This document provides an overview of biophysics and various physics concepts relevant to measurements and units, coordinate systems, vectors, and other foundational topics. It discusses standardized measurement systems, dimensional analysis, vector notation and properties, and methods for adding and subtracting vectors graphically and algebraically. The purpose is to introduce basic physics concepts and mathematical tools used in biophysics.

1. A Rapid trip Through Physics ToA Rapid trip Through Physics To
BiophysicsBiophysics
Umed Aruzery (PhDc)
2016-2017
Biophysics

2. 1. Measurements1. Measurements
Measurement is :Measurement is :
►Basis ofBasis of testingtesting theories in sciencetheories in science
►Need to have consistentNeed to have consistent systems of unitssystems of units forfor
the measurementsthe measurements
►UncertaintiesUncertainties are inherentare inherent
►NeedNeed rules for dealing with the uncertaintiesrules for dealing with the uncertainties

3. Systems of MeasurementSystems of Measurement
►Standardized systemsStandardized systems
 agreed upon by some authority, usually aagreed upon by some authority, usually a
governmental bodygovernmental body
►SI -- SystSI -- Systééme Internationalme International
 agreed to in 1960 by an international committeeagreed to in 1960 by an international committee
 main system used in this coursemain system used in this course
 also calledalso called mksmks for the first letters in the units offor the first letters in the units of
the fundamental quantitiesthe fundamental quantities

4. Systems of MeasurementsSystems of Measurements
►cgscgs -- Gaussian system-- Gaussian system
 named for the first letters of the units it uses fornamed for the first letters of the units it uses for
fundamental quantitiesfundamental quantities
►US CustomaryUS Customary
 everyday units (ft, mile, etc.)everyday units (ft, mile, etc.)
 often uses weight, in pounds, instead of massoften uses weight, in pounds, instead of mass
as a fundamental quantityas a fundamental quantity

5. Basic Quantities and Their DimensionBasic Quantities and Their Dimension
►Length [L]Length [L]
►Mass [M]Mass [M]
►Time [T]Time [T]
Why do we need standards?

6. LengthLength
►UnitsUnits
 SI -- meter, mSI -- meter, m
 cgs -- centimeter, cmcgs -- centimeter, cm
 US Customary -- foot, ftUS Customary -- foot, ft
►Defined in terms of a meter -- the distanceDefined in terms of a meter -- the distance
traveled by light in a vacuum during a giventraveled by light in a vacuum during a given
time (1/299 792 458 s)time (1/299 792 458 s)

7. MassMass
►UnitsUnits
 SI -- kilogram, kgSI -- kilogram, kg
 cgs -- gram, gcgs -- gram, g
 USC -- slug, slugUSC -- slug, slug
►Defined in terms of kilogram, based on aDefined in terms of kilogram, based on a
specific Pt-Ir cylinder kept at thespecific Pt-Ir cylinder kept at the
International Bureau of StandardsInternational Bureau of Standards

8. Standard KilogramStandard Kilogram
Why is it hidden under two glass domes?

9. TimeTime
►UnitsUnits
 seconds, sseconds, s in all three systemsin all three systems
►Defined in terms of the oscillation ofDefined in terms of the oscillation of
radiation from a cesium atomradiation from a cesium atom
(9 192 631 700 times frequency of light emitted)(9 192 631 700 times frequency of light emitted)

10. Time MeasurementsTime Measurements

11. USUS “Official” Atomic Clock“Official” Atomic Clock

12. 2. Dimensional Analysis2. Dimensional Analysis
► DimensionDimension denotes thedenotes the physical naturephysical nature of aof a
quantityquantity
► Technique toTechnique to check the correctnesscheck the correctness of anof an
equationequation
► Dimensions (length, mass, time, combinations)Dimensions (length, mass, time, combinations)
can be treated as algebraic quantitiescan be treated as algebraic quantities
 add, subtract, multiply, divideadd, subtract, multiply, divide
 quantities added/subtracted only if have same unitsquantities added/subtracted only if have same units
► Both sides of equation must have the sameBoth sides of equation must have the same
dimensionsdimensions

13. Dimensional AnalysisDimensional Analysis
► Dimensions for commonly used quantitiesDimensions for commonly used quantities
Length L m (SI)
Area L2
m2
(SI)
Volume L3
m3
(SI)
Velocity (speed) L/T m/s (SI)
Acceleration L/T2
m/s2
(SI)
 Example of dimensional analysisExample of dimensional analysis
distance = velocity · time
L = (L/T) · T

14. 3. Conversions3. Conversions
►WhenWhen units are not consistentunits are not consistent, you may, you may
need toneed to convertconvert to appropriate onesto appropriate ones
►Units can be treated like algebraicUnits can be treated like algebraic
quantities that canquantities that can cancel each other outcancel each other out
1 mile = 1609 m = 1.609 km 1 ft = 0.3048 m = 30.48 cm
1m = 39.37 in = 3.281 ft 1 in = 0.0254 m = 2.54 cm

15. Example 1Example 1. Scotch tape:. Scotch tape:
Example 2Example 2. Trip to Canada:. Trip to Canada:
Legal freeway speed limit in Canada is 100 km/h.
What is it in miles/h?
h
miles
km
mile
h
km
h
km
62
609.1
1
100100 ≈⋅=

16. PrefixesPrefixes
►Prefixes correspond to powers of 10Prefixes correspond to powers of 10
►Each prefix has a specificEach prefix has a specific
name/abbreviationname/abbreviation
Power Prefix Abbrev.
1015
peta P
109
giga G
106
mega M
103
kilo k
10-2
centi P
10-3
milli m
10-6
micro µ
10-9
nano n
Distance from Earth to nearest star 40 Pm
Mean radius of Earth 6 Mm
Length of a housefly 5 mm
Size of living cells 10 µm
Size of an atom 0.1 nm

17. Example: An aspirin tablet contains 325 mg of acetylsalicylic acid.
Express this mass in grams.
Solution:
3
325 325 10 0.325m mg g g−
= = × =
Given:
m = 325 mg
Find:
m (grams)=?
Recall that prefix “milli” implies 10-3
, so

18. Math Review:Math Review: Coordinate SystemsCoordinate Systems
►Used to describe the position of a point inUsed to describe the position of a point in
spacespace
►Coordinate system (frame)Coordinate system (frame) consists ofconsists of
 a fixed reference point called thea fixed reference point called the originorigin
 specificspecific axes with scales and labelsaxes with scales and labels
 instructions on how to label a pointinstructions on how to label a point relative torelative to
the origin and the axesthe origin and the axes

19. Types of Coordinate SystemsTypes of Coordinate Systems
►CartesianCartesian
►Plane polarPlane polar

20. Cartesian coordinate systemCartesian coordinate system
► also called rectangularalso called rectangular
coordinate systemcoordinate system
► x- and y- axesx- and y- axes
► points are labeled (x,y)points are labeled (x,y)

21. Plane polar coordinate systemPlane polar coordinate system
 origin and referenceorigin and reference
line are notedline are noted
 point is distance r frompoint is distance r from
the origin in thethe origin in the
direction of angledirection of angle θθ, ccw, ccw
from reference linefrom reference line
 points are labeled (r,points are labeled (r,θθ))

22. Math Review:Math Review: TrigonometryTrigonometry
sin
sideadjacent
sideopposite
hypotenuse
sideadjacent
hypotenuse
sideopposite
=
=
=
θ
θ
θ
tan
cos
sin
 PythagoreanPythagorean
TheoremTheorem 222
bac +=

23. Example: how high is the building?Example: how high is the building?
Slide 13
Fig. 1.7, p.14
Known: angle and one side
Find: another side
Key: tangent is defined via two
sides!
mmdistheight
dist
buildingofheight
3.37)0.46)(0.39(tantan.
,
.
tan
==×=
=

α
α
α

24. Math Review:Math Review: Scalar and VectorScalar and Vector
QuantitiesQuantities
► ScalarScalar quantities are completely described byquantities are completely described by
magnitude only (magnitude only (temperature, lengthtemperature, length,…),…)
► VectorVector quantities need both magnitude (size) andquantities need both magnitude (size) and
direction to completely describe themdirection to completely describe them
((force, displacement, velocityforce, displacement, velocity,…),…)
 Represented by an arrow, theRepresented by an arrow, the lengthlength of the arrowof the arrow isis
proportional to the magnitudeproportional to the magnitude of the vectorof the vector
 Head of the arrow represents the directionHead of the arrow represents the direction

25. Vector NotationVector Notation
►WhenWhen handwrittenhandwritten, use an arrow:, use an arrow:
►WhenWhen printedprinted, will be in bold print:, will be in bold print: AA
►When dealing with just the magnitude of aWhen dealing with just the magnitude of a
vector in print, an italic letter will be used:vector in print, an italic letter will be used: AA
A


26. Properties of VectorsProperties of Vectors
►Equality of Two VectorsEquality of Two Vectors
 Two vectors areTwo vectors are equalequal if they have theif they have the samesame
magnitudemagnitude and theand the same directionsame direction
►Movement of vectors in a diagramMovement of vectors in a diagram
 Any vector can be movedAny vector can be moved parallel to itselfparallel to itself
without being affectedwithout being affected

27. More Properties of VectorsMore Properties of Vectors
►Negative VectorsNegative Vectors
 Two vectors areTwo vectors are negativenegative if they have theif they have the
same magnitude but are 180° apart (oppositesame magnitude but are 180° apart (opposite
directions)directions)
► AA = -= -BB
►Resultant VectorResultant Vector
 TheThe resultantresultant vector is the sum of a given setvector is the sum of a given set
of vectorsof vectors

28. Adding VectorsAdding Vectors
►When adding vectors,When adding vectors, their directions musttheir directions must
be taken into accountbe taken into account
►Units must be the sameUnits must be the same
►Graphical MethodsGraphical Methods
 Use scale drawingsUse scale drawings
►Algebraic MethodsAlgebraic Methods
 More convenientMore convenient

29. Adding Vectors GraphicallyAdding Vectors Graphically
(Triangle or Polygon Method)(Triangle or Polygon Method)
► Choose a scaleChoose a scale
► Draw the first vector with the appropriate lengthDraw the first vector with the appropriate length
and in the direction specified, with respect to aand in the direction specified, with respect to a
coordinate systemcoordinate system
► Draw the next vector with the appropriate lengthDraw the next vector with the appropriate length
and in the direction specified, with respect to aand in the direction specified, with respect to a
coordinate system whose origin is the end ofcoordinate system whose origin is the end of
vectorvector AA and parallel to the coordinate systemand parallel to the coordinate system
used forused for AA

30. Graphically Adding VectorsGraphically Adding Vectors
► Continue drawing theContinue drawing the
vectorsvectors “tip-to-tail”“tip-to-tail”
► The resultant is drawnThe resultant is drawn
from the origin offrom the origin of AA to theto the
end of the last vectorend of the last vector
► Measure the length ofMeasure the length of RR
and its angleand its angle
 Use the scale factor toUse the scale factor to
convert length to actualconvert length to actual
magnitudemagnitude

31. Graphically Adding VectorsGraphically Adding Vectors
► When you have manyWhen you have many
vectors, just keepvectors, just keep
repeating the processrepeating the process
until all are includeduntil all are included
► The resultant is stillThe resultant is still
drawn from the origindrawn from the origin
of the first vector to theof the first vector to the
end of the last vectorend of the last vector

32. Alternative Graphical MethodAlternative Graphical Method
► When you have only twoWhen you have only two
vectors, you may use thevectors, you may use the
Parallelogram MethodParallelogram Method
► All vectors, including theAll vectors, including the
resultant, are drawn from aresultant, are drawn from a
common origincommon origin
 The remaining sides of theThe remaining sides of the
parallelogram are sketchedparallelogram are sketched
to determine the diagonal,to determine the diagonal, RR

33. Notes about Vector AdditionNotes about Vector Addition
► Vectors obey theVectors obey the
Commutative LawCommutative Law
of Additionof Addition
 The order in which theThe order in which the
vectors are addedvectors are added
doesndoesn’t affect the result’t affect the result

34. Vector SubtractionVector Subtraction
► Special case of vectorSpecial case of vector
additionaddition
► IfIf AA –– BB, then use, then use AA+(+(--
BB))
► Continue with standardContinue with standard
vector additionvector addition
procedureprocedure

35. Multiplying or Dividing a VectorMultiplying or Dividing a Vector
by a Scalarby a Scalar
► TheThe resultresult of the multiplication or division is aof the multiplication or division is a vectorvector
► TheThe magnitudemagnitude of the vector is multiplied or divided by theof the vector is multiplied or divided by the
scalarscalar
► If the scalar isIf the scalar is positivepositive, the, the directiondirection of the result is theof the result is the
samesame as of the original vectoras of the original vector
► If the scalar isIf the scalar is negativenegative, the, the directiondirection of the result isof the result is
oppositeopposite that of the original vectorthat of the original vector

36. Components of a VectorComponents of a Vector
► AA componentcomponent is ais a
partpart
► It is useful to useIt is useful to use
rectangularrectangular
componentscomponents
 These are theThese are the
projections of the vectorprojections of the vector
along the x- and y-axesalong the x- and y-axes

37. Components of a VectorComponents of a Vector
►The x-component of a vector is theThe x-component of a vector is the
projection along the x-axisprojection along the x-axis
►The y-component of a vector is theThe y-component of a vector is the
projection along the y-axisprojection along the y-axis
►Then,Then,
cosxA A θ=
sinyA A θ=
x yA A= +A
ur ur ur

38. More About Components of aMore About Components of a
VectorVector
► The previous equations are validThe previous equations are valid only ifonly if θ isθ is
measured with respect to the x-axismeasured with respect to the x-axis
► The components can be positive or negative andThe components can be positive or negative and
will have the same units as the original vectorwill have the same units as the original vector
► The components are the legs of the right triangleThe components are the legs of the right triangle
whose hypotenuse iswhose hypotenuse is AA
 May still have to find θ with respect to the positive x-axisMay still have to find θ with respect to the positive x-axis
x
y12
y
2
x
A
A
tanandAAA −
=θ+=

39. Adding Vectors AlgebraicallyAdding Vectors Algebraically
►Choose a coordinate system and sketch theChoose a coordinate system and sketch the
vectorsvectors
►Find the x- and y-components of all theFind the x- and y-components of all the
vectorsvectors
►Add all the x-componentsAdd all the x-components
 This gives RThis gives Rxx::
∑= xx vR

40. Adding Vectors AlgebraicallyAdding Vectors Algebraically
►Add all the y-componentsAdd all the y-components
 This gives RThis gives Ryy::
►Use the Pythagorean Theorem to find theUse the Pythagorean Theorem to find the
magnitude of the Resultant:magnitude of the Resultant:
►Use the inverse tangent function to find theUse the inverse tangent function to find the
direction of R:direction of R:
∑= yy vR
2
y
2
x RRR +=
x
y1
R
R
tan−
=θ

41. Problem Solving StrategyProblem Solving Strategy
Slide 13
Fig. 1.7, p.14
Known: angle and one side
Find: another side
Key: tangent is defined via two sides!
mmdistheight
dist
buildingofheight
3.37)0.46)(0.39(tantan.
,
.
tan
==×=
=

α
α

42. Problem Solving StrategyProblem Solving Strategy
►Read the problemRead the problem
 identify type of problem, principle involvedidentify type of problem, principle involved
►Draw a diagramDraw a diagram
 include appropriate values and coordinateinclude appropriate values and coordinate
systemsystem
 some types of problems require very specificsome types of problems require very specific
types of diagramstypes of diagrams

43. Problem Solving cont.Problem Solving cont.
►Visualize the problemVisualize the problem
►Identify informationIdentify information
 identify the principle involvedidentify the principle involved
 list the data (given information)list the data (given information)
 indicate the unknown (what you are looking for)indicate the unknown (what you are looking for)

44. Problem Solving, cont.Problem Solving, cont.
►Choose equation(s)Choose equation(s)
 based on the principle, choose an equation orbased on the principle, choose an equation or
set of equations to apply to the problemset of equations to apply to the problem
 solve for the unknownsolve for the unknown
►Solve the equation(s)Solve the equation(s)
 substitute the data into the equationsubstitute the data into the equation
 include unitsinclude units

45. Problem Solving, finalProblem Solving, final
► Evaluate the answerEvaluate the answer
 find the numerical resultfind the numerical result
 determine the units of the resultdetermine the units of the result
► Check the answerCheck the answer
 are the units correct for the quantity being found?are the units correct for the quantity being found?
 does the answer seem reasonable?does the answer seem reasonable?
► check order of magnitudecheck order of magnitude
 are signs appropriate and meaningful?are signs appropriate and meaningful?