CHAPTER 1PHYSICAL QUANTITIES &

MEASUREMENTS (3 hours)

1.1 Physical Quantities and Units

1.2 Scalars and Vectors

1.3 Measurement and Errors (Laboratory work)

How fast does light travel? How much do you weigh ?What is the radius of the Earth?What temperature does ice melt at?

We can find the answers to all of these questions by measurement.Speed, mass, length and temperature are all examples of physical quantities.

Measurement of physical quantities is an essential part of Physics.

LEARNING OUTCOMESa) State basic quantities and their respective SI units: length (m), time (s), mass (kg), electrical current (A),

temperature (K), amount of substance (mol) and luminosity (cd).

b) State derived quantities and their respective units and

symbols: velocity (m s-1), acceleration (m s-2), work (J), force (N), pressure (Pa), energy (J), power (W) and frequency (Hz).

c) State and convert units with common S.I. prefixes.

1.1 Physical Quantities and Units (1 hour)

Physical Quantities

- quantities that are measurable.

- consists of a numerical value & a unit.

Two categories:1. Base Quantities2. Derived Quantities

- standard for measurement of physical quantities that need clear definition to be useful.

Physical Unit

- ex: metre (m) – unit for length / distancesecond (s) – unit for timeKelvin (K) – unit for temperature

SI Unit- International System of Units

- has been agreed internationally.

Base Quantity- fundamental quantity that can not be derived in

terms of other physics quantities.

candela

mole

kelvin

ampere

second

kilogram

meter

Name of SI unit

KTemperature, T

mLength , l

kgMass, m

sTime, t

cdLuminous intensity

molAmount of substance, n

AElectric current, I

Unit symbol

Base

Quantity

- Some physical quantities have no units.- Example: refractive index, strain

Derived Quantity

- physical quantity which can be expressed in term of base quantity.

Q = It

W = Fs

P = F / A

F = ma

f = 1 / T

ρ = m / V

v = s / t

Defining equation

A s

kg m2 s-2

kg m-1 s-2

kg m s-2

s-1

kg m-3

m s-1

SI unit

Pa (pascal)Pressure

--Velocity

--Density

Hz (hertz)Frequency

C (coulomb)Charge

J (joule)Work

N (newton)Force

Special

name

Physical Quantity

Prefix

- can be added to SI base & derived units to make larger or smaller units

Wavelength of an X-ray = 0.000 000 001 m= 1 × 10−9 m= 1 nm

For example:

10-1 deci- (d)

10-2 centi- (c)

10-3 milli- (m)

10-6 micro- (μ)

10-9 nano- (n)

10-12 pico- (p)

10-15 femto- (f)

Multiple Prefix ( & abbreviation )

1012 tera- (T)

109 giga- (G)

106 mega- (M)

103 kilo- (k)

102 hecto- (h)

Unit Conversions- S.I. unit system is predominant throughout the world.

- Units can be expressed in the same quantity.

- It is necessary to change from one set of units to another.

Example--------------------------------------------------------------------------------

Express the speed limit of 65 km/hour in terms of meters/second.

Solution

Knowing that 1 km = 1000 m1 hour = 3600 s

Speed = )s3600

m1000(65

1s m06.18

Example of Conversion of Units in SI system

11 sm17.4s)60(60

m100015hkm15

39333 m105)m10(5mm5

332

33 mkg7000

)m10(

kg107cmg7

K15.303K )15.27330(C30

Remember: Every answer for physics problem solution must followed with unit of that quantity otherwise mark will be deducted.

Follow Up Exercise--------------------------------------------------------------------------------1. The largest diamond ever found had a size of 3106

carats. One carat is equivalent to a mass of 0.200 g. Determine the mass of this diamond in kg.

2. A hall bulletin board has an area of 2.5 m2. What is this area in square centimeters (cm2)?

3. A football field is 110 m long and 90 m wide. What is the area of the field kilometers ?

4. The density of metal mercury is 13.6 g/cm3. What is this density as expressed in kg/m3?

LEARNING OUTCOMES

kji ˆ,ˆ,ˆ

a) Define scalar and vector quantities, unit vectors in Cartesian coordinate.

b) Perform vector addition and subtraction operations graphically.

c) Resolve vector into two perpendicular components (2-D) and three perpendicular components (3-D): i) Components in the x, y and z axes. ii) Components in the unit vectors.

1.2 Scalars and Vectors (1 hour)

LEARNING OUTCOMES

e) Define and use cross (vector) product;

A B

= A (B sin θ) = B (A sin θ).

Direction of cross product is determined by corkscrew method or right hand rule.

A B d) Define and use dot (scalar) product;

= A (B cos θ) = B (A cos θ)

If you know the library is 5 m from you, it could be anywhere on a circle of radius 5.0 m. Instead, if you are told the library is 0.5 m northwest, you knows it precise location.

Scalar Quantity- Quantity which has only magnitude.- Example: mass, distance, speed, work.

Vector Quantity- Quantity which has both magnitude and direction.- Example: displacement, velocity, force, momentum

- Magnitude of the vector is written as

- Symbols for vectors are printed bold or use an arrow over a letter,

- A vector can be represented by arrow where its length indicates the magnitude & direction of the arrow represents direction of the vector.

A

Representing vectors

A

A

Equality of two vectors

- 2 vectors & are equal if they have the same magnitude and point in the same direction.

A

B

Negative of a vector- The vector is a vector with the same

magnitude as but points in opposite direction.

B

B

Multiplying a vector by a scalar

When a vector is multiplied by a scalar k, the product is a vector of magnitude .The direction of the vector is same as that of if k is positive, and opposite if k is negative.

A

Ak

kAAk

A

Unit vectors

A unit vector is a dimensionless vector having a magnitude of exactly 1 and points in a particular direction.

Are use to specify a given direction in space.

, & is used to represent unit vectors pointing in the positive x, y & z directions.

i j k

| | = | | = | | = 1i j k

Vector Addition & Subtraction

Addition- The addition of 2 vector, and will result in a

third vector called resultant vector.

A

B

R

- Resultant vector is a single vector which produces the same effect ( in both magnitude and direction ) as the vector sum of 2 or more vectors.

- 2 methods of vector addition:

(1) Drawing / Graphical method - tail to head & Parallelogram

(2) Mathematic Calculation – unit vector & trigonometry

Addition of vectors in the same directions

Addition of vectors in the opposite directions

The direction of resultant vector S is in the direction of the bigger vector

Recall

(a) Tail to head graphical method

For two or more coplanar vectors point in different directions can be added by using the tail to head method or parallelogram method.

Placing the tail of each successive arrow at the head of the previous one. The resultant vector is the arrow drawn from the tail of the first vector to the head of the last vector.

(b) Parallelogram method

Resultant vector, : diagonal of a parallelogramdiagonal of a parallelogram

formed with & as two of its 4 sides.

R

A

B

Vectors SubtractionThe subtraction of 2 vectors ( ) can be written as the addition of two vectors ( ).

BA

)( BA

Resolving vector into 2 perpendicular components (2D)

Any type of vector may be expressed in terms of its component.

with the aid of trigonometry:

22|| yx AAA

A

Axcos cosAAx

A

Aysin sinAAy

Magnitude of vector A :

Direction of vector A :

x

y

A

Atan

* θ is always measured from x axis.

consider a vector lying in the xy plane ( 2D vector ) :A

jAiAA yx ˆˆ

The vector can be written as:A

Unit vector form

Example

A force of 800 N is exerted on a bolt A as shown in Fig. below. Determine the horizontal and vertical components of the force.

Solution

with the aid of trigonometry:

cosFFx 35cos800

NF x 655sinFFy

35sin800

NF y 459

We may write in the unit vector formF

jNiNF ˆ)459(ˆ)655(

Resolving vector into 3 perpendicular components (3D)

kAjAiAA zyx ˆˆˆ

In 3D space, vector can be written as :

A

Magnitude of vector A :

222|| zyx AAAA

64.9)8()5()2(|| 222 A

Example : Given vector kjiA ˆ8ˆ5ˆ2

vector can be resolved into 3 components : Ax, Ay & Az

A

xAAx cos

yAAy cos

zAAz cos

where θx, θy and θz are the angles that vector A forms

with x, y & z axes respectively

Addition of vectors by means of components

Suppose we have 2 vectors A and B and we want to find their resultant, R. The components of a vector provide the most convenient & accurate way of adding (or subtracting) any number of vectors.

1. Resolve each vector into its x and y components.Pay careful attention to signs:any component that points along the negative x or y axis get a − sign.

2. Add all the x components together to get the x component of resultant. Ditto for y:

* do not add x components to y components

Adding vectors using components

otherany xxx BAR

otherany yyy BAR

3. The magnitude of the resultant vector, R is given by:

Direction of the resultant vector :

* vector diagram drawn help to obtain the

correct position of the angle θ

22|| yx RRR

x

y

R

R

tan

θ θ

θ θ

Example

The magnitudes of the 3 displacement vectors shown in drawing. Determine the magnitude & directional angle for the resultant that occurs when these vectors are added together.

Solution

Cy= − 8Cx=0

By=+5 sin 30Bx=−5 cos 30

Ay =+10 sin 45Ax=+10 cos 45Component yComponent x

Resultant vector along x axis:

Rx = Ax + Bx + Cx

= +7.07 + (− 4.33 )+0 = + 2.74 m

Resultant vector along y axis:

Ry = Ay + By + Cy

= +7.07 + 2.50+ (−8) = + 1.57 m

22yx RRR

m16.3

)57.1()74.2( 22

Magnitude of resultant vector

x

y

R

R

tan 573.074.2

57.1

x above 81.29

Direction of resultant vector

Resultant of the displacement write in unit vector form

jmimR ˆ)57.1(ˆ)74.2(

Follow Up Exercise

Four forces act on bolt A shown. Determine the resultant of the forces on the bolt .

Answer :R = (199.1N )i + (14.3N)j

or R = 199.6 N at 4.1˚ above positive x axis.

Example

Let :

jib

jia

ˆ3ˆ5

ˆ5ˆ2

Find : (a) (b) (c)

ba

ba

32 |2| a

Solution

)ˆ3ˆ5()ˆ5ˆ2( jijiba

(a)

ji ˆ2ˆ7

To find the magnitude of , 1st we have to calculate

(b) )ˆ3ˆ5(3)ˆ5ˆ2(232 jijiba

jiji ˆ9ˆ15ˆ10ˆ4

ji ˆ19ˆ11

(c) |2| a

a

2

)ˆ5ˆ2(22 jia

)ˆ10ˆ4 ji

22 104|2| a

77.10

Check your understanding

1. Find the sum of two vectors A and B lying in the xy plane and given by

m)ˆ0.2ˆ0.2( jiA

m)ˆ0.4ˆ0.2( jiB

and

cm 40R cm; 7.0R ; cm 31R cm; 25R (2)

x- from 27 anglean at m 4.5 is Ror m)ˆ0.2ˆ0.4()1(:ans

zyx jiR

2. A particle undergoes three consecutive displacements:

Find the components of the resultant displacement and its magnitude.

cm)ˆ12ˆ30ˆ15(1 kjid

cm)ˆ5ˆ14ˆ23(2 kjid

and cm)ˆ15ˆ13(3 jid

Multiplying a vector by a vector

Scalar product ( dot product )Vector product ( cross product )

BA

BA

Dot Product ( ) BA

cos|||| BABA

where |A| : magnitude of vector |B|: magnitude of vector θ : angle between &

A

B

A

B

0° ≤ θ ≤ 180°

= zero when θ = 90°BA

= maximum value when θ = 0°BA

Commutative law applied to dot product :

ABBA

Example of physical quantity : sFW

Dot product Calculation

)ˆˆˆ()ˆˆˆ( kBjBiBkAjAiABA zyxzyx

zzyyxx BABABABA

All 3 vectors are perpendicular to each other

090cos)1)(1(ˆˆˆˆˆˆ

10cos)1)(1(ˆˆˆˆˆˆ

kjkiji

kkjjii

Example

Given 2 vectors :

)285(

)423(

kjiB

kjiA

Calculate (a) the value of (b) the angle θ between 2 vectors

BA

Solution

)285()423( kjikjiBA

)2)(4()8)(2()5)(3(

9 BA

(a)

Dot product is a scalar quantity

(b)

||||cos

BA

BA

)64.9)(39.5(

9

cos : from BABA

03.80

39.5)4()2()3(|| 222 A

64.9)2()8()5(|| 222 B

Cross Product ( ) BA

sin|||||| BABA

- create a new vector

- The magnitude of the cross product is given by:

0° ≤ θ ≤ 180°

is equals the magnitude of multiplied by the component of perpendicular to .

|| BA

A

B

A

A

B

sinB

q

A

B

sinA

-- if is parallel @ anti parallel ( θ=0° @

180° )

BA

&0BA

-- if is 90°

max|| BA

BA

&

Example of physical quantity :

BvqFm

Force acting on a charge moving in magnetic field

Moment or Torque, Fr

is equals the magnitude of multiplied by the component of perpendicular to .

|| BA

B

A

B

Alternatively:

)( ABBA

- the direction of new vector ( ) is normal to the plane that contain vector & given by Right Hand Rule

BA

A

B

Example

Given 2 vector :

Calculate : (a)

30

)085(

)423(

kjiB

kjiA

|| BA

Solution

40.25

30sin8929

89

85

29

423

22

222

BA

B

A

Follow Up Exercise

1. A force is acting on an object. The displacement of the object is given byFind(a) the work done by this force(b) the angle between the force & the displacement.

2. Given 2 vector as below :

jiBjiA ˆ2ˆ5ˆ3ˆ3

)ˆ5ˆ( jiF

)ˆˆ10( jix

Find the cross product of the two vector State its magnitude & draw the vector diagram to shows the direction of the new vector ( ).

BA

BA