A Rapid trip Through A Rapid trip Through Physics To BiophysicsPhysics To Biophysics

Umed Aruzery (PhDc)

2016-2017

Biophysics

1. Measurements1. Measurements

Measurement is :Measurement is :►Basis of Basis of testingtesting theories in theories in sciencescience►Need to have consistent Need to have consistent systems systems of unitsof units for the measurements for the measurements►UncertaintiesUncertainties are inherent are inherent►Need Need rules for dealing with the rules for dealing with the uncertaintiesuncertainties

Systems of MeasurementSystems of Measurement►Standardized systemsStandardized systems

agreed upon by some authority, agreed upon by some authority, usually a governmental bodyusually a governmental body

►SI -- SystSI -- Systééme Internationalme International agreed to in 1960 by an agreed to in 1960 by an international committeeinternational committee

main system used in this coursemain system used in this course also called also called mksmks for the first for the first letters in the units of the letters in the units of the fundamental quantitiesfundamental quantities

Systems of MeasurementsSystems of Measurements►cgscgs -- Gaussian system -- Gaussian system

named for the first letters of the named for the first letters of the units it uses for fundamental units it uses for fundamental quantitiesquantities

►US CustomaryUS Customary everyday units (ft, mile, etc.)everyday units (ft, mile, etc.) often uses weight, in pounds, often uses weight, in pounds, instead of mass as a fundamental instead of mass as a fundamental quantityquantity

Basic Quantities and Their Basic Quantities and Their DimensionDimension

►Length [L]Length [L]►Mass [M]Mass [M]►Time [T]Time [T]

Why do we need standards?

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 -- Defined in terms of a meter -- the distance traveled by light the distance traveled by light in a vacuum during a given time in a vacuum during a given time (1/299 792 458 s)(1/299 792 458 s)

MassMass►UnitsUnits

SI -- kilogram, kgSI -- kilogram, kg cgs -- gram, gcgs -- gram, g USC -- slug, slugUSC -- slug, slug

►Defined in terms of kilogram, Defined in terms of kilogram, based on a specific Pt-Ir based on a specific Pt-Ir cylinder kept at the cylinder kept at the International Bureau of StandardsInternational Bureau of Standards

Standard KilogramStandard Kilogram

Why is it hidden under two glass domes?

TimeTime►UnitsUnits

seconds, sseconds, s in all three systems in all three systems►Defined in terms of the Defined in terms of the oscillation of radiation from a oscillation of radiation from a cesium atom cesium atom

(9 192 631 700 times frequency of light emitted)(9 192 631 700 times frequency of light emitted)

Time MeasurementsTime Measurements

US US ““OfficialOfficial”” Atomic Atomic ClockClock

2. Dimensional Analysis2. Dimensional Analysis

► DimensionDimension denotes the denotes the physical naturephysical nature of a of a quantity quantity

► Technique to Technique to check the correctnesscheck the correctness of an of an equationequation

► Dimensions (length, mass, time, Dimensions (length, mass, time, combinations) combinations) can be treated as algebraic can be treated as algebraic quantitiesquantities add, subtract, multiply, divideadd, subtract, multiply, divide quantities added/subtracted only if have same quantities added/subtracted only if have same unitsunits

► Both sides of equation must have the same Both sides of equation must have the same dimensionsdimensions

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 analysis Example of dimensional analysis distance = velocity · time L = (L/T) · T

3. Conversions3. Conversions

►When When units are not consistentunits are not consistent, you , you may need to may need to convertconvert to appropriate to appropriate onesones

►Units can be treated like algebraic Units can be treated like algebraic quantities that can quantities that can cancel each other cancel each other outout

1 mile = 1609 m = 1.609 km 1 ft = 0.3048 m = 30.48 cm1m = 39.37 in = 3.281 ft 1 in = 0.0254 m = 2.54 cm

Example 1Example 1. Scotch . Scotch tape: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?

hmiles

kmmile

hkm

hkm 62

609.11100100

PrefixesPrefixes►Prefixes correspond to powers of 10Prefixes correspond to powers of 10►Each prefix has a specific name/abbreviationEach prefix has a specific name/abbreviation

Power Prefix Abbrev.

1015 peta P109 giga G106 mega M103 kilo k10-2 centi P10-3 milli m10-6 micro 10-9 nano n

Distance from Earth to nearest star 40 PmMean radius of Earth 6 MmLength of a housefly 5 mmSize of living cells 10 mSize of an atom 0.1 nm

Example: An aspirin tablet contains 325 mg of acetylsalicylic acid. Express this mass in grams.

Solution:Given:

m = 325 mg

Find:

m (grams)=?

Recall that prefix “milli” implies 10-3, so

Math Review: Math Review: Coordinate Coordinate SystemsSystems

►Used to describe the position of Used to describe the position of a point in spacea point in space

►Coordinate system (frame)Coordinate system (frame) consists ofconsists of a fixed reference point called the a fixed reference point called the originorigin

specific specific axes with scales and labelsaxes with scales and labels instructions on how to label a pointinstructions on how to label a point relative to the origin and the axesrelative to the origin and the axes

Types of Coordinate Types of Coordinate SystemsSystems

►Cartesian Cartesian ►Plane polarPlane polar

Cartesian coordinate Cartesian coordinate systemsystem

► also called also called rectangular rectangular coordinate systemcoordinate system

► x- and y- axesx- and y- axes► points are points are labeled (x,y)labeled (x,y)

Plane polar coordinate Plane polar coordinate systemsystem

origin and origin and reference line reference line are notedare noted

point is distance point is distance r from the origin r from the origin in the direction in the direction of angle of angle , ccw , ccw from reference from reference lineline

points are points are labeled (r,labeled (r,))

Math Review: Math Review: TrigonometryTrigonometry

sin

sideadjacentsideopposite

hypotenusesideadjacent

hypotenusesideopposite

tan

cos

sin

Pythagorean Pythagorean TheoremTheorem

Example: how high is the Example: how high is the building?building?

Slide 13

Fig. 1.7, p.14

Known: angle and one sideFind: another side

Key: tangent is defined via two sides!

mmdistheightdistbuildingofheight

3.37)0.46)(0.39(tantan.

,.

tan

Math Review: Math Review: Scalar and Scalar and Vector Vector

QuantitiesQuantities► ScalarScalar quantities are completely described quantities are completely described by magnitude only (by magnitude only (temperature, lengthtemperature, length,…),…)

► VectorVector quantities need both magnitude quantities need both magnitude (size) and direction to completely (size) and direction to completely describe themdescribe them((force, displacement, velocityforce, displacement, velocity,…),…)

Represented by an arrow, the Represented by an arrow, the lengthlength of the of the arrow arrow is proportional to the magnitudeis proportional to the magnitude of the of the vectorvector

Head of the arrow represents the directionHead of the arrow represents the direction

Vector NotationVector Notation►When When handwrittenhandwritten, use an arrow:, use an arrow:►When When printedprinted, will be in bold , will be in bold print: print: AA

►When dealing with just the When dealing with just the magnitude of a vector in print, magnitude of a vector in print, an italic letter will be used: an italic letter will be used: AA

A

Properties of VectorsProperties of Vectors►Equality of Two VectorsEquality of Two Vectors

Two vectors are Two vectors are equalequal if they have if they have the the same magnitudesame magnitude and the and the same same directiondirection

►Movement of vectors in a diagramMovement of vectors in a diagram Any vector can be moved Any vector can be moved parallel to parallel to itselfitself without being affected without being affected

More Properties of More Properties of VectorsVectors

►Negative VectorsNegative Vectors Two vectors are Two vectors are negativenegative if they if they have the same magnitude but are have the same magnitude but are 180° apart (opposite directions)180° apart (opposite directions)

► AA = - = -BB►Resultant VectorResultant Vector

The The resultantresultant vector is the sum of vector is the sum of a given set of vectorsa given set of vectors

Adding VectorsAdding Vectors►When adding vectors, When adding vectors, their their directions must be taken into directions must be taken into accountaccount

►Units must be the sameUnits must be the same ►Graphical MethodsGraphical Methods

Use scale drawingsUse scale drawings►Algebraic MethodsAlgebraic Methods

More convenientMore convenient

Adding Vectors Adding Vectors Graphically (Triangle or Graphically (Triangle or

Polygon Method)Polygon Method)► Choose a scale Choose a scale ► Draw the first vector with the Draw the first vector with the appropriate length and in the direction appropriate length and in the direction specified, with respect to a coordinate specified, with respect to a coordinate systemsystem

► Draw the next vector with the appropriate Draw the next vector with the appropriate length and in the direction specified, length and in the direction specified, with respect to a coordinate system whose with respect to a coordinate system whose origin is the end of vector origin is the end of vector AA and and parallel to the coordinate system used parallel to the coordinate system used for for AA

Graphically Adding Graphically Adding VectorsVectors

► Continue drawing the Continue drawing the vectors vectors ““tip-to-taitip-to-taill””

► The resultant is The resultant is drawn from the drawn from the origin of origin of AA to the to the end of the last end of the last vectorvector

► Measure the length Measure the length of of RR and its angle and its angle Use the scale factor Use the scale factor to convert length to to convert length to actual magnitudeactual magnitude

Graphically Adding Graphically Adding VectorsVectors

► When you have many When you have many vectors, just keep vectors, just keep repeating the repeating the process until all process until all are includedare included

► The resultant is The resultant is still drawn from still drawn from the origin of the the origin of the first vector to first vector to the end of the the end of the last vectorlast vector

Alternative Graphical Alternative Graphical MethodMethod

► When you have only When you have only two vectors, you may two vectors, you may use the use the Parallelogram MethodParallelogram Method

► All vectors, All vectors, including the including the resultant, are drawn resultant, are drawn from a common originfrom a common origin The remaining sides of The remaining sides of the parallelogram are the parallelogram are sketched to determine sketched to determine the diagonal, the diagonal, RR

Notes about Vector Notes about Vector AdditionAddition

► Vectors obey the Vectors obey the Commutative Law Commutative Law of Additionof Addition The order in The order in which the vectors which the vectors are added doesnare added doesn’’t t affect the resultaffect the result

Vector SubtractionVector Subtraction► Special case of Special case of vector additionvector addition

► If If AA – – BB, then , then use use AA+(+(-B-B))

► Continue with Continue with standard vector standard vector addition addition procedureprocedure

Multiplying or Dividing Multiplying or Dividing a Vector by a Scalara Vector by a Scalar

► The The resultresult of the multiplication or division of the multiplication or division is a is a vectorvector

► The The magnitudemagnitude of the vector is multiplied or of the vector is multiplied or divided by the divided by the scalarscalar

► If the scalar is If the scalar is positivepositive, the , the directiondirection of of the result is the the result is the samesame as of the original as of the original vectorvector

► If the scalar is If the scalar is negativenegative, the , the directiondirection of of the result is the result is oppositeopposite that of the original that of the original vectorvector

Components of a VectorComponents of a Vector► A A componentcomponent is a is a partpart

► It is useful to It is useful to use use rectangular rectangular componentscomponents These are the These are the projections of projections of the vector along the vector along the x- and y-axesthe x- and y-axes

Components of a VectorComponents of a Vector►The x-component of a vector is The x-component of a vector is the projection along the x-axisthe projection along the x-axis

►The y-component of a vector is The y-component of a vector is the projection along the y-axisthe projection along the y-axis

►Then, Then,

cosxA A

sinyA A

x yA A A

More About Components of More About Components of a Vectora Vector

► The previous equations are valid The previous equations are valid only if only if θ is measured with respect to the x-axisθ is measured with respect to the x-axis

► The components can be positive or The components can be positive or negative and will have the same units as negative and will have the same units as the original vectorthe original vector

► The components are the legs of the right The components are the legs of the right triangle whose hypotenuse is triangle whose hypotenuse is AA

May still have to find θ with respect to the May still have to find θ with respect to the positive x-axispositive x-axis

x

y12y

2x A

AtanandAAA

Adding Vectors Adding Vectors AlgebraicallyAlgebraically

►Choose a coordinate system and Choose a coordinate system and sketch the vectorssketch the vectors

►Find the x- and y-components of Find the x- and y-components of all the vectorsall the vectors

►Add all the x-componentsAdd all the x-components This gives RThis gives Rxx::

xx vR

Adding Vectors Adding Vectors AlgebraicallyAlgebraically

►Add all the y-componentsAdd all the y-components This gives RThis gives Ryy: :

►Use the Pythagorean Theorem to Use the Pythagorean Theorem to find the magnitude of the find the magnitude of the Resultant:Resultant:

►Use the inverse tangent function Use the inverse tangent function to find the direction of R:to find the direction of R:

yy vR

2y

2x RRR

x

y1

RR

tan

Problem Solving Problem Solving StrategyStrategy

Slide 13

Fig. 1.7, p.14

Known: angle and one sideFind: another sideKey: tangent is defined via two sides!

mmdistheightdistbuildingofheight

3.37)0.46)(0.39(tantan.

,.

tan

Problem Solving StrategyProblem Solving Strategy►Read the problemRead the problem

identify type of problem, principle identify type of problem, principle involvedinvolved

►Draw a diagramDraw a diagram include appropriate values and include appropriate values and coordinate systemcoordinate system

some types of problems require very some types of problems require very specific types of diagramsspecific types of diagrams

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 indicate the unknown (what you are looking for)looking for)

Problem Solving, cont.Problem Solving, cont.►Choose equation(s)Choose equation(s)

based on the principle, choose an based on the principle, choose an equation or set of equations to equation or set of equations to apply to the problemapply to the problem

solve for the unknownsolve for the unknown►Solve the equation(s)Solve the equation(s)

substitute the data into the substitute the data into the equationequation

include unitsinclude units

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 are the units correct for the quantity being found?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?