12 13 h2_measurement_ppt

JJ H1/H2 Physics 2012JJ H1/H2 Physics 2012MeasurementsMeasurements

Expensive error in HistoryExpensive error in History

Entertainment TimeEntertainment Time

Learning OutcomesLearning Outcomes

Recall the following base quantities and their units: mass (kg), length (m), time (s), current (A), temperature (K), amount of substance (mol).

Express derived units as products or quotients of the base units.

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unit

It defines some measurable feature of many different items. It consists of a numerical magnitude and a unit of measure.Area of the school compound, A = 5000 m2

Physical quantity magnitude

Numbers are not physical quantities. Without a unit, numbers cannot be a measure of any physical quantity.

What is a Physical QuantityWhat is a Physical Quantity 1

Types of Physical QuantitiesTypes of Physical Quantities

There are 2 types of physical quantities:

• Base (fundamental) quantities

• Derived quantities

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1.1 What is a Base Quantity1.1 What is a Base Quantity

A base quantity is

chosen and arbitrarily defined rather than being derived from a combination of other physical

quantities.

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7 chosen Base Quantities7 chosen Base QuantitiesBase Quantity Symbol SI unit

length

mass

time

electric current

temperature

amt of substance

luminous intensity*

m

kg

s

A

K

mol

cd

metre

kilogram

second

ampere

kelvin

mole

candela

* - Not in syllabus

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1.2 What is a Derived Quantity1.2 What is a Derived Quantity

A derived quantity is defined based on

combination of base quantities and has a derived unit that is the product and/or quotient of these base units.

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Derived QuantityDerived Quantity

Example

Velocity = Displacement Time

Unit of Velocity = unit of Displacement unit of Time

= m s = m s1

Base quantitiesDerived quantity

Derived unit

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Example Force = Mass x Acceleration

Since F = ma Therefore [ F ] = [ m ] x [ a ]

= kg x ms2

= kg m s2 = N (Newton)

Derived QuantityDerived Quantity 3

ExampleThe unit of Energy is Joule ( J ). Can you try expressing Joule in terms of its base units?

[ E ] = J = kg m2 s-2



Worked Example 1

(Pg 3)

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Worked Example 2

(Pg 4)

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1.3 Homogeneity of equation1.3 Homogeneity of equation

An equation is An equation is homogenoushomogenous/ / dimensionally dimensionally consistentconsistent if: if:

The term has the same unitsThe term has the same units Only quantities of the same units can be Only quantities of the same units can be

added/ subtracted/ equated in an added/ subtracted/ equated in an equation.equation.

3

Homogeneity TestHomogeneity Test

The units of the terms on the right hand The units of the terms on the right hand (RHS) of the equation must be (RHS) of the equation must be equalequal to the to the units of the terms on the LHS.units of the terms on the LHS.

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Beware!!!Beware!!!

The units for the various terms in an The units for the various terms in an equation are the same, it equation are the same, it does not imply does not imply that the equation is physically correct that the equation is physically correct

Why!!!Why!!! Incorrect CoefficientIncorrect Coefficient Missing termsMissing terms Extra termsExtra terms

8

The base unit on the L.H.S. must be equal to the base unit of the terms on the right hand side.


Worked Example 3

(Pg 5)

5

Worked Example 4

(Pg 5)



Worked Example 5

(Pg 5)

5


Worked Example 6

(Pg 5)

e-bt/2m and the index bt/2m are numbers and hence have no unit.

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Show an understanding of and use the conventions for labelling graph axes and table columns as set out in the ASE publication SI units, Signs, Symbols and Abbreviations, except where these have been superseded by Signs, Symbols and Systematics (The ASE Companion to 5-16 Science, 1995).

(to be covered during practical)

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Use the following prefixes and their symbols to indicate decimal sub-multiples or multiples of both base and derived units: pico (p), nano (n), micro (), milli (m), centi (c), deci (d), kilo (k), mega (M), giga (G), tera (T).

Make reasonable estimates of physical quantities included within the syllabus.

3.Prefixes3.Prefixes

Prefixes are used to simplify the writing of very large or very small orders of magnitude of physical quantities.

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Fraction/multipleFraction/multiple PrefixPrefix SymbolSymbol

1010-12-12 picopico pp

1010-9-9 nanonano nn

1010-6-6 micromicro

1010-3-3 millimilli mm

1010-2-2 centicenti cc

1010-1-1 decideci dd

101033 kilokilo kk

101066 megamega MM

101099 gigagiga GG

10101212 teratera TT

Examples:

1500 m = 1.5 x 103 m = 1.5 km

0.00077 V = 0.77 x 10-3 V = 0.77 mV

100 x 10-9 m3 = 100 x (10-3)3 m3 = 100 mm3

PrefixesPrefixes 7

Estimates of physical quantitiesEstimates of physical quantities

The following are examples of estimated values of some physical quantities:

Diameter of an atom ~ 10-10 mDiameter of a nucleus ~ 10-15 mAir pressure ~ 100 kPaWavelength of visible light ~ 500 nmResistance of a domestic lamp ~ 1000

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Worked Example 7

(Pg 7)

PrefixesPrefixes

From today onwards, you must learn to be sensitive to your surrounding.

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Show an understanding of the distinction between systematic errors (including zero errors) and random errors.

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4. Measurements in Physics4. Measurements in Physics

Measuring any physical quantity requires a measuring instrument. The reading will always have an uncertainty.

This arises because

a) experimenter is not skilled enoughb) limitations of instruments

c) environmental fluctuations

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As a result, measurements can become unreliable if we do not use good measurement techniques.

Some common ways to minimize errors are:a) taking average of many readingsb) avoiding parallax errors

c) take readings promptly

Uncertainty in measurementsUncertainty in measurements 8

Analogue & Digital displays

Half the smallest scale division

Often when we measure a quantity with an instrument, we can make an estimate of the uncertainty with the following rule:

Estimating uncertainty Estimating uncertainty 8

5.35Reading =

Uncertainty = 0.05

5

6

Reading =

Uncertainty =

2.28

0.005

Estimating uncertainty Estimating uncertainty

2.2

2.3

8

Even when instruments with digital displays are used, there are still uncertainties in the measurements.

For example, when a digital ammeter shows 358 mA, it does not mean that the current is exactly 358 mA.

Estimating uncertainty Estimating uncertainty 8

5. Errors & Uncertainties5. Errors & Uncertainties

Errors or uncertainties fall generally into 2 categories :

Systematic errors

Random errors

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Random errors are errors without a fixed pattern, resulting in a scattering of readings about the mean value.

5.1 Random errors5.1 Random errors

xx x

x

x

xx

x

x

x

9

The readings are equally likely to be higher or lower than the mean value.

Example: Measuring the diameter of a awire due to its non-uniformity

Random errorsRandom errors

Random errors are of varying sign and magnitude and cannot be eliminated. Averaging repeated readings is the best way to minimize random errors.

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Systematic errors are ones that occurs with a fixed pattern, resulting in a consistent over-estimation or underestimation of the actual value.

5.2 Systematic errors5.2 Systematic errors

xx

x

xx

xx

xx

xx

x

xx

xx

xx

9

The readings are consistently higher or lower than the actual value.

Examples: zero error, wrong calibration, a clock running fast

Systematic errorsSystematic errors

Systematic errors cannot be reduced or eliminated by taking the average of repeated readings. It could be reduced by techniques such as making a mathematical correction or correcting the faulty equipment.

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Show an understanding of the distinction between precision and accuracy.

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Measurements are often described as accurate or precise.

But in Physics, accuracy and precision have different meanings. It is possible to have

5.3 Precision and Accuracy5.3 Precision and Accuracy

precise but inaccurate measurements

accurate but not precise measurements

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No. of readings, n

Value of reading, x

Expected 9.81

Precision and AccuracyPrecision and AccuracySuppose we do an experiment to find g. Expected result is 9.81 ms-2.

10

precise, not accurate

Accurate & precise

accurate but not precise

neither precise nor accurate

Precision and AccuracyPrecision and Accuracy8.63, 8.78, 8.82, 8.59, 8.74, 8.88 9.76, 9.79, 9.83, 9.85, 9.88, 9.90

9.64, 9.81, 9.95, 10.02, 9.77, 9.68 7.65, 8.92, 10.00, 9.12, 8.41, 9.45

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Who is the best shooter??Who is the best shooter??

xxxx

precise, not accurate

xxx

x

accurate & precise

x x

x x

accurate but not precise

x

xx

x

neither precise nor accurate

Mr Low Mr Tan

Mr KwokMr Phang

A set of measurements is precise if

b) there are small random errors in the measurements

a) the measurements have a small spread or scatter

PrecisionPrecision 10

A set of measurements is accurate if

b) there are small systematic errors in the measurements

a) the measurements are close to the actual value

AccuracyAccuracy 10


Assess the uncertainty in a derived quantity by simple addition of actual, fractional or percentage uncertainties

11

If we denote the uncertainty or error as P, then we write the measured quantity as

P ± P

Fractional error of P = P / P

Percentage error of P = P / P 100%

Absolute, Fractional & Percentage Absolute, Fractional & Percentage UncertaintyUncertainty 11

Worked Example 8

(Pg 11)

UncertaintyUncertainty 11

The length of a piece of paper is measured as 297 1 mm. Its width is measured as 209 1 mm.

(a) What is the fractional uncertainty in its length?(b) What is the percentage uncertainty in its length?

Note : 297 + 1 mm

Mean value

Absolute error

Worked Example 8Worked Example 8 11

Percentage uncertainty in its length =

= 0.337 %

1/ 297 100 %

Fractional uncertainty in its length = 1/ 297

= 0.00337

Worked Example 8Worked Example 8 11

Addition and Subtraction

If C = A + B

B A C

Uncertainty in derived quantityUncertainty in derived quantity

B A D If D = A - B

Suppose A and B are measured with uncertainties A and B respectively.

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Multiplication and Division

B

B

A

A

E

E

If E = A B

If F = A/B

A B

F A B

F

Uncertainty in derived quantityUncertainty in derived quantity 11

If A = Bn, then

If A = Bm Cn , then

If A = Bm / Cn , then

Uncertainty in derived quantityUncertainty in derived quantity

A Bn

A B

A B Cm n

A B C

A B Cm n

A B C

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Worked Example 9

(Pg 12)



To find the uncertainty of a quantity, always make it the subject of the given equation before finding its associated uncertainty. Answers should always be rounded off to 3 significant figures except for absolute errors, which are to be rounded up to 1 s.f. The mean value is always rounded off to the same number of decimal places of the absolute error when expressed with in scientific notation.

Worked Example 10

(Pg 12)


Make g the subject of the given equation before finding its associated uncertainty.

Worked Example 11

(Pg 14)


Worked Example 12

(Pg 14)



Distinguish between scalar and vector quantities, and give examples of each. Add and subtract coplanar vectors Represent a vector as two perpendicular components.

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6. Scalars & Vectors6. Scalars & Vectors

A scalar quantityA scalar quantity isis specified by specified by its magnitude aloneits magnitude alone

AA vector quantityvector quantity isis specified by its specified by its magnitude and directionmagnitude and direction

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Examples of Scalars & VectorsExamples of Scalars & Vectors

Some examples:Some examples:

• displacement

Vectors

Scalars

• velocity• acceleration• force• momentum

• distance• speed• time• frequency• density

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Notes for VectorsNotes for VectorsNote:Note: A vector can be placed anywhere as long as A vector can be placed anywhere as long as

it keeps its it keeps its same length and directionsame length and direction..

Two vectors with the Two vectors with the same length but same length but different directionsdifferent directions are different. are different.

Direction for vectors must be given Direction for vectors must be given clearlyclearly without ambiguity.without ambiguity.

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Direction for VectorsDirection for Vectors3 different ways to give directions clearly:3 different ways to give directions clearly:

i) Compass pointse.g. due east, 75o north of west, 20o east of south

iii) X-Y planee.g. positive x-axis, 75o above the negative x-axis, 70o below the positive x-axis

ii) Bearings e.g. bearing of 090o, 345o, 160o

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i) Due East ii ) Bearing of 090

i) 75 north of west ii) Bearing of 345iii) 75 above the -ve x-axis

i) 40 south of eastii) Bearing of 130iii) 40 below the +ve x-axis

Direction for VectorsDirection for Vectors

75

40

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6.1 Additio6.1 Addition of vn of vectorsectors When vectors are added, the When vectors are added, the

result is result is NOTNOT just the sum of the just the sum of the numbers.numbers.

The The directionsdirections of the vectors of the vectors must be considered, especially must be considered, especially when they point in different when they point in different directions.directions.

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AdditioAddition of vn of vectorsectors

Triangle LawTriangle Law

Parallelogram LawParallelogram Law

16

A

BB

A+B

A

B

A

B

A+B

A

B

6.2 Subtraction6.2 Subtraction of v of vectorsectors

A – B = A + (-B)A – B = A + (-B)

16

- B

A

B

A

A - B

During a subtraction, the orientation of the second vector B is reversed before addition is applied

Vector addition/ subtrVector addition/ subtractionaction 16

Scale drawing

Mathematical formula

6.3 Mathematical 6.3 Mathematical requirementsrequirements 16

Worked Example 13

(Pg 17)

Mathematical RequirementMathematical Requirement 17

Adding (Calculating the resultant of vectors)Adding (Calculating the resultant of vectors)

When 2 perpendicular vectors are added, they give When 2 perpendicular vectors are added, they give a resultant as shown:a resultant as shown:

6.4 Resolution of vectors6.4 Resolution of vectors 17

V + H = R

H

V

R

ResolvingResolving

the the reverse processreverse process of vector addition. Instead of of vector addition. Instead of combining 2 vectors into one, a vector can be combining 2 vectors into one, a vector can be spilt spilt into 2 components.into 2 components.

Resolution of vectorsResolution of vectors 17

Rx

Ry

R

Rx = R cos

Ry = R sin

tan = Ry / Rx

6.5 Change in physic6.5 Change in physical quantityal quantity

Change in Physical quantity Change in Physical quantity

= Final Quantity- Initial Quantity= Final Quantity- Initial Quantity

16

Scalar Change

Direction is not important

Involves just the subtraction of magnitudes

Vector Change

Both direction and magnitude is important

Involves subtraction of vectors

Worked Example 15

(Pg 18)

Change in physical quantityChange in physical quantity 17

ENDEND

12 13 h2_measurement_ppt

Education

quantity derived unit

physical quantity unit

base units

derived quantity example

following base quantities

express derived units

base fundamental quantities

types of physical quantities