l• Which should have a smaller error, an individualmeasurement or the mean?

• The answer ….. the mean, if you do more than one

σ

, ymeasurement:

σm =

N

Applying this in the laboratory.Measuring g.

Student 1: 9.0 m/s2

Student 2: 8.8 m/s2

Student 3: 9.1 m/s2

Student 4: 8.9 m/s2

What is the best estimate of theStudent 5: 9.1 m/s2 What is the best estimate of the

gravitational acceleration measured by these students?

1998198809 m++++20.9

51.99.81.98.80.9

sma =

++++=

Applying this in the laboratory.Measuring g.

St d t 1 9 0 / 2Student 1: 9.0 m/s2

Student 2: 8.8 m/s2

Student 3: 9 1 m/s2 Wh i h b i f hStudent 3: 9.1 m/s2

Student 4: 8.9 m/s2

Student 5: 9 1 m/s2

What is the best estimate of the standard deviation of the gravitational accelerationStudent 5: 9.1 m/s gravitational acceleration measured by these students?

22222 )0.91.9()0.99.8()0.91.9()0.98.8()0.90.9( −+−+−+−+−=σ

120

15m

=

−=σ

212.0s

=

Applying this in the laboratory.Measuring g.

Student 1: 9.0 m/s2

Student 2: 8.8 m/s2

Student 3: 9.1 m/s2

Student 4: 8.9 m/s2

Student 5: 9.1 m/s2

2054.0512.0

sm

m ==σ

Note: this procedure is valid if you can assume that all your measurements have the same measurement error.

Final result: g = 9.0 ± 0,05 m/s2. Does this agree with the accepted value of 9,8 m/s2?

How does an error in one measurable affect the error in another measurable?

y

y = F(x)

y1

y1+δy

y1-δy

xx1x1-δx x1+δx

How does an error in one measurable affect the error in another measurable?

The degree to which an error in one meas rabley error in one measurable affects the error in another is driven by the functional dependence of the

y

dependence of the variables (or the slope: dy/dx)y = F(x)

y1+δy

y1y1-δy

xx1x1-δx x1+δx

How does an error in one measurable affect the error in another measurable?

• But …… Most physical relationships involve multiple measurables!multiple measurables!

x = xo + vot +12

at2

• We must take into account the dependence

2F = Ma

• We must take into account the dependence of the parameter of interest, f, on each of h b fthe contributing quantities, x, y, z, ….: f = F(x, y, z,…)

Error Propagation.Partial Derivatives.

• The partial derivative with respect to a certain variable is the ordinary derivative of the function with respect to that variable pwhere all the other variables are treated as constantsconstants.

zyxdFzyxF ...),,(,...),,( ⎤∂

constzydxy

xy

...,

),,(, ),,(⎥⎦⎤=

∂

Error Propagation.Partial Derivatives: an example.

32),,( yzxzyxF =

32xyzF=

∂ 2xyzx=

∂32F∂ 32 zx

yF=

∂∂

F∂ 22 3zyxzF=

∂∂

z∂

Error Propagation.The formula!

• Consider that a parameter of interest f = F(x, y, z, …)depends on the measured parameters x y zdepends on the measured parameters x, y, z, ….

• The error in f, σf, depends on the function F, measuredd h i dparameters x, y, z, …, and their errors, σx, σy, σz, …, and

can be calculated using the following formula:

...22

22

22

+⎠⎞

⎜⎝⎛ ∂+

⎞⎜⎜⎛ ∂

+⎠⎞

⎜⎝⎛ ∂= zyxf

FFF σσσσ⎠

⎜⎝ ∂⎠

⎜⎝ ∂⎠

⎜⎝ ∂

zyxf zyx

The formula for error propagation.An Example.

A pitcher throws a baseball at a distance of 30 ± 0,5 m at 40 ± 3 m/s .From this data, calculate the time of flight of the baseball.

t = d σ t =1⎛

⎝⎜⎞⎠

2

σ d2 + −

d2

⎛⎝⎜

⎞⎠

2

σ v2 =t

v∂F

=1

t v⎝⎜ ⎠ d v2⎝⎜ ⎠ v

0 5⎛ ⎞2

30⎛ ⎞2

∂d=

v∂F d

=0.540

⎛⎝⎜

⎞⎠⎟

+30402

⎛⎝⎜

⎞⎠⎟

32 = 0.058⇒

∂F∂v

= −dv2

t = 0.75 ± 0.058s

Another example of error propagationAnother example of error propagation.

v = at: determine a and its errorv = at: determine a and its error.

Suppose we just had 1 dataSuppose we just had 1 datapoint, which data point would provide the best estimate of a?

Another example of error propagationAnother example of error propagation.

• For each data point we can determinea (= v/t) and its error:

σ a =1σ v

⎛⎝⎜

⎞⎠

2

+v2 σ t

⎛⎝⎜

⎞⎠

2

=a t v⎝⎜ ⎠ t2 t⎝⎜ ⎠

=v σ v⎛

⎜⎞

2

+σ t⎛

⎜⎞

2

=

t v⎝⎜ ⎠+

t⎝⎜ ⎠

• We see that the error in a is differentfor different points. Simpleaveraging will not be the proper wayaveraging will not be the proper wayto determine a and its error.

The weighted meanThe weighted mean.

Wh h d h diff d h• When the data have different errors, we need to use theweighted mean to estimate the mean value.

• This procedure requires you to assign a weight to each datap q y g gpoint:

w =1

• Note: when the error decreases the weight increases.

wi = σ i2

• The weighted mean and its error are defined as:

w yN

∑y =

wi yii=1∑

wN

∑σ y =

1

wN

∑wii=1∑ wi

i=1∑