FAP0015 Ch01 Measurement

Chapter 1: Measurement  

Quantities



Units, Standards & SI System

 Prefixes



 Dimensions & Dimensional Analysis Analysis



 Errors & Accuracy

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 Lesson Outcomes At the end of the lesson, students should be able to: 1. state the meaning & give examples of  physi  physical cal quanti quantitie tiess 2.

dist distin ingu guis ish h between base ase quan quanti titi ties es & derived derived quanti quantitie tiess

3.

state what the standards are measured against

4.

write values in prefix or  standard forms

5.

apply dimensio dimensional nal analys analysis is to solve equations

6.

state the different type ypes of errors ors

7.

define the terms µ precision¶ and µaccuracy¶

8.

measurement.. dete de term rmiine the uncertainty or  error of a measurement

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 Lesson Outcomes At the end of the lesson, students should be able to: 1. state the meaning & give examples of  physi  physical cal quanti quantitie tiess 2.

dist distin ingu guis ish h between base ase quan quanti titi ties es & derived derived quanti quantitie tiess

3.

state what the standards are measured against

4.

write values in prefix or  standard forms

5.

apply dimensio dimensional nal analys analysis is to solve equations

6.

state the different type ypes of errors ors

7.

define the terms µ precision¶ and µaccuracy¶

8.

measurement.. dete de term rmiine the uncertainty or  error of a measurement

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 Physical Quantities is a quantitative science based on measurement. Physics

A physical quantity is quantity with a numerical value and units. Physical quantities are assigned to measurements taken.

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 Base and Derived Quantities There are so many physical quantities and they can be categorised as base and derived  quantities.  Base quantities are the ones that you can measure directly by using suita ble instruments.

 Mass, length and time are examples of  base quantities.

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The International System of Units or  SI (Sy stème  International ), is a name adopted by the Eleventh General Conference on Weights and Measures , held in Paris in 1960, for a universal , unified , selfconsistent system of measurement units based on the mks (meter-kilogram-second) system *.

*

Microsoft® Encarta® Encyclopedia 2003. © 1993-2002 Microsoft Corporation.

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T able 1.  Base Quantities QUANTITY

NAME OF BASE SI UNIT

SYMBOL

Length

meter

m

Mass

kilogram

kg

Time

second

s

Electric current

ampere

A

Temperature

Kelvin

K

Amount of substance Luminous intensity

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mole

mol

candela

cd

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 Any physical quantity will comprise of certain base quantities. If you com bine two or  more base quantities accordingly, you will get a derived quantity. For example, if you com bine length and time accordingly, you might find the speed , which is a derived quantity. Other  derived quantities include area, acceleration, density, energy and power . FAP0015 PHYSICS I

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T able 2 : Derived Quantities QUANTITY

NAME OF DERIVED Sl UNIT SYMBOL 2

Area

square metre

m

Volume

cubic metre

m

Velocity Acceleration

metres per second

3

m/s

metres per second squared

m/s

2 3

Density

kilograms per cubic metres

kg/m

Current density

amperes per square metre

A/m

Magnetic field strength

amperes per metre

A/m

Specific volume

cubic metres per kilogram

m  /kg

Luminance

candelas per square metre

cd/m

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3

2

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V ector and Scalar Quantities A physical quantity is categorized either as a vector or  scalar quantity. A scalar quantity is a quantity with magnitude only. Examples are distance, mass and energy. A vector quantity is a quantity with both magnitude and direction. Examples include displacement , velocity and force.

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V ector and Scalar Quantities

 Arrows are used to represent vectors. The direction of the arrow gives the direction of  the vector. By

convention, the length of a vector  arrow is proportional to the magnitude of the vector. 8 lb 4 lb

 Standards Every unit used as measurement of a certain quantity has a standard which is accepted by international  agreement . For example, the standard of length... 1 meter = 1650763.73 times the wavelength of  light emitted by krypton-86 (1960). 1 meter = path travelled by light in vacuum in 1/299792458 second (1983). FAP0015 PHYSICS I

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Similarly, the standard of mass and time... 1 kilogram = mass of  1 cu bic decimetre of  pure

water at the temperature of its maximum density (4.0° C/39.2° F) 1 second = 1/86,400 of a mean solar day or one complete rotation of the Earth on its axis in relation to the Sun. (redefined in 1967 in terms of the resonant frequency of the caesium atom, that is, the frequency at which this atom a bsor  bs energy).

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 Prefix  When dealing with very large or  very small  quantities, a prefix to the unit name is used that has the effect of multiplying the unit by some power of ten. An example is the millisecond (103 s)

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T able 3 : Prefixes

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 Dimension and Dimension Analysis Every quantity has a dimension expressed in terms of the basic units.

The sym bols for the dimensions of the basic units mass, length and time are M, L and T respectively. The dimension of any derived quantity can be expressed in terms of  M, L and T.

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Quantity

Symbol

Units

Dimension

Mass

m

kg

M

Distance

x

m

L

Time

t 

s

T

Velocity

v

m s-1

LT-1

Momentum

p

kg ms-1

MLT-1

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1.3 The Role of Units in Problem Solving 

Is the following equation dimensionally correct?

 x ! vt  «L» ?LA! ¬ ¼?TA! ?LA T½

Examples: 1-A table is 41.5 inches wide, express this in centimeters and also feet?

2-Convert 1342 meter to feet?

3-A silicon chip has an area of 8.42 square inches. Express this in square centimeters?

4-What is the speed 100 (mi/h) a)in meter per second(m/s) b) in kilometers per hour(km/h)

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Dimensional analysis: One useful technique is the use of diminutions to check if a relation is incorrect. Example: By using dimensional analysis check below¶s equation is correct or incorrect. Dimension of speed: [L/T] Dimension of acceleration: [L/T 2]

Dimension of left side must be equaled Dimension of right side Dimension of left side=[L/T]

Dimension of right side=[L/T] + [L/T2] [ T2]= [L/T] + [L]

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Example: Three students derive the following equations in which x refer to distance traveled, v the speed and a acceleration, and t refer to time. which of these could possibly be correct according to the dimensional check.

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 Example: Use dimensional analysis to determine how the  period, T   , of a simple pendulum depends on the length & mass of pendulum, and gravity ( l, m, and/or g)

T  = k l w m  x g  z

[ T  ] = [ k l w m  x g  z ] = Lw M  x (L/T2)  z T = Lw+z M x T

2 z

 

w + z = 0,  x = 0,

2 z = 1

 

 z =   ½ , w = ½ ,  x = 0

T  = k l ½ g 

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½

T  ! k 

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 Errors The usual meaning of the word error is mistake. However the term error is used in experimental   physics to descri be the quantity by which result  obtained  by o bservation differs from an accurate determination (µactual value¶).

 Error is also called uncertainty.

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So in every practical measurement there is some degree of  error or  uncertainty. In assessing errors, whether human or  instrumental, there are two types of  error:- random and systematic errors.

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 Random errors R andom error results from unknown and unpredictable variations in experimental situations. R andom errors can be also referred to as accidental errors and are at times beyond the control of the o bserver.

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 Random errors will cause the measured value to  be sometimes higher or  lower than the actual value.

Taking a large number of readings and then  finding the mean value can reduce the effect of  random errors. Source of random errors can be mechanical  vi brations of the experimental setup or  unpredictable fluctuations in temperature, etc.

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 Systematic errors A reading consistently shifted in one direction

is called a systematic error. Systematic errors are usually associated with  particular measurement instruments or  techniques such as an improperly cali brated instrument.

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Other examples include zero error and  parallax error .

 H uman reaction time can also be classified under this category.

Systematic errors are more serious form of error  since they cannot be reduced b y taking  repeated readings or  by any other form of  averaging.

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 Accuracy and Precision The accuracy of a measurement signifies how close it comes to the true value.  Precision refers to the agreement among repeated measurements, the measure of  how close together they are.

The more precise the measurements, the closer  together they are.

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Good precision but poor accuracy

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Poor precision

and poor accuracy. (average reading has good accuracy)

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Good precision and good accuracy

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Example: What, roughly, is the percent uncertainty in the volume of a spherical beach ball whose radius is r = 2.86 ± 0.09m? 

 Answer: To find the approximate uncertainty in the volume, calculate the volume for the specified radius, the minimum radius, and the maximum radius. Subtract the extreme volumes. The uncertainty in the volume is then half this variation in volume. Example: Express the following sum with the correct number of significant figures: 1.80 m+142.5 cm+ 5.34×105 m..

 Answer: To add values with significant figures, adjust all values so that their units are all the same.

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If  both types of error are small, then the measurement is accurate and precise. One point to note is that the degree of accuracy or  uncertainty of a measurement largely depend on the quality of the instrument and the skills of the  person carrying out the experiment.

The degree of accuracy or  uncertainty of a measurement can usually be indicated by the number of significant figures used.

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Examples: A visitor to a Museum was admiring a Tyrannosaurus  fossil, and asked a nearb y museum em plo yee how old it was.   at skeleton's sixty-five million and three years, two " Th  ,"  the em plo yee re plied . months and eighteen days old 

"H ow can you know it that well?"   she asked . " Well, when I started working here, I asked a scientist the e xact same question, and he said it was sixty-five million years old  ± and that was three years, two months and   eighteen days ago."  In the above example, the humor is that the employee fails to understand the scientist's implication of the uncertainty in the age of the fossil. FAP0015 PHYSICS I

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