12 13 h2_measurement_ppt

JJ H1/H2 Physics 2012 Measurements

Learning 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. 1

What is a Physical Quantity 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 m 2 Physical quantity magnitude Numbers are not physical quantities. Without a unit, numbers cannot be a measure of any physical quantity. 1

Types of Physical Quantities There are 2 types of physical quantities: Base (fundamental) quantities Derived quantities 1

1.1 What is a Base Quantity A base quantity is chosen and arbitrarily defined rather than being derived from a combination of other physical quantities. 1

7 chosen Base Quantities Base Quantity Symbol SI unit * - Not in syllabus 1 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

1.2 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. 2

Derived Quantity Example Velocity = Displacement  Time Unit of Velocity = unit of Displacement  unit of Time = m  s = m s  1 3 Base quantities Derived quantity Derived unit

Derived Quantity Example Force = Mass x Acceleration Since F = ma Therefore [ F ] = [ m ] x [ a ] = kg x ms  2 = kg m s  2 = N (Newton) 3

Derived Quantity Example The unit of Energy is Joule ( J ). Can you try expressing Joule in terms of its base units? [ E ] = J = kg m 2 s -2 3

Derived Quantity Worked Example 1 (Pg 3) 3

1.3 Homogeneity of equation An equation is homogenous / dimensionally consistent if: The term has the same units Only quantities of the same units can be added/ subtracted/ equated in an equation. 3

Homogeneity Test The units of the terms on the right hand (RHS) of the equation must be equal to the units of the terms on the LHS. 4

Beware!!! The units for the various terms in an equation are the same, it does not imply that the equation is physically correct Why!!! Incorrect Coefficient Missing terms Extra terms 8

Derived Quantity 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

Derived Quantity Worked Example 6 (Pg 5) e -bt/2m and the index bt/2m are numbers and hence have no unit. 5

Learning Outcomes 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) 6

Learning Outcomes 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.Prefixes Prefixes are used to simplify the writing of very large or very small orders of magnitude of physical quantities. 7

d deci 10 -1 m milli 10 -3 Symbol Prefix Fraction/multiple T tera 10 12 G giga 10 9 M mega 10 6 k kilo 10 3 c centi 10 -2  micro 10 -6 n nano 10 -9 p pico 10 -12

Prefixes Examples: 1500 m = 1.5 x 10 3 m = 1.5 km 0.00077 V = 0.77 x 10 -3 V = 0.77 mV 100 x 10 -9 m 3 = 100 x (10 -3 ) 3 m 3 = 100 mm 3 7

Estimates of physical quantities The following are examples of estimated values of some physical quantities: Diameter of an atom ~ 10 -10 m Diameter of a nucleus ~ 10 -15 m Air pressure ~ 100 kPa Wavelength of visible light ~ 500 nm Resistance of a domestic lamp ~ 1000  7

Prefixes Worked Example 7 (Pg 7) From today onwards, you must learn to be sensitive to your surrounding. 7

Learning Outcomes Show an understanding of the distinction between systematic errors (including zero errors) and random errors. 8

4. 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 enough b) limitations of instruments c) environmental fluctuations 8

Uncertainty in measurements 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 readings b) avoiding parallax errors c) take readings promptly 8

Estimating uncertainty 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: 8

Estimating uncertainty 5.35 Reading = Uncertainty = 0.05 Reading = Uncertainty = 2.28 0.005 8 5 6 2.2 2.3

Estimating uncertainty 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 . 8

5. Errors & Uncertainties Errors or uncertainties fall generally into 2 categories : Systematic errors Random errors 8

5.1 Random errors Random errors are errors without a fixed pattern , resulting in a scattering of readings about the mean value. 9 x x x x x x x x x x

Random errors 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 errors are of varying sign and magnitude and cannot be eliminated . Averaging repeated readings is the best way to minimize random errors. 9

5.2 Systematic errors Systematic errors are ones that occurs with a fixed pattern , resulting in a consistent over-estimation or underestimation of the actual value. 9 x x x x x x x x x x x x x x x x x x

Systematic errors The readings are consistently higher or lower than the actual value. Examples: zero error, wrong calibration, a clock running fast 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. 9

Learning Outcomes Show an understanding of the distinction between precision and accuracy. 10

5.3 Precision and Accuracy Measurements are often described as accurate or precise. But in Physics, accuracy and precision have different meanings . It is possible to have precise but inaccurate measurements accurate but not precise measurements 10

Precision and Accuracy Suppose we do an experiment to find g . Expected result is 9.81 ms -2 . 10 No. of readings, n Value of reading, x Expected 9.81

Precision and Accuracy precise, not accurate Accurate & precise accurate but not precise neither precise nor accurate 8.63, 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 10

Who is the best shooter?? x x x x precise, not accurate x x x x accurate & precise x x x x accurate but not precise x x x x neither precise nor accurate Mr Low Mr Tan Mr Kwok Mr Phang

Precision 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 10

Accuracy 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 10

Learning Outcomes Assess the uncertainty in a derived quantity by simple addition of actual, fractional or percentage uncertainties 11

Absolute, Fractional & Percentage Uncertainty 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% 11

Uncertainty Worked Example 8 (Pg 11) 11

Worked Example 8 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 11

Worked Example 8 11 Percentage uncertainty in its length = = 0.337 % 1/ 297  100 % Fractional uncertainty in its length = 1/ 297 = 0.00337

Uncertainty in derived quantity Addition and Subtraction If C = A + B If D = A - B Suppose A and B are measured with uncertainties  A and  B respectively. 11

Uncertainty in derived quantity Multiplication and Division If E = A  B If F = A/B 11

Uncertainty in derived quantity If A = B n , then If A = B m  C n , then If A = B m / C n , then 11

Uncertainty 13 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.

Uncertainty Worked Example 10 (Pg 12) 13 Make g the subject of the given equation before finding its associated uncertainty.

Learning Outcomes  Distinguish between scalar and vector quantities, and give examples of each.  Add and subtract coplanar vectors  Represent a vector as two perpendicular components. 15

6. Scalars & Vectors A scalar quantity is specified by its magnitude alone A vector quantity is specified by its magnitude and direction 15

Examples of Scalars & Vectors Some examples: displacement Vectors Scalars velocity acceleration force momentum distance speed time frequency density 15

Notes for Vectors Note: A vector can be placed anywhere as long as it keeps its same length and direction . Two vectors with the same length but different directions are different. Direction for vectors must be given clearly without ambiguity. 15

Direction for Vectors 3 different ways to give directions clearly: i) Compass points e.g. due east, 75 o north of west, 20 o east of south iii) X-Y plane e.g. positive x-axis, 75 o above the negative x-axis, 70 o below the positive x-axis ii) Bearings e.g. bearing of 090 o , 345 o , 160 o 15

Direction for Vectors i) Due East ii ) Bearing of 090  i) 75  north of west ii) Bearing of 345  iii) 75  above the -ve x-axis i) 40  south of east ii) Bearing of 130  iii) 40  below the +ve x-axis 15 75  40 

6.1 Additio n of v ectors When vectors are added, the result is NOT just the sum of the numbers. The directions of the vectors must be considered, especially when they point in different directions. 16

Additio n of v ectors Triangle Law Parallelogram Law 16 A B B A+B A B A B A+B A B

6.2 Subtraction of v ectors 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/ subtr action 16 Scale drawing Mathematical formula

6.3 Mathematical requirements 16

Mathematical Requirement Worked Example 13 (Pg 17) 17

Adding (Calculating the resultant of vectors) When 2 perpendicular vectors are added, they give a resultant as shown: 6.4 Resolution of vectors 17 V + H = R H V R

Resolving the reverse process of vector addition. Instead of combining 2 vectors into one, a vector can be spilt into 2 components. Resolution of vectors 17 R x R y R R x = R cos  R y = R sin  tan  = Ry / Rx

6.5 Change in physic al quantity Change in Physical 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

Change in physical quantity Worked Example 15 (Pg 18) 17

12 13 h2_measurement_ppt

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