c1 pysical quantities

8/3/2019 C1 Pysical Quantities

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FOSEE , MULTIMEDIA UNIVERSITY (436821-T)MELAKA CAMPUS, JALAN AYER KEROH LAMA, 75450 MELAKA, MALAYSIA.

Tel 606 252 3594 Fax 606 231 8799URL: http://fosee.mmu.edu.my/~asd/

PPH 0095

Mechanics

Foundation in Engineering

ONLINE NOTES

Chapter 1

Physical Quantities

Applied Science Department

(ASD)

Centre for Foundation Studies and Extension Education

(FOSEE)


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ASD 2011/12 PHYSICAL QUANTITIES 1/ 20

Contents

1.0 Introduction to SI Units.

1.1 Basic Quantities

1.2 Derived Quantities

1.3 Conversion of Units

1.4 Scalar and Vector Quantity

1.5 Factor of ten Simplification (Standard Prefixes)

1.6 Dimension: Homogeneity of Physical Equations.

1.7 Error and Accuracy and Significant Figures.

1.8 Vectors.

1.8.1 Component of a vector.

1.8.2 Vector addition and Subtraction.

1.8.2.1 Graphical Methods.

1.8.2.2 Component Methods.1.8.3 Dot Product and Cross Product.

Mind Map


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OBJECTIVES

Upon completion of this chapter, you should be able to:

1) list the base quantities and their SI units.

2) use common standard prefixes3) explain the advantage of and apply dimensional analysis and unit analysis.4) determine the number of significant figures in a numerical value and report the

proper number of significant figures after performing simple calculation.

5) distinguish between scalars and vectors.

6) Add and subtract vectors graphically and by the component method.7) determine the dot and cross products of given vectors.

1.0 INTRODUCTION TO SI UNITS

A physical property that can be measured is called a physical quantity.The most commonly encountered physical quantities are length, mass,

time, current, temperature, light intensity and amount of substance. Theseseven quantities are known as the base quantities.

Other physical quantities is described by a numeral value and a

combination of these quantities.

A physical quantity is described by a numerical value and a unit. A unit isthe standard size for a physical quantity.

Different units can be used to describe the same quantity. For example, theheight of a person can be expressed in feet and inches, or in meters and

centimeters.

The units of a certain quantity can be converted from one system ofmeasurement to another. Such conversion are sometimes necessary.

It is, however, most practical to work consistently within the same systemof units.


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1.1 BASIC QUANTITIES

The international system of units commonly used in science an engineering is theSysteme International dUnites, or SI in short.

The SI system of units defines the base units for seven base quantities as givenin Table 1.

Various derived units for different derived quantities are obtained from thesebase quantities.

Table 1

QUANTITY SI UNITS SYMBOL

Length meter m

Mass kilogram kg

Time second s

Electric current ampere A

Temperature kelvin K

Light Intensity candela cd

Amount of Substance Mole mol


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1.2 DERIVED QUANTITIES

Derived quantities are quantities that are not base quantities.

The units for these derived quantities are known as derived units. These

units are formed from the base units using the known links between the

physical quantities involved. A list of the more common derived quantitiesand their units is given in Table 2.

QUANTITY SYMBOL ABBREVIATION IN TERMS OF BASE

UNITS

Area

A - m 2

Volume

V - m 3

Speed or velocityv -

ms

1

Accelerationa - ms 2

Density

- kg.m 3

Force F Newton (N) kg ms 2

Energy & Work E or W Joule (J) kg.m 2 s 2

Power P Watt (W) kg.m2

s3

Pressure p Pascal (Pa) kg/(m.s 2 )

Electric Charge C Coulomb (C) A.s

Frequency f Hertz (Hz) s 1

Electric Resistance R Ohm ()(

kg.m 2/(A 2 .s 3 )

Capacitance C Farad (F) A

2

s

4

kg

1

m

1

Table 2


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1.3 CONVERSION of UNITS

Any quantity we measure, such as a length, a speed, or an electric current,consists of a number anda unit.

Often we are given a quantity in one set of units, but we want it expressed inanother set of units. For example, suppose we measure that a table is 20.5 inches

wide, and we want to express this in centimeters. We must use a conversion

factor.

In any conversion, if the units do not combine algebraically to give the desiredresult, the conversion has not been carried out properly.

cm km , m 2 mm 2 , kg/m 3 g / cm 3

1.4 SCALAR and VECTOR QUANTITY

A scalar quantity is a physical quantity that is represented by magnitudetogether with the relevant units. It can be positive, negative or zero.Example: Length, mass, speed an time.

A vector quantity is a physical quantity that is represented by both magnitude

and direction together with the relevant units.

Example:: Velocity, acceleration, and force.

1.5 FACTOR of TEN SIMPLIFICATION (STANDARD PREFIXES)

Standard prefixes are used to designate common multiples in powers of ten. Thismakes calculation easier.

The most common prefixes are listed in Table 3.


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Table 3

Example: 1.2 x 10 12 F = 1.2 pF

2.9 x 10 6 Hz = 2.9 Hz

1.6 DIMENSIONS : HOMOGENEITY of PHYSICAL EQUATIONS.

The dimensions of a physical quantity are a combination of the basic physicalquantities, raised to the appropriate powers, which are used to define the physical

quantity.

Many physical quantities can be expressed in terms of a combination of

fundamental dimensions such as length,( L ), time, ( T ) and mass, ( M ).

The dimension of a physical quantity refer to the type of quantity in questions,

regardless of the units used in the measurements. In other words, dimensions are

not units.

Dimension of a physical quantity is an algebraic combination of L, T and Mfrom which the quantity is found.

FACTOR PREFIX SYMBOL FACTOR PREFIX SYMBOL

10 1 deci d 10 1 deka da

10 2 centi c 10 2 hecto h

10 3 milli m 103 kilo k

10 6 micro 10 6 mega M

109 nano n 10

9 giga G

10 12 pico p 10 12 tera T

10 15 femto f 1015 peta P

10 18 atto a 1018 exa E


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Area can be measure in various units such as square metres, square feet, acres andhectares. However, regardless of the units used, area is always length multiplied

by length. The dimensions on area are therefore the square of length, usually

stated as L 2

Symbol for dimension of a physical quantity is quantity

Table 4

There is no dimension for,

Numerical value

Ratio between the same quantityExample:

specific density = density of substance / density of water

= ML 3/ ML 3

= 1

QUANTITY DIMENSION SYMBOL

Mass mass M

Length length L

Time time T

Densitymass / length

3

ML

3

Velocity length / time LT 1

Acceleration velocity / time LT 2

Force mass acceleration MLT 2

Work/

Power Energyforce distance ML 2 T 2

Power work / time ML 2 T 3


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Angle has no dimension because it is a comparison between two position oflength measurement.

B

C A

tan = BC /AC = L/L = 1

= tan 1 1

Known constant

Example: ln, lg and

But, there are some constant has a dimension.

Example: Modulus YoungGravitational acceleration

Uses of Dimensions

a) To determine the dimensions and units of a quantity.

Example 1:

Given equation, ( p + a/ V 2 ) ( V - b ) = nRt,where : p = pressure, V = volume

Determine the dimension and units for constant a and b.

Solution:

Quantity ofa / V 2 must be the same with quantity of p , pressure

p =ML 1 T 2

a / V 2 = p

= ML 1 T 2

a = ML 1 T 2 V 2

= ML 1 T 2 (L 3 ) 2

= ML 5 T 2

Unit ofa is kgm 5 s 2


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Constant for b must be the same with quantity volume, V

b = V = L 3

Unit ofb is m3

b) Check whether an equation is dimensionally correct, i.e.. if an equation has the

same dimension ( unit ) on both sides. (Homogeneity of equations). This is usefulfor checking the correctness of an equation.

Note : Dimensionally correct does not necessarily mean the equation is correct.

Example 2:An object move linearly with initial velocity, u and constant acceleration, a to a final

velocity, v in a time, t. The final velocity is given by equation : v = u + at. Show that theequation is dimensionally correct

Solution:v = u + atThe dimensions ofv, u and a:

v = LT 1

u = LT 1

at = a t =LT 2 T= LT 1

Dimension analysis:

Dimension of left side is LT 1

Dimension of right side is = LT 1 + LT 1 = 2 LT 1

So the dimension of the left side is equal to the dimension of the right side and theequation is dimensionally correct.

c) Derive an equation.

Example 3:

When a sphere moves through a liquid at steady speed, the drag force is thought to

depend on

the viscosity of the liquid

the speed v of the sphere

the radius rof the sphere.

Assuming that F = k a v

br

c, where k, a, b and c are dimensionless constants,determine the values ofa,b and c. Then express the formula in its simplest form.

The dimension ofare M L-1

T-1


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Solution:

F = MLT 2

kav

br

c= (ML

-1T

-1)ax (LT

-1)b

x (L)c

MLT-2 = ML-1T-1)a x (LT-1)b x (L)c

= Ma L-a+b+c T-a-b

Equating the indices of each dimension gives,M: 1 = a

a = 1

T: -2 = -ab

b = 2a

= 2-1 = 1

L: 1 = -a + b + c

c = 1 + ab= 1 + 11 = 1

So , a = 1, b = 1, c = 1.

Hence ,F = kv r

1.7 ERRORS and ACCURACY and SIGNIFICANT FIGURES.

1.7.1 ERRORS and ACCURACYThe measured thickness of a book is 2.4 cm using an ordinary ruler. It is wrong tostate the result as 2.40 cm because the measurement is reliable up to the nearest

centimeter only. Given the limitations of the ordinary ruler, it is not possible to

determine whether the actual thickness is 2.40 cm, 2.37 cm or 2.45 cm.

If vernier calipers are used, the result will be 2.43 cm. Vernier calipers can

measure lengths reliably to the nearest 0.01 cm.The difference between these two measurement is in their uncertainty. Themeasurements obtained using vernier calipers have smaller uncertainties.

Therefore, a more accurate reading for the thickness can be obtained.

The uncertainty is also called the error, because it points out the maximumdifference that is likely to exist between the measured value and the actual value.

So, when it is said that there is an error or uncertainty in a measurement, this

does not mean there is a mistake or that the measurement value cannot be

confirmed.

The accuracy of a measured value, that is how close the measurement is to theactual value, is indicated by writing the number, the symbol and a second

number which indicates the uncertainty of the measurement.If the diameter of a copper wire is measured as 0.50 0.01 mm, this means thatthe actual value is unlikely to be less than 0.49 mm or greater than 0.51 mm. The

value 0.01 mm represents the absolute error of the measurement.

Accuracy can also be expressed in terms of the maximum likely fractional error or

percentage error. A resistor labeled 12 10% probably has a true resistance

value differing from 12 by no more than 10% of 12 , that is about 1 . The

resistance is probably between 11 and 13 .


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For the diameter of the copper wire mentioned earlier, the fractional error is (0.01mm) / ( 0.50 mm ) , or about 0.02. The percentage error is 0.02 x 100% = 2.00%

% Uncertainty = Uncertainty of a quantity x 100 %Central Value

Example 4:Width of a page of paper = 21.6 0.1 cm

Length of a page of paper = 27.9 0.1 cm% Uncertainty ( width ) = 0.1/21.6 x 100%

= 0.5%%Uncertainty ( length ) = 0.1/27.9 x 100%

= 0.4 %

Area, A = ( 21.6 x 27.9 ) cm 2

= 603 cm 2

The total area uncertainty = 0.9 %

This means an uncertainty of ( 0.009 ) ( 603 cm 2 ) = 5 cm 2

The area of the paper is ( 603 5 ) cm 2

RULER: Uncertainty of 1 mm ( 0.1 cm )

Example: (45.0 0.1) cm

VERNIER CALIPERS : Uncertainty of 0.01cm

Example: ( 8.16 0.01 ) cm

MICROMETER SCREW GAUGE: Uncertainty of 0.01mm

Example: ( 8.20 0.01 ) mm

Parallax Error: Position of an object had been altered because of observers eye.Effect: Accuracy decrease, Error increase.

1.7.2 SIGNIFICANT FIGURES.In many cases, the error or uncertainty of a measurement or number is not statedexplicitly. Instead, the uncertainty is indicated by the number of meaningful

digits, or significant figures, in the measured value.

The number of digits known with certainty in the number is called the number ofsignificant figures. Thus, there are four significant figures in the number 45.83

cm and two in the number 0.029 cm.

Addition /subtraction process:When numbers are added/subtracted, the number of decimal places in the result

should equal the smallest number of decimal places of any term (or quantity) in

the summation/subtraction.

Example 5: 23.1 +45 +0.68 +100 = 169 (result with zero decimal place)

23.5 + 0.567 + 0.85 = 24.9 (result with one decimal place)


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Multiplication /division process:In multiplying/dividing two or more quantities, the number of significant figures

in the final result should be the same as the number of significant figures of any

term (or quantity) having the lowest number of significant figures.

Example 6: 0.424 x 3.4 = 1.4416

= 1.4 (result rounded down two significant figures)

13.90 0.580 = 23.9655= 24.0 (result rounded up to three significant figures)

1. Zeros at the beginning of a number are not significant.

They merely locate the decimal point.0.254 m 3 s.f

2. Zeros within a numberare significant.

104.6 m 4 s.f

3. Zeros at the endof a number after the decimal point are significant.

2705.0 .5 s.f

1.8 VECTORS

IMPORTANT TERMS

SCALAR QUANTITY: A quantity which can be described by a single number

VECTOR QUANTITY : A quantity which can be adequately described by a number

(magnitude) and a direction.

VECTOR COMPONENT : Two perpendicular vectors which added together producethe original vector.

THE RESULTANT or sum of a number of vectors of a particular type ( force vector, for

example ) is that single vector that would have the same effect as all the original vectors

taken together.

A SCALAR QUANTITY

A scalar quantity, or scalar, is one that has nothing to do with spatial direction. Many

physical concepts such as length, time temperature, mass, density, charge and volume arescalars; each has a scale or size but no associated direction. The number of students in a

class, the quantity of sugar in a jar, and the cost of a house are familiar scalar quantities.

Scalars are specified by ordinary number and add and subtract in the usual way. Two

candies in one box plus seven in another will give total nine candies.


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A VECTOR QUANTITYVector quantities have magnitude and direction. Physical quantities that have both

numerical and directional properties are represented by vectors. Examples of vectorquantities are force, momentum, velocity, displacement and acceleration. In printed

material, vectors are often represented by boldface type, such as F. When written by

hand, the designation F is commonly used. The magnitude of vector ais written as a ora .

Two vectorsa and b are equal only if:

a = b

direction ofa = direction ofb

When a vectorP is multiplied by a scalar k, the result is a vector kP with magnitude kP.

Ifk= 0 , the result is a zero vector 0

1.8.1 COMPONENTS of a VECTORSA component of a vector is its effective value in a given direction. For example,the x-component of a displacement is the displacement parallel to the x-axis

caused by the given displacement. A vector in three dimensions may be

considered as the resultant of its component vectors resolved along any three

mutually perpendicular directions. Similarly, a vector in two dimensions may beresolved into two component vectors acting along any two mutually perpendicular

directions.

Figure 1

Figure 1 shows the vector R and its x and y vector components R x and R y ,

which have magnitudes

xR = R Cos and yR = R Sin

or equivalently

R x = R Cos and R y = R Sin

R

xRx

Ry


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Angle, = tan-1

(x

y

R

R)

Conversely, ifRxandRyare given, Pythagoras theorem is used to findR.

R= 22yx RR and = tan

-1(

x

y

R

R)

1.8.2 VECTOR ADDITION and SUBTRACTION .

1.8.2.1 GRAPHICAL METHOD

a) ADDING VECTORSTwo vectors can be added by arranging them such that the tail of the secondvector is at the head of the first vector.

The sum of these vectors is another vector which extends from the tail of the firstvector to the head of the second vector. It is called the resultant vector.

Triangle Method.

Draw vectorA with its magnitude represented by a convenient scale.

Draw vectorB to the same scale , its tail start from tip ofA

Resultant vector ,R =A +B drawn from the tail ofA to the tip ofB ( Figure2 )

Figure 2

Parallelogram Method.

For adding two vectors: the resultant of two vectors acting at any anglemay be represented by the diagonal of a parallelogram.

The two vectors are drawn as the sides of the parallelogram and theresultant is its diagonal, as shown in Figure 3.

The direction of the resultant is away from the origin of the two vectors.

A

B

R

Start


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Figure 3

Polygon Method.

This method for finding the resultant R of several vectors ( A ,B and C)consists in beginning at any convenient point and drawing ( to scale and in

the proper directions ) each vector arrow in turn.

They may be taken in any order of succession : A + B + C = C +B + A =

R . The tail end of each arrow is positioned at the tip end of the precedingone, as shown in Figure 4.

Figure 4

The resultant is represented by an arrow with its tail end at the startingpoint and its tip end at the tip of the last vector added. If R is the resultant,

R = R is the size or magnitude of the resultant.

A

B

RRes

ultant

C

B

A

Start

End

Resulta

nt


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b) SUBTRACTING VECTORS

To subtract a vector B from A , reverse the direction of B and add individually

to vector A , that is , A - B = A + (-B )

Figure 5

R = A + ( B) = AB

1.8.2.2 COMPONENTS METHODS.

Each vector is resolved into itsx-,y-, andz- components, with negatively directed

components taken as negative. The scalarx-componentR x of the resultant R is the

algebraic sum of all the scalarx-components. The scalary- andz- components of

the resultant are found in a similar way. With the components known, the

magnitude of the resultant is given byR = 222 zyx RRR

In two dimension, the angle of the resultant with thex-axis can be found from the

relation tan =x

y

R

R.

Example:The three forces shown in Figure 6 act on an object at point O. Calculate the

resultant force.

Figure 6


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Solution:

Sum of components along Ox

Rx = 19 + 15cos 60 16cos 45= 15.19 N

Sum of components along OyRy = 0 + 15sin 60 + 16 sin 45

= 24.30 N

Magnitude of resultant forc,

R = [(15.19)2 + 24.30)2]1/2

= 28.66 N

Direction of resultant force is given by

= tan 1 (24.30/15.19) = 57.99

1.8.3 DOT PRODUCT and CROSS PRODUCT1.8.3.1 UNIT VECTOR

A unit vector is a dimension vector having a magnitude of one. Units vectors areused in describing direction in place. The symbols i, j, k are usually used for unit

vectors in the positive x, y and z respectively. ( Figure 7 )

Figure 7

A vectora in a three dimensional coordinate system can be written as:

a = a x i + a y j + a zk

y

x

i

jk


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Suppose, a = a x i + a y j + a zk

b = b x i + b y j + b zk

r = a + b

=(a x i+ a y j + a zk) + (b x i + b y j + b zk)= (a x + b x ) i + (a y + b y )j + (a z + b z )k

1.8.3.2 DOT PRODUCT

Dot product of two vectors a.b yield a scalar. ( Figure 8 )

a.b = ab cos

Figure 8

Since i,jandk are all one unit in length and they are all mutually perpendicular, we have,

i) i i =j j =k k = 1 [cos 0o = 1]ii) i j=j i = i k =k i=j k =k j= 0 [cos 90o = 0]

For example, ifr1 = ai+ bj + ck and r2 = di + ej+ fk, then,

r1 r2 = (ai+ bj + ck) (di + ej + fk) = ad + be + cf

a

b


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1.8.3.3 CROSS PRODUCT

Cross product of two vectors A.B yield another vector C, where C = A x B( Figure 9 )

A B = (|A| |B| sin )

Figure 9

Note: A B = B A

Vector Product of Parallel Vectors

A IfA and B are parallel vectors, |A B| = |A| |B| sin 0o = 0

B In the case of unit vectors i,jandk,

i i=j j =k k = 0, (the zero vector).

Vector Product of Perpendicular Vectors

In the case of unit vectors i, j and k, which are

perpendicular to each other, sin 90o

= 1. Thus,

i j=k and j i = k

j k = i and k j = i

k i =j and i k = j

If a =x1i +y1j+z1k and b =x2i +y2j +z2k,

a b = (x1i+y1j +z1k) (x2i +y2j +z2k)

=x1x2 (i i) +x1y2 (i j) +x1z2 (i k) +y1x2 (j i) +y1y2 (j j)


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+y1z2 (j k) +z1x2 (k i) +z1y2 (k j) +z1z2 (k k)

=x1y2kx1z2jy1x2k +y1z2 i +z1x2jz1y2 i

= (y1z2z1y2) i + (z1x2x1z2)j+ (x1y2y1x2)k

i j k

This expression is the expansion of the determinant222

111

zyx

zyx

i j k

Therefore (x1i+y1j+z1k) (x2i +y2j +z2k) =222

111

zyx

zyx

END OF CHAPTER 1.

c1 pysical quantities

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