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Chapter 1 Physical Quantities and Units

 

Introduction

What is physic ?

• Definition of physics – derives from Greek word means nature.

• Each theory in physics involves:

(a) Concept of physical quantities.

(b) Assumption(andaian) to obtain mathematical model.

(c) Relationship between physical concepts.

- proportional (berkadar langsung)

(d) Procedures to relate mathematical models to actual measurements from experiments.

(e) Experimental proofs to devise explanation to nature phenomena.

1.1 Basic Quantities and International System of Units (SI units)

> Physical quantity

A physical quantity is a quantity that can be measured. Physical quantity consist of a

numerical magnitude and a unit.

Example:

250 ml (magnitude and unit)

> Basic quantity

Quantity that cannot be derived (terbit) from other quantities. This quantity is important because it

- can be easily produced

- does not change its magnitude

- is internationally accepted

> SI units

The unit of a physical quantity is the standard size used to compare different magnitudes

of the same physical quantity.

> Systems of units

Several systems of units have been in use. Example:

- The MKS (meter-kilogram-second) system

- The cgs (centimeter-gram-second) system

- British engineering system: foot for length, pound for mass and second for time.

Today the most important system of unit is the Systems International or Sl units.

Basic Quantity and the SI Base Units

• Physical quantities can be divided into two categories:

1. basic quantities and

2. derived quantities.

The corresponding units for these quantities are called base units and derived units.

Basic Quantities

• In the interest of simplicity, seven basics quantities1, consistent with a full description of the

physical world, have been chosen.

Basic quantity

Symbol

Dimension

(base quantity symbol)

Definition2

SI units

Length

l

L

length most commonly refers to the longest dimension of an object

Meter

Mass

m

M

Mass , more specifically inertial mass, can be defined as a quantitative measure of an object’s resistance to acceleration

Kilogram

Time

t

T

Time is a dimension in which events can be ordered from the past through the present into the future, and also the measure of durations of events and the intervals between them

Second

Electric

current

I

A

Electric current is a flow of electric charge through a conductive medium

Ampere

Thermodynamic

temperature

T

q

Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold.

Kelvin

Amount of substances, Quantity of

matter

n

N

Amount of substance is a standards-defined quantity that measures the size of an ensemble of elementary entities, such as atoms, molecules, electrons, and other particles

Mole

Luminous intensity

Iv

J

luminous intensity is a measure of the wavelength-weighted power emitted by a light source in a particular direction per unit solid angle

Candela

Base Units

There are only seven base unit3 in SI system.

SI Base units

Symbol

Definition1

Metre

m

"The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second."
17th CGPM (1983, Resolution 1, CR, 97)

Kilogram

kg

"The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram."
3rd CGPM (1901, CR, 70)

Second

s

"The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom."
13th CGPM (1967/68, Resolution 1; CR, 103)
"This definition refers to a caesium atom at rest at a temperature of 0 K."
(Added by CIPM in 1997)

Ampere

A

"The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 × 10−7 newton per metre of length."
9th CGPM (1948)

Kelvin

K

"The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water."
13th CGPM (1967/68, Resolution 4; CR, 104)
"This definition refers to water having the isotopic composition defined exactly by the following amount of substance ratios: 0.000 155 76 mole of 2H per mole of 1H, 0.000 379 9 mole of 17O per mole of 16O, and 0.002 005 2 mole of 18O per mole of 16O."
(Added by CIPM in 2005)

Mole

Mol

"1. The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12; its symbol is ‘mol.’

2. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles."
14th CGPM (1971, Resolution 3; CR, 78)
"In this definition, it is understood that unbound atoms of carbon 12, at rest and in their ground state, are referred to."
(Added by CIPM in 1980)

Candela

cd

"The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian."
16th CGPM (1979, Resolution 3; CR, 100)

Prefixes(Imbuhan)

•For very large or very small numbers, we can use standard prefixes with the base units.

Prefix

tera

giga

mega

kilo

deci

centi

mili

micro

nano

pico

Factor

102

109

106

103

10-1

10-2

10-3

10-6

10-9

10-12

Symbol

T

G

M

k

d

c

m

µ

n

P

Example:

2 x 10-7 —- 2 x 10-6 x 10-1 —- 2m x 10-1 – 0.2m


Derived quantities and derived units

•Derived quantity

Quantity that derived from basic quantities through multiplication and division.

•For example,

Derived quantity

Derive from base quntity of

Derived unit

Area

length x length

m2

Volume

length x length x length

m3

Density

clip_image002

kg m-3

Velocity

clip_image004

m s-1

Acceleration

clip_image006

m s-2

Frequency

clip_image008

s-1/hz

Momentum

Mass x velocity

Kg ms-1

Force

Mass x acceleration

Kg ms-2

Pressure

clip_image010

N m-2

Energy

clip_image012

Kg m2 s-2

Ø The derived unit change

Example :

7854 kg m-3 change into g cm-3

clip_image014 —- clip_image016 —– clip_image018– 7.854 g cm-3

1.2 Dimensions(Dimensi) and Physical Quantities

> The dimension of a physical quantities is the relation between the physical quantity and the base quantities.

‘[ ]’ The dimension of (pronounce its loudly) or the base quantity of

Example [v] “the dimension of velocity” , this means that the base quantities in the velocity.

Example 1

Write the dimensions for the following physical quantity

(a) Acceleration(pecutan),a

(b) Density

(c) Force

Use of dimensions

•To check the homogeneity of physical equations

Concept of homogeneous

•The dimensions on both sides of an equation are the same.

•Those equations which are not homogeneous are definitely wrong.

•However, the homogeneous equation could be wrong due to the incomplete or has extra terms.

•The validity of a physical equation can only be confirmed experimentally.

•In experiment, graphs have to be drawn then. A straight line graph shows the correct equation and the non linear graph is not the correct equation.

•Deriving a physical equation

•An equation can be derived to relate a physical quantity to the variables that the quantity depends on.

Example 2

Determine the homogeneous of the equation.

v2 =u2 +2as

Example 3

(a)Diberi persamaan yang berikut tentang aliran cecair dalam sebatang paip mendatar

(a) clip_image020

(b) clip_image022

(c) clip_image024

Dimana W,X,Y berdimensi sama dengan dimensi tekanan, A,B,C mewakili pemalar tak berdimensi

g mewakili pecutan graviti

T mewakili tegangan permukaan cecair (dimensinya MT-2)

r mewakili ketumpatan cecair

v mewakili halaju cecair

p mewakili perubahan tekanan

Tentukan kehomogenan persamaan di atas.

(b) Dibawah ialah bacaan-bacaan bagi p dan v :

p (Nm-2)

2.0 ´ 103

1.5 ´ 103

1.2 ´ 103

0.7 ´ 103

0.3 ´ 103

v (m s-2)

1.0

1.4

1.6

1.9

2.1

Dengan menggunakan bacaan-bacaan ini,

(i) cari persamaan yang betul

(ii) cari pemalar bagi persamaan yang betul itu dengan menggunakan maklumat berkenaan

[r = 1.0 ´ 103 kg m-3, T = 7.4 ´ 10-2 N m-1]

Example 4

Use the dimension analysis to obtain an expression which shows how the momentum p depends on the force; F, mass; m and the length, l.

1.3 Scalar and Vectors

> A scalar quantity is a physical quantity which has only magnitude. For example, mass, speed (laju), density, pressure, ….

> A vector quantity is a physical quantity which has magnitude and direction. For example, force, momentum, velocity (halaju), acceleration ….

In most cases in physic, the physic quantity is express in vector. If the number(magnitude) can be operated through Subtract, Add, multiplication and fraction. Then the vector also can be threat the same way except fraction, but it’s have to follow the rule that govern them.

Graphical representation of vectors

•A vector can be represented by a straight arrow,

clip_image025

The length of the arrow represents the magnitude of the vector.

The vector points in the direction of the arrow.

Basic principle of vectors

• Two vectors P and Q are equal if:

a) Magnitude of P = magnitude of Q

(b) Direction of P = direction of Q

• When a vector P is multiplied by a scalar k, the product is k P and the direction remains the same as P.

The vector -P has same magnitude with P but comes in the opposite direction.

Principles of vectors

(a) Substitute of Vector (Relative of)

Relative velocity

Let us look at two cases: VA = 10 ms-1 (faster) VB = 3 ms-1. (slower)

Case one

The velocity of A relative to B = (VA – VB) (comparing faster toward slower)

= (10- 3) ms­

= 7 ms -1 (in forward direction).(mean that A is 7 ms -1 faster than B)

Case two

The velocity of B relative to A = (VB – VA)

= (3 – 10) ms­

= -7 ms -1 (in backwards direction).

We observe that(VB – VA) and (VA – VB) are same magnitude but different direction.

(b) Sum of vectors (Resultant of)

If there are two or more vector , these vector can be add to form a single vector called a Resultant vector.

To solve the problem involving vectors in two dimension, we usually used any one of these method depend on the information given.

Method 1: Parallelogram of vectors

It’s the drawing method. The drawing of the parallelogram need to be draw according scale and angle given in the question. The instrument used for this drawing are:

(a) ruler

(b) protractor

(c) sharp pencil

It two vectors clip_image027 and clip_image029 are represented in magnitude and direction by the adjacent sides OA and OB of a parallelogram OABC, then OC represents their resultant(paduan).

clip_image031

This method is used when there are information about angle and magnitudes of the vector.

Method 2: Triangle of vectors and polygon of vector

It’s the drawing method. The drawing of the vectors need to be draw according scale and angle given in the question. The instrument used for this drawing are:

(d) ruler

(e) protractor

(f) sharp pencil

•Use a suitable scale to draw the first vector.

•From the end of first vector, draw a line to represent the second vector. (attaching the head with the it’s tail)

•Complete the triangle/polygon. The line from the beginning of the first vector to the end of the second vector represents the sum in magnitude and direction.

clip_image033

Example 5

A kite flies in still air is 4.0 ms-1. Find the magnitude and direction of the resultant velocity of the kite when the air flows across perpendicularly(serenjang) is 2.5 ms-1. If the distance of the kite is 30 m, what is the time taken for the kite to fly? Calculate the height of the kite from the ground.

Example 6

clip_image035

Figure 1.8.1

Five coplanar forces act on a particle, as drown in Figure 1.8.1. Draw a scaled force polygon for these forces. State the magnitude and direction of the resultant of these forces.

Method 3 : Component Method

It’s is a calculation method , because every vector can be replace into x-component and y-component. Replacing a single vector into it’s components is called Resolving.

To determine the resultant of the vector using this method, it’s need to follow these four keyword carefully.

1. Axis

2. Resolve vector

3. add vector component

4. Resultant

Axis

Need to be determine before resolving the vector.

•Resolving(leraian) vector

The vector that is not on any axis have to be resolve into it’s component. Resolving vector mean resolving :

(a) magnitude

(b) Direction

A vector R can be considered as the two vectors. R refers to the resultant vectors. There are two mutually perpendicular component Rx and Ry

clip_image036

Add Vector Component

clip_image038 and clip_image040

Only the same axis component can be added.

Resultant

Magnitude, clip_image042 and Direction of R, clip_image044

Example 7

The figure shows 3 forces F1, F2 and F3 acting on a point O. Calculate the resultant force and the direction of resultant.

clip_image046

(d) Multiplication of vector

It’s have been discuss about subtraction and addition of the vector. From subtraction and addition of vector we can explain most of the physical quantity. Now is about multiplication of vectors. When two vectors were multiply the result is called product.

There are two kind of product produced :

1. Dot Product

2. Cross Product

Dot Product

The dot product is fundamentally a projection.

clip_image048

The dot product of a vector with a unit vector is the projection of that vector in the direction given by the unit vector. This leads to the geometric formula

clip_image050

Furthermore,it follows immediately from the geometric definition that two vectors are orthogonal if and only if their dot product vanishes, that is

clip_image052

Cross Product

The cross product is fundamentally a directed area.

clip_image054

whose magnitude is defined to be the area of the parallelogram?. The direction of the cross product is given by the right-hand rule, so that in the example shown clip_image056 points into the page.

In mathematics and physics, the right-hand rule is a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 19th century.

clip_image058clip_image060

Unit vector

When comes into multiplying vector it’s easier to used component method. The basis for the coordinate system used in vector notation is unit vector.

in mathematics, a unit vector in a normed vector space is a vector whose length is 1 (the unit length). A unit vector is often denoted by a lowercase letter with a "hat", like this: clip_image061(pronounced "i-hat"),clip_image063and clip_image065. The clip_image061[1],clip_image063[1]and clip_image065[1]is use in 3D or cartesian coordinate and : clip_image061[2] andclip_image063[2] is use in Euclidean space.

The operation on the vector will be much more faster compared to the drawing method.

A vector can be represent in component method as clip_image069 meaning that a vector A is stretch from origin to point (2,3) in Euclidean space.

clip_image073

1.4 Metrology

Metrology is the science of measurement and its application.

Terminology related to measurement uncertainty is not used consistently among experts. To avoid further confusions lets refer to BIPM-VIM(International Vocabulary of Basic and General Terms in Metrology) and GUM (Guide to the expression of uncertainty in measurement).

1.4.1 Error

VIM define the error as below:

error (of measurement) [VIM 3, 2.16] – measured quantity value minus a reference quantity value

there are two type of error

(a) Systematic Error

Characteristics of systematic error in the measurement of a particular physical quantity:

-Its magnitude is constant.

-It causes the measured value to be always greater or always less than the true value.

Corrected reading = direct reading – systematic Error

Sources of systematic Error:

- Zero Error of instrument.

- Incorrectly calibrated scale of instrument.

- Personal error of observer, for example reaction time of observer.

- Error due to certain assumption of physical conditions of surrounding for example, g = 9.81 ms-2

Systematic error cannot be reduced or eliminated by taking repeated readings using the same method, instrument and by the same observer.

(b) Random Error

Characteristics of Random Error :

- It’s magnitude is not constant.

- It causes the measured value to be sometimes greater and sometimes less than the true value.

Corrected reading = direct reading ± Random Error

The main source of random Uncertainty is the observer.

The surroundings and the instruments used are also sources of random error.

Example of random Error:

- Parallax Error due to incorrect position of the eye when taking reading

Parallax Error can be reduced by having the line of sight perpendicular to the scale reading.

- Error due to the inability to read an instrument beyond some fraction of the smallest division

Reading are recorded to a precision of half the smallest division of the scale.

Random Error can be reduced by taking several readings and calculating the mean.

Error contributes to but is different from Uncertainty

1.4.2 The Uncertainty of the Instrumental

VIM define the Uncertainty as below

uncertainty of measurement [VIM 3, 2.6] non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand (quantity intend to measure), based on the information used and it’s have a statistical concept of standard deviation means.

Instrumental Measurement

When handling the experiment the reading is given by the apparatus used, these apparatus have their own uncertainty.

instrumental measurement uncertainty(VIM 3, 4.24) – the amount (often stated in the form ±dx) that along with the measured value, indicates the range in which the desired or true value most likely lies. Instrumental measurement uncertainty is used in a Type B evaluation of measurement uncertainty

Here the magnitude of ±dx is called the absolute Uncertainty. Absolute Uncertainty is the smallest scale of the instrument or half of the smallest scale if it’s can be determine “easily”.

Instruments

Absolute Uncertainty

Example of readings

Millimeter ruler

0.1 cm

(50.1 ± 0.1)cm

Vernier caliper

0.01 cm

(3.23 ± 0.01)cm

Micrometer screw gauge

0.01 mm

(2.63 ± 0.01)mm

Stopwatch (analogue)

0.1 s

(1.4 ± 0. 1 )s

Stopwatch(Digital)

0.01 s

(1.452 ± 0.01)s

Thermometer

0.5 °C

(28.0 ± 0.5)°C

Ammeter (0 – 3A)

0.05 A

(1.70 ± 0.05)A

Voltmeter (0 – 5V)

0.05 V

(0.65 ± 0.05)V

The smaller absolute uncertainty of the instrument is contribute to the high accuracy, precision and sensitivity of the measuring system of the experiment.

1.4.2 Analysing Uncertainty of the data

- specifically Uncertainty analysing is refer to Uncertainty that cause by repetition measurement to produce more accurate data.

- Meaning that if we want to measure a mass of cube, of course we cannot just used a single measurement then we will get the answer. We have to measure the mass with the triple balance beam more than one time for example 3 time.

- While doing the measurement actually we have continually increasing the Uncertainty.

- It is a good idea to mention the Uncertainty for every measurement and calculation.

- In this subtopic we deal with the repetition reading or data. It’s known that if we have more than one reading so the true value is the mean of the reading.

- Mean value for a is clip_image075

- Mean value of Uncertainty of a, clip_image077should be caculated this way

1. Calculated the deviation of every data given:

clip_image079

clip_image081

clip_image083

clip_image083[1]

clip_image083[2]

clip_image087

2. Find the sum of deviation

clip_image089

3. find the mean of deviation clip_image091

It’s known that the mean deviataion is equally the same as the Uncertainty of the mean value(true value).

Or

clip_image093

Working example on a single quantity :

1. Aim : to determine the diameter, d of a wire

2. Theory : used outer jaw of vernier caliper

3. Precaution : measure more than one reading

4. Choosing Apparatus and Determine the absolute uncertainty:

Instruments

Uncertainty

(Absolute/actual)

Vernier caliper

0.01 cm

5. Manage the reading/data:

Diameter ,d of a wire was measured several time to reduce the Uncertainty and the reading is given in the table below. Find the true value(mean value) and the Uncertainty of the diameter.

 

i

ii

iii

iv

v

vi

(d±0.01)/cm

1.55

1.52

1.54

1.53

1.54

1.53

6. Determine the quantity and it’s uncertainty

a. Calculating the true value of diameter (mean value) <d>:

clip_image095

b. Calculating the uncertainty of diameter:

clip_image097

So the diameter of a wire should be written (1.54 ± 0.01)cm

Note: calculating the uncertainty this way is refer to a single quantity and not involving with the graph.

Primary data and secondary data

• Primary data are raw data or readings taken in an experiment. Primary data obtained using the same instrument have to be recorded to the same degree of precision i.e to the same number of decimal places.

• Secondary data are derived from primary data. Secondary data have to be recorded to the correct number of significant figures. The number of significant figures for secondary data may be the same (or one more than) the least number of significant figures in the primary data. Measurement play a crucial role in physics, but can never be perfectly precise.

It is important to specify the Uncertainty or Uncertainty of a measurement either by stating it directly using the ± notation, and / or by keeping only correct number of significant figures.

Example: 51.2 ± 0.1

Processing significant figures

• Addition and subtraction

When two or more measured values are added or subtracted, the final calculated value must have the same number of decimal places as that measured value which has the least number , of decimal places.

Example

1. a = 1.35 cm + 1.325 cm

= 2.675 cm

= 2.68 cm

2. b = 3.2 cm – 0.3545 cm

= 2.8465 cm

= 2.8 cm

3. c = clip_image099

= 1.142 cm

= 1.14 cm

· Multiplication and division

• When two or more measured values are multiplied and/or divided, the final calculated value must have as many significant figures as that measured value which has the least number of significant figures.

Example

1. Volume of a wooden block = 9.5 cm x 2.36 cm x 0.515 cm

= 11.5463 cm3

= 12 cm3

2. If the time for 50 oscillations of a simple pendulum is 43.7 s, then the period of oscillation = 43.7 ÷ 50 = 0.874 s

3. The gradient of a graph clip_image101

Note: Sometimes the final answer may be obtained only after performing several intermediate calculations. In this case, results produced in intermediate calculations need not be rounded off. Round only the final answer.

1.4.3 Analysing Uncertainty of combining measurement or equation.

1. Actual Value

- is in the scale reading (pointer reading) of an instrument.(single reading)

Or

- is in the mean value.(of the repetition reading)

2. Fractional and percentage Uncertainty,

(a) The fractional Uncertainty of R : clip_image103

(b) The percentage Uncertainty of R : clip_image105

3. Consequential Uncertianties/Uncertainty- to state the Uncertainty of a derive quantities

Given

R 1 ± DR1 = Data ± Absolute Data Uncertainty = 51.2 ± 0.1

R 2 ± DR2 = Data ± Absolute Data Uncertainty = 30.1 ± 0.1

(a) Addition

W = R1 + R2 = 51.2 + 30.3 = 81.3

DW = DR1 + DR2 = 0.1 + 0.1 = 0.2

So W ± DW = 81.3 ± 0.2

(b) Subtraction

S = R1 – R2 = 51.2 – 30.3 = 21.1

DS = DR1 + DR2 = 0.1 + 0.1 = 0.2

So S ± DS = 21.1 ± 0.2

(c) Product

P = R1 ´ R2 = 51.2 ´ 30.3 =1541.12

From clip_image107

clip_image109

clip_image111

clip_image113

P ± DP = 1541.12 ± 7.71

(d) Quotient

clip_image115

From clip_image117

clip_image119

clip_image121

clip_image123

Q ± DQ = 1.70 ± 0.01

Working example:

1. Aim : to determine the value of B

2. Theory :

B is given by

clip_image125

3. Precaution : B have a combine uncertainty from various apparatus (quantity)

4. Choosing Apparatus and Determine the absolute uncertainty:


Quantity

Instruments

Uncertainty

(Absolute/actual)

a,b

meter ruler

1 cm

q

Stopwatch(Digital)

0.01 s

5. Manage the reading/data:

After the measuring and calculating the uncertainty of the quantity a,b,d,q and T(refer 1.4.2). The true value (mean value) and the uncertainty of the quantities are witten as below :

a =(1.83±0.01)m,

b=(1.65 ±0.01) m,

d=(0.00106±0.00003)m,

q = (4.28 ± 0.05) s

T = (3.7 ± 0.1) x 103 s.T is

6. Determine the quantity and it’s uncertainty

(a) Find B use the equation given

clip_image125[1]

clip_image128

B = 7.8 x 10-11 m3 s-

(b) Find the uncertainty of B

1. Fisrt check the equation for addition and subtraction, by applying 1.4.3 no 3 (b) , subtraction so (a – b) = (0.18±0.02)m

2. Second calculate the percentage uncertainties in each of the 4 terms:

Term

Magnitude and uncertainty

Fractional Uncertainty

Uncertainty percentage

(a – b)

= (0.18±0.02)m

clip_image130

11%

d

= (0.001 06 ± 0.000 03) m

clip_image132

3%

q

= (4.28 ± 0.05) s

clip_image134

1.2%

T

= (3.7±0.1) x 103 s

clip_image136

3%

- The Uncertainty in (a – b) is now very large, although the readings them­selves have been taken carefully. This is always the effect when subtracting two nearly equal numbers.

- The percentage Uncertainty in d2 will be twice the percentage Uncertainty in d;

- The percentage Uncertainty in clip_image138 will be half the percentage Uncertainty in T because a square root is a power of clip_image140.

This gives:

Uncertainty percentage in B = 11% + 2(3%) + 1.2% + clip_image142(3%) = 19.7% ≈ 20%

This gives B = (7.8 ± 1.6) x 10-11 m3 s-1.

the rules for uncertainties therefore :

Operator

Uncertainty

addition and subtraction

ADD absolute uncertainties

multiplication and division

ADD percentage uncertainties

powers

Multiply the percentage Uncertainty by the power

Example 8

The diameter of a cone is (98 ± 1)mm and the height is (224 ± 1 )mm. What is:

(a) The absolute Uncertainty of the diameter.

(b) The percentage Uncertainty of the diameter.

(c) state the volume of the cone and it’s uncertainty. Give your answer to the correct number of significant number.

Example 9

Discuss the ways of minimizing systematic and random Error

Example 10

The period of a spring is determined by measuring the time for 10 oscillations using a stopwatch. State a source of:

(a) Systematic Error

(b) Random Error

1.4.4. Method to find Uncertainty/Uncertainty from a graph

clip_image144

Figure 1

where n is the number of points plotted.

1. The usual quantities that are deduced from a straight line graph are

(a) the gradient of the graph m, and the intercept on the y-axis or the x-axis

(b) the intercepts on the axes.

First calculate the coordinates of the centroid using the formula

clip_image146 where n is the number of sets of readings5,6.

2. The straight line graph that is drawn must pass through the centroid Figure . The best line is the straight line which has the plotted points closest to it. This line will give clip_image148the best gradient together with c.

3. Two other straight lines, one with the maximum gradient clip_image150 and another with the least gradient clip_image152, are then drawn. For a straight line graph where the intercept is not the origin , the three lines drawn must all pass through the centroid. Here also we can find clip_image154 and clip_image156

4. To find the Uncertainty for the gradient and intercept used this equation

clip_image158 and clip_image160

Working Example

1. Aim

To determine the acceleration due to gravity using a simple pendulum.

2. Theory : the theory of the simple pendulum, the period T is related to the length l, and the acceleration due to gravity g by the equation

clip_image162

Hence, the acceleration due to gravity, clip_image164

A straight line graph would be obtained if a graph of clip_image166 against clip_image168 is plotted.

3. Precaution :

The time t for 50 oscillations of the pendulum is measured for different lengths l of the pendulum. The period T is calculated using

clip_image170

4. Choosing Apparatus and Determine the absolute uncertainty:

Instruments

Uncertainty

(Absolute/actual)

Millimeter ruler

0.1 cm

Stopwatch (analogue)

0.1 s

5. Manage the table

Note the various important characteristics when tabulating the data as shown in Table

image

Table 1

(a) Name or symbol of each quantity and its unit are stated in the heading of each column. Example: Length and cm, and T(s). The Uncertainty for the primary data, such as length and t time for 50 oscillations, is also written. Example: (l ± 0.05) cm and (t ± 0.1)s.

(b) All primary data, such as length and time, should be recorded to reflect the precision (absolute uncetainty) of the instrument used.

For example, the length of the pendulum l is measured using a metre rule. hence it should be recorded to two decimal places of a cm, that is 10.00 cm, and not 10 cm or 10.0 cm.

The time for 50 oscillations t is recorded to 0.1 s, that is 32.0 s and not 32 s.

The average value of t is also calculated to 0.1 s. The average value of 31.9 s and 32.0 s is recorded as 32.0 s and not 31.95 s.

(c) The secondary data such as T and T2, are calculated from the primary data. Secondary data should be calculated to the same number of significant figures as I hat in the least accurate measurement. For example, T and T2, are calculated to three significant figures, the same number of significant figures as the readings of t.

(d) For a straight line graph, there should be at least six point plotted. If the graph is a curve, then more points should be plotted, especially near the maximum and minimum points.

Note that the graph is plotted with the assumption that the origin (0, 0) is a point.

The x-coordinate of the centroid = clip_image174

= clip_image176

= 50 cm

The y-coordinate of the centroid =clip_image178

= clip_image180

= 2.00s2

The coordinate for the centroid is (50cm, 2.00s2)

clip_image182

Graph 1

from the equation clip_image164[1]

clip_image162[1]

Hence a graph of T 2 against l is a straight line, passing through the origin, and gradient,

clip_image184

From the graph,

gradient of best line, clip_image186

Maximum gradient, clip_image188

Minimum gradient, clip_image190

Absolute Uncertainty in the gradient,

clip_image192

Fractional Uncertainty in the gradient

clip_image194

percentage Uncertainty in gradient

clip_image196

Acceleration due to gravity, clip_image198

Hence the percentage Uncertainty in g is the sum of the percentage Uncertainty in m only because 4p2 is a constant.

Therefore percentage Uncertainty in gravity,Dg = S Uncertainty percentage = 1.88% according to above equation

Hence acceleration due to gravity,

Written in percentage Uncertainty

g = (9.870±1.88%) m s2

also can be write in absolute Uncertainty

clip_image200

g = (9.9 ± 0.2) m s2 Since there is Uncertainty in the second significant figure, the value of g is given to two significant figures.

Activities:

Experiment number 1 *

To determine the density of a substance

Rujukan:

1 BIPM, Www.bipm.org (2011).

2 Wikipedia contributors, Wikipedia, the Free Encyclopedia (2012).

3 Wikipedia contributors, Wikipedia, the Free Encyclopedia (2012).

4 R. Hutching, Physics (Macmillan Education Ltd, Hong Kong, 1990).

5 H. Ahmad, R.H. Raja Mustapaha, and D. Bradley, Panduan Kaedah Ujikaji (Dewan Bahasa dan Pustaka, kuala lumpur, 1986).

6 S. Zainal Abidin, Fizik Amali (Dewan Bahasa dan Pustaka, kuala lumpur, 1992).

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