3. Physical Measurements

8/8/2019 3. Physical Measurements

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PHYSICAL MEASUREMENTS

Prof. Dr. Metin TULGAR


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Physical Measurement

A concept that is necessary for physical evaluation of

measurements related to creatures in the nature.

Scalar Quantities Vector Quantities


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Scalar Measurement

Meaningful with number and unit, only.

e.g.e.g.

1 kg sugar, 500 g tomatoe (mass),

3 m2 carpet (surface),

50 m3 wood, 2 lt milk (volume),

4 h, 5 min, 10 s (time),

18 C, 300 K (temperature).


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Vector Measurement

T

o describe some quantities,

number and unit are not enough;

other specifications have to be determined.


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e.g.e.g.Wheat in a land.

Its starting point where it locates on earth,

its direction depending on wind,

its line

and its length

must be known for a complete description.


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This example shows thata vector measurement necessitates

the determination of four parameters:

- starting point (application point),

- line

- direction,

- amplitude


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Physical quantities such as

force,force,

velocityvelocity,,

accelerationacceleration,,

moment,moment,

impulsimpulsee

can only be explained with vectors.


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VV

..Asymbol of a vector


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If the lines of certain vectors are in parallel, and

their directions and amplitudes are the same,

then these vectors will be called asequal vectorsequal vectors.

V1

A1 V2

A2


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If the lines and amplitudes of certain vectors

are the same, but their directions are opposite,

then these vectors will be called asopposite vectorsopposite vectors.

A1A2

V2

V1


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Components of a Vector

In two dimension co-ordinate system;

y

Vy V

j x

i Vx


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i andj: unit vectors

V = Vx i + Vyj

V = Vx2 + Vy

2

Vx = VCos

Vy= VSin


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In three dimension co-ordinate system;z

Vz V

k

i j Vy y

Vx V


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i , j and k : unit vectors

V = Vx i + Vyj + Vz k

V = Vx

2 + Vy

2 + Vz

2


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Addition of Vectors

Total vector is a vector that is equal to addition of more thanone vector.

Addition of vectors which are

on the same line and in the same direction;

V2 VA V1

V1 V2

V = V1 + V2


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Additionofvectorswhich are

onthesameplaneandinthedifferentdirection;

V3

A V2 V

V1 V3

V1 V2

V = V1 + V2 + V3


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Additionofvectorswhich are

onthes

ame

line

and

in

theo

pposite

direction

;

V V2A A

V2 V1 V1

V = V1 + (-V2) = V1 - V2


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Total of the opposite vectors is equal to zero.

V2A V1 V = 0


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Subtraction of Vectors

A V1 V V2

V2 V1

V = V1 - V2

A

-


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Multiplication of Vectors

Scalar multiplication :A . B = ABCos

A = Ax i + Ay j + Az k

B = Bx i + By j + Bz k

A = Ax2 + Ay

2 + Az

2

B = Bx

2 + By

2 + Bz

2

in case of unit vectors;

i . i = 1 . 1 Cos0 = 1 . 1 . 1 = 1

i . i = j . j = k . k = 1

Vectoral multiplication :

A B = ABSin

in case of unit vectors:

i i = 1 . 1 Sin0 = 1 . 1 . 0 = 0

i i = j . j = k . k = 0


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Problem 1

Length of a vector is 7.3 unit and it makes 250 with the positive

horizontal axis in the direction of counter clockwise. Find the

components of this vector.

Solution:A = 7,3 units

= 250Ax = A Cos = 7,3 Cos 250 = -2,5 unit

Ay = A Sin = 7,3 Sin 250 = -6,9 unit

y

Ax 2,5 250

x

6,9

A Ay

7,3


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Problem 2

Horizontal component of a vector is 25 unit, and its vertical

component is 40 unit. Find amplitude and angle made with the

horizontal axis of this vector.

Solution:A = A

x

2 + Ay

2 = (25)2 + (40)2 = 47,1

tg = 25/40 ===> = arc tg 25/40 = 32

= 32 + 90 = 122

y

A Ay

40

Ax 25 x


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Problem 3

Write vectors whose components are A (3,5) and B (4,6,8)using unit vectors and calculate their total vector.

Solution:

A (3,5) ===> A = 3 i + 5 j

B (4,6,8) ===> B = 4 i + 6 j + 8 k

A + B = 7 i + 11 j + 8 k


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Problem 4

The bistoury of a surgeon goes to east by 3 cm first, then

to north by 4 cm starting from a point, S. Calculate the total

transposition (replacement) of the bistoury regarding tostarting point, S.

Solution:

4 cm

S 3 cm

transposition = (3)2 + (4)2 = 5 cm

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