Pharmaceutical Physics I

Dr. A. Búcsi room No O-3bucsi@fpharm.uniba.sk

Physics –Exam (test) 80%Exercises 20%

Minimum requirement 60%

Problem solving

• Physics - measurable quantities

- units

International System of Units (Le Système Internationald'Unités) – SI

Seven dimensionally independent,

fundamental,

irreducible

and complete system

New definitions valid from May 2019!

The International System of Units, the SI is the system of units based

on physical constants;

________________________________________________________________________

Defining constant Symbol Numerical value Unit________________________________________________________________hyperfine transition frequency of Cs ΔνCs 9 192 631 770 Hzspeed of light in vacuum c 299 792 458 m s−1

Planck constant h 6.626 070 15×10−34 J selementary charge e 1.602 176 634 ×10−19 CBoltzmann constant k 1.380 649 × 10−23 J K−1

Avogadro constant NA 6.022 140 76 × 1023 mol−1

luminous efficacy Kcd 683 lm W−1

___________________________________________________________________

• second – unit of time

The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency ΔνCs, the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, to be 9 192 631 770 when expressed in the unit Hz, which is equal to s−1.

1𝑠 =9 192 631 770

∆𝜈𝐶𝑠

𝐸 = ℎ𝜈𝜆 = 𝑐 /ν

History - the fraction 1/86 400 of the mean solar day

s

• meter - unit of length

The metre, symbol m, is the SI unit of length. It is defined bytaking the fixed numerical value of the speed of light invacuum c to be 299 792 458 when expressed in the unitms−1, where the second is defined in terms of the caesiumfrequency ΔνCs.

1𝑚 =𝑐

299 792 458𝑠 𝑐 = 𝑚𝑠−1

• The current definition had the effect of fixing the speed of light in a vacuum at exactly 299 792 458 m/s.

• History – derived using the distance of the North pole from the Equator, later length of a platinum-iridium bar

m

• kilogram - unit of mass

The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant h to be 6.626 070 15 × 10−34 when expressed in the unit J s, which is equal to kg m2 s−1, where the metre and the second are defined in terms of c and ΔνCs.

ℎ = 6.626070𝑥10−34 kg𝑚2𝑠−1

relation gives an exact expression for the kilogram in terms of the three defining constants h, ΔνCS and c:

1𝑘𝑔 =ℎ

6.626 070 15𝑥10−34𝑚−2𝑠

kg

History

• platinum-iridium cylinder - the international prototype of the kilogram

kg

• Ampere – unit of electric current

The ampere, symbol A, is the SI unit of electric current. It isdefined by taking the fixed numerical value of theelementary charge e to be 1.602 176 634 × 10−19 whenexpressed in the unit C, which is equal to A s, where thesecond is defined in terms of ΔνCs.

This definition implies the exact relation e = 1.602 176 634 x10−19 As. Inverting this relation gives an exact expression forthe unit ampere in terms of the defining constants e

and ΔνCs:

1𝐴 =𝑒

1.602 176 364 𝑥 10−19𝑠−1

1𝐶 = 1𝐴𝑠

A

History

• The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 meter apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 newton per meter of length.

A

• Kelvin – unit of thermodynamic temperatureIt is defined by taking the fixed numerical value of the Boltzmann constant k to be 1.380 649 × 10−23 when expressed in the unit JK−1, which is equal to kg m2 s−2 K−1, where the kilogram, metreand second are defined in terms of h, c and ΔνCs.

K

1𝐾 =1.308 649

𝑘𝑥10−23𝑘𝑔𝑚2𝑠−2

𝐸 = 𝑘𝑇1𝐾 → 1.380 649𝑥10−23𝐽

• History

• The kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.

Celsius temperature is the degree Celsius - °C, which is by definition equal in magnitude to the kelvin

t/°C = T/K - 273.15.

• degree Kelvin

K

• mole – unit of the amount of substance

The mole, symbol mol, is the SI unit of amount of substance. Onemole contains exactly 6.022 140 76 × 1023 elementary entities.This number is the fixed numerical value of the Avogadroconstant, NA, when expressed in the unit mol−1 and is called the

Avogadro number.

• elementary entities - atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

1𝑚𝑜𝑙 =6.022 140 76𝑥1023

𝑁𝐴

mol

• History

• The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12. (NA=6.022x1023 Avogadro’s number)

mol

• candela – luminous intensityThe candela, symbol cd, is the SI unit of luminous intensity in agiven direction. It is defined by taking the fixed numerical valueof the luminous efficacy of monochromatic radiation offrequency 540 × 1012 Hz, Kcd, to be 683 when expressed in theunit lm W−1, which is equal to cd sr W−1, or cd sr kg−1 m−2 s3,where the kilogram, metre and second are defined in terms of h,c and ΔνCs.

1𝑐𝑑 =𝐾𝑐𝑑

683𝑘𝑔 𝑚2𝑠−3𝑠𝑟 −1

cd

• History

Luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

cd

Derived units.

radian 0-2 π [rad] = (m/m)=1

Arc length 𝑙 = 𝛼 𝑟 [α]=rad

circumference= 2π radius

steradian (square radian) 0-4 π

[sr] = (m2 /m2)=1

Surface area S = 𝛽𝑟2 [β]=sr

sphere surface area = 4 π radius2

degree Celsius

t [°C] = T – 273.15 [K]

Multiplication and/or division of fundamental units

𝑑𝑖𝑚 𝑋 = 𝑠𝛼𝑚𝛽𝑘𝑔𝛾𝐴𝛿𝐾 𝑚𝑜𝑙𝜖𝑐𝑑𝜃

Examples

Speed: length/time -> [v] = m/sAcceleration: speed/time -> [a] = m/s2

Force: mass x acceleration -> [F]= kg m/s2 = NPressure: force/area => [P] = N / m2 = kg /ms2 = Paetc…

Prefix Label Decimal value Numerical factor

yocto y 0.000 000 000 000 000 000 000 001 10-24

zepto z 0.000 000 000 000 000 000 001 10-21

atto a 0. 000 000 000 000 000 001 10-18

femto f 0. 000 000 000 000 001 10-15

pico p 0. 000 000 000 001 10-12

nano n 0.000 000 001 10-9

micro μ 0.000 001 10-6

milli m 0.001 10-3

centi c 0.01 10-2

deci d 0.1 10-1

---- --- 1 100

deka da 10 101

hecto h 100 102

kilo k 1 000 103

mega M 1 000 000 106

giga G 1 000 000 000 109

tera T 1 000 000 000 000 1012

peta P 1 000 000 000 000 000 1015

exa E 1 000 000 000 000 000 000 1018

zetta Z 1 000 000 000 000 000 000 000 1021

yotta Y 1 000 000 000 000 000 000 000 000 1024

Conversion of units

• Example1 v= 28 m/s ? km/h

28𝑚

𝑠

1𝑘𝑚

1000𝑚

3600𝑠

1ℎ= 28

3600

1000ൗ𝑘𝑚ℎ

28 Τ𝑚𝑠 = 100.8 ൗ𝑘𝑚

ℎ

• Example2 κ=3400 μS cm-1 ? S m-1

3400 𝜇𝑆𝑐𝑚 −11𝑆

106𝜇𝑆

100 𝑐𝑚

1𝑚= 0.34𝑆𝑚−1

Scalars and vectorsScalars - magnitudeTemperature, speed, mass, volume... Vectors – magnitude, direction and orientation. force, displacement, velocity, acceleration...

Vector notation

Vector equality

Vector representation –a=(x,y,z)a=(r,θ,ϕ)

Components(2D)

• Vector operations

Addition

Graphical

• Calculation

• Subtraction

B(Bx,By) => - B(-Bx,-By)

A – B = A + (-B)

𝐴 + 𝐵

= 𝐴 2

+ 𝐵 2

+ 2 𝐴 𝐵 𝑐𝑜𝑠𝜃

0 4 8 12 160

7

14

21

28

3 A (15,21) y

x

A (5,7)parallel

Multiplication with scalarA (Ax,Ay)

k A = (kAx,kAy)

result - vector

• Scalar (dot) product of two vectors

result – scalar !!

𝐷 = 𝐴 ∙ 𝐵 cos 𝜃

𝐷 = 𝐴𝑥𝐵𝑥 + 𝐴𝑦𝐵𝑦

• Vector product of two vectors

Magnitude

|A x B| = |A| |B| sinθ

A x B = ‒ B x A

Result – vector!!

A x B (AyBz-AzBy, AxBz-AzBx, AxBy-AyBx)

Newtonian MechanicsKinematics, dynamics

Masspoint mechanics

• Dimensionless object with mass – point

trajectory - s (d,h) [m]speed – velocity v (u) [m/s]acceleration a [m/s2]time t [s]

Uniform linear motion

• Trajectory – along straight line

• Speed – constant

• Acceleration - zero

s = s0 + v0 . t

v = v0 = const.

a = 0

v0= (s-s0)/t

𝑣 = න 𝑎 𝑑𝑡 = 0 + 𝑐𝑜𝑛𝑠𝑡

𝑠 = න 𝑣0𝑑𝑡 = 𝑣𝑡 + 𝑐𝑜𝑛𝑠𝑡

Uniformly accelerated linear motion

a=constant

a = const.

𝑡 = 0 ⟹ 𝑣 = 𝑣0

𝑡 = 0 ⟹ 𝑠 = 𝑠0

𝑣 = න 𝑎 𝑑𝑡 = 𝑎𝑡 + 𝑣0

𝑠 = න 𝑣 𝑑𝑡 = න 𝑎𝑡 + 𝑣0 𝑑𝑡 =1

2𝑎𝑡2 + 𝑣0𝑡 + 𝑠0

• Freefall – special case

standard acceleration due

to gravity - standard gravity

(Earth)

a=g=9,81 m/s2

𝑡 =2ℎ

𝑔

Summary

SI units

Vectors 𝐴 + 𝐵 = 𝐴 2

+ 𝐵 2

+ 2 𝐴 𝐵 𝑐𝑜𝑠𝜃

𝐴 + 𝐵 = 𝐴𝑥 + 𝐵𝑥 , 𝐴𝑦 + 𝐵𝑦 tan 𝜃 =𝐴𝑦 +𝐵𝑦

𝐴𝑥 +𝐵𝑥

𝑘𝐴 = (𝑘𝐴𝑥 , 𝑘𝐴𝑦 )

𝐷 = 𝐴 𝐵 𝑐𝑜𝑠𝜃 𝐷 = 𝐴𝑥𝐵𝑥 + 𝐴𝑦𝐵𝑦

𝐴 𝑥𝐵 = 𝐴 𝐵 𝑠𝑖𝑛𝜃 𝐴 𝑥𝐵 = −𝐵 𝑥𝐴

Uniform linear motion

𝑠 = 𝑠0 + 𝑣0𝑡 𝑣 =Δ𝑠

Δ𝑡

Uniformly accelerated linear motion

𝑠 = 𝑠0 + 𝑣0𝑡 + 12ൗ 𝑎𝑡2 𝑣 =

𝑑𝑠

𝑑𝑡= 𝑣0 + 𝑎𝑡 𝑎 =

𝑑2𝑠

𝑑𝑡 2 =𝑑𝑣

𝑑𝑡

Freefall ℎ = 12ൗ 𝑔𝑡2