مسائل في المساحه

ﺔﻳﻮﺘﺴﻤﻟا ﺔﺣﺎﺴﻤﻟا ﻰﻓ ﻞﺋﺎﺴﻣ

1 ﻪﻟﻮﻃ نﺎﻜﻓ ﻞﻴﺘﻟا ﻦﻣ ﻂﻳﺮﺸﺑ ﻂﺧ ﺲﻴﻗ (93.75 m) ﻮ ه ﻂﻳﺮﺸ ﻟا اﺬ ﻬﻟ ﻰﻤ ﺳﻹا لﻮ ﻄﻟا نأ ﻢ ﻠﻋ اذﺈﻓ ، (20.0 m) لﻮ ﻄﻟاو ، ﻪﻟ ﻰﻘﻴﻘﺤﻟا (20.01 m) ﻂﺨﻠﻟ ﻰﻘﻴﻘﺤﻟا لﻮﻄﻟا ﺐﺴﺣا . ﻞﺤﻟا ﻟو ،ﺐﻠﺼ ﻟا ﻦ ﻣ وأ نﺎ ﺘﻜﻟا ﻦ ﻣ نﻮ ﻜﻳ نأ ﻦ ﻜﻤﻳو ،ﺮﺷﺎﺒﻤﻟا سﺎﻴﻘﻠﻟ ﻂﻳﺮﺸﻟا مﺪﺨﺘﺴﻳ تارﺎ ﻴﺗ ﻰ ﻓ ﻪﻣاﺪﺨﺘ ﺳا ﺐﻌﺼ ﻳ ﻪ ﻨﻜ ًﺎﻴﻘﻓأ ﻩﺪﺷ ﺔﺑﻮﻌﺼﻟ ةﺪﻳﺪﺸﻟا ءاﻮﻬﻟا . ﻂﻳﺮﺸﻠﻟ ﻰﻘﻴﻘﺤﻟا لﻮﻄﻟا ) ﺮﻳﺰﻨﺠﻟا وأ ( ﻂﺨﻠﻟ ﻰﻘﻴﻘﺤﻟا لﻮﻄﻟا = سﺎﻘﻤﻟا لﻮﻄﻟا × ﻂﻳﺮﺸﻠﻟ ﻰﻤﺳﻹا لﻮﻄﻟا ) ﺮﻳﺰﻨﺠﻟا وأ True length = 93.75 * 0 . 20 01 . 20 = 93.7968 m

∗∗∗∗∗∗

2 ﺮﺘﻤﻟا ﻦﻣ ﺔﻠﻘﻋ تﺪﻘﻓ سﺎﻘﻤﻟا ﻂﺨﻟا لﻮﻃ نﺎآو ،ﺮﻳﺰﻨﺠﻟا ﻦﻣ ﻦﻣﺎﺜﻟا (88.50 m) . ﻂﺨﻠﻟ ﻰﻘﻴﻘﺤﻟا لﻮﻄﻟا ﺐﺴﺣا . ﻞﺤﻟا ﺮ ﻳﺰﻨﺠﻟا ﺎ ﻓﺮﻃ ﻰ ﻬﺘﻨﻳو ،تﺎ ﻘﻠﺣ ﻖ ﻳﺮﻃ ﻦ ﻋ ﺎﻬﻀ ﻌﺒﺑ ﻞﺼ ﺘﺗ ﺐﻠﺼ ﻟا وأ ﺪ ﻳﺪﺤﻟا ﻦ ﻣ ﺔ ﻌﻴﻓر نﺎﺒﻀ ﻗ ﻦﻋ ةرﺎﺒﻋ ﺮﻳﺰﻨﺠﻟا ﻮه ﺮﻳﺰﻨﺠﻠﻟ فوﺮﻌﻤﻟا لﻮﻄﻟاو ،سﺎﺤﻨﻟا ﻦﻣ ﻦﻴﻀﺒﻘﻤﺑ (20.0 m) ﻮه ﺔﻠﻘﻋ ﻞآ لﻮﻃو ، (20.0 cm) تﺎﻘﻠﺣ ﻦﻣ ﺎﻬﻌﺒﺘﻳ ﺎﻤﺑ . ﺮﻳﺰﻨﺠﻠﻟ ﻰﻤﺳﻹا لﻮﻄﻟا = 20.0 m ﺮﻳﺰﻨﺠﻠﻟ ﻰﻘﻴﻘﺤﻟا لﻮﻄﻟا ، = 20.0 – 0.20 = 19.80 m true length = 88.50 * 0 . 20 80 . 19 = 87.615 m

∗∗∗∗∗∗

3 ﻪ ﻟﻮﻃ ﻂﻳﺮ ﺷ ماﺪﺨﺘ ﺳﺎﺑ ضرأ ﺔﺣﺎﺴ ﻣ ﺖﺴ ﻴﻗ (30.0 m) ﺔﺣﺎﺴ ﻤﻟا ﺖ ﻧﺎﻜﻓ (656.0 m2) ا نأ ﻢ ﻠﻋ اذﺈ ﻓ ، ﻰ ﻘﻴﻘﺤﻟا لﻮ ﻄﻟ ﻮه ﻂﻳﺮﺸﻠﻟ (29.99 m) ضرﻷا ﺔﻌﻄﻘﻟ ﺔﻴﻘﻴﻘﺤﻟا ﺔﺣﺎﺴﻤﻟا ﺪﺟوﺄﻓ . ﻞﺤﻟا ) ﺮﻳﺰﻨﺠﻟا وأ ﻂﻳﺮﺸﻠﻟ ﻰﻘﻴﻘﺤﻟا لﻮﻄﻟا ( 2 ﺔﻴﻘﻴﻘﺤﻟا ﺔﺣﺎﺴﻤﻟا = ﺔﺳﺎﻘﻤﻟا ﺔﺣﺎﺴﻤﻟا × ) ﺮﻳﺰﻨﺠﻟا وأ ﻂﻳﺮﺸﻠﻟ ﻰﻤﺳﻹا لﻮﻄﻟا ( 2 True area = 656.0 * 2 0 . 30 99 . 29 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ _{= 655.562 m}2_.

∗∗∗∗∗∗

4 -ا ﺔﻟﻻﺪﺑ مﺎﻈﺘﻧﺎﺑ ﺔﻠﺋﺎﻣ ضرأ ﻰﻠﻋ سﺎﻴﻘﻟا ﻦﻋ ﺞﺗﺎﻨﻟا ﺄﻄﺨﻟا ﺢﻴﺤﺼﺘﻟ ﺔﻗﻼﻌﻟا ﻖﺘﺷ : أ ضرﻷا ﻞﻴﻣ ﺔﻳواز θ

H

₁ 2

H

1

V

V

2

L

1

L

₂

θ

ﻞﺤﻟا أ ﻞﻴﻤﻟا ﺔﻳواز ﺔﻟﻻﺪﺑ θ راﺪﺤﻧﻹا ﺔﺟرد ) ضرﻷا ﻞﻴﻣ وأ ( 1:n ﺚﻴﺣ 1 ﺔﻓﺎﺴﻤﻟا ، ﺔﻴﺳأﺮﻟا n ﺔﻴﻘﻓﻷا ﺔﻓﺎﺴﻤﻟا . E (Correction) = ₂ . 2 nL

Or the Horizontal distance (H) = L . cos θ

ب بﻮﺴﻨﻤﻟا قﺮﻓ ﺔﻟﻻﺪﺑ ) ﻰﺳأﺮﻟا ﺪﻌﺒﻟا :(

Data is the inclined distance (L), and level difference (V). Assume that L2 = V2 + (L-E)2, where E is the correction value.

⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − − − = 2 4₃ . 8 . 2 L V L V L L E 3 4 2 . 8 . 2 L V L V E = + 5 ﻪﻟﻮﻃ ﺐﻠﺼﻟا ﻦﻣ ﻂﻳﺮﺸﺑ ﻂﺧ ﺲﻴﻗ (20.0 m) ﻪﻟﻮﻃ نﺎﻜﻓ (100.0 m) ﻢﻴﺧﺮﺘﻟا راﺪﻘﻣ نﺎآو (5.0 cm) ﻰ ﻘﻴﻘﺤﻟا لﻮﻄﻟا ﻦﻣ ﻂﻳﺮﺸﻠﻟ (20.01m) سﺎﻴﻘﻟا ءﺎﻨﺛأ ةراﺮﺤﻟا ﺔﺟرد ﺖﻧﺎآو (18.0 0C) ةﺮﻳﺎﻌﻤﻟا ءﺎﻨﺛأو (20.0 0C) ﻂﻳﺮﺸﻟا دﺪﻤﺗ ﻞﻣﺎﻌﻣو (9*10 -7₎ ﺌﻣ ﺔﺟرد ﻞﻜﻟ ﻂﻳﺮﺸﻟا ﻞﻴﻣ ﺔﻳوازو ،ﺔﻳﻮ (60 0) ﻂﺨﻠﻟ ﻰﻘﻴﻘﺤﻟا لﻮﻄﻟا ﺐﺴﺣا . ﻞﺤﻟا 1 -ﻂﻳﺮﺸﻟا ﻰﻓ ﻢﻴﺧﺮﺘﻟا ﺢﻴﺤﺼﺘﻟ مﺪﺨﺘﺴﺗ ﻰﺘﻟا ﺔﻗﻼﻌﻟا : .... . 15 . 32 . 3 . 8 2 2 2 − − − = D S D S D H Where: H Horizontal distance

D length of the measuring tape

ﻦﻣ ﺞﺗﺎﻨﻟا ﺄﻄﺨﻟا بﺎﺴﺣ ﻦﻜﻤﻳ ﻰﻟﺎﺘﻟﺎﺑو ﺔﻟدﺎﻌﻤﻟا ﻦﻣ ﻢﻴﺧﺮﺘﻟا : 3 4 2 . 15 . 32 . 3 . 8 D S D S H D E = − = + ًاﺪﺟ ﺮﻴﻐﺻ ﻪﻧﻷ ﺔﻟدﺎﻌﻤﻟا ﻦﻣ ﻰﻧﺎﺜﻟا ﺪﺤﻟا لﺎﻤهإ ﻦﻜﻤﻳو . Correction of sag: 0 . 20 * 3 ) 05 . 0 ( * 8 2 = E = 0.033 cm. 2 -ةراﺮﺤﻟا ﺔﺟرد فﻼﺘﺧا ﻦﻣ ﺞﺗﺎﻨﻟا ﺄﻄﺨﻟا : ) ( .H d₁ d E =α − Where:

E Correction due to temperature change

α coefficient of thermal expansion

d1 degree of temperature at measuring d degree of temperature at calibration. E = 9*10-7* 6 ﺚﻠﺜﻤﺑ ﺖﻄﻴﺣأ ﻩﺎﻴﻣ ﺔآﺮﺒﻟ ﻊﻓر ﻞﻤﻋ ﺪﻨﻋ ABC ﻰﺗﻵﺎآ ﻪﻋﻼﺿأ لاﻮﻃأ : AB = 400.0 m , AC = 250.0 m. ﺔﻄﻘﻨﻟا تﺬﺧأ D ﻋ ﻂﺨﻟا ﻰﻠ BC لﻮﻃ نﺎآ ﺚﻴﺤﺑ BD = 125.0 m ، CD = 150.0 m ﻂﺨﻟا لﻮﻃ ﺐﺴﺣا AD نﺎآ اذأ ، : أ – ًﺎﺤﻴﺤﺻ ﻞﻤﻌﺘﺴﻤﻟا ﺮﻳﺰﻨﺠﻟا لﻮﻃ . ب – ﺮﺘﻤﻟا ﻰﻓ ةدﻮﻘﻔﻣ ﺔﻠﻘﻋ ﻪﺑ ﻞﻤﻌﺘﺴﻤﻟا ﺮﻳﺰﻨﺠﻟا (15) . ﻞﺤﻟا

A

D

B

C

400.0 m

250

.0 m

12

5.

0

m

15

0.

0

m

θ

DC BD BC DB AC DC AB AD ( ) . ( ) . . 2 2 − + = 0 . 150 * 0 . 125 275 125 . ) 250 ( 150 . ) 400 ( 2 2 − + = AD = 311.339 m or, BC AC AB BC AC * * 2 ) ( ) ( ) ( ) cos(θ = 2 + 2 − 2 275 * 250 * 2 ) 400 ( ) 275 ( ) 250 ( ) cos(θ = 2 + 2 − 2 =

(AD)2 = (AC)2 + (CD)2 – 2*AC*CD*cos (θ)

7 ﺔﻌﻴﺒﻄﻟا ﻰﻓ ﺎﻬﻬﺟاﻮﻧ ﻰﺘﻟا تﺎﺒﻘﻌﻟا ﻰﻠﻋ ﺐﻠﻐﺘﻟا ﺔﻴﻔﻴآ ﺢﺿو ب -ﺔﻴﺗﻵا قﺮﻄﻟا ماﺪﺨﺘﺳا ﻦﻜﻤﻳ سﺎﻴﻘﻟا ﻰﻓ ﻖﺋاﻮﻌﻟا ﻰﻠﻋ ﺐﻠﻐﺘﻠﻟ : 1 -A B e c f d 90. 0° B A c d e f A B c d _e A B c d 1 2 4 3

A

B

C

D

E

F

G

H

8 ﻪﺗﺪﻋﺎﻗ ﻰﻟإ لﻮﺻﻮﻟا ﻦﻜﻤﻳ جﺮﺑ سﺎﻴﻗ ﺔﻴﻔﻴآ ﺢﺿو ﻰﻟﻮﻄﻟا سﺎﻴﻘﻟا تاودأ ماﺪﺨﺘﺳﺎﺑ ﻪﺘﻤﻗ ﻰﻟإ لﻮﺻﻮﻟا ﻦﻜﻤﻳ ﻻو . 9 -ﻂﻘﻓ ﻰﻟﻮﻄﻟا سﺎﻴﻘﻟا تاودأ ماﺪﺨﺘﺳﺎﺑ ﻪﻋﺎﻔﺗرإ ﻰﻟإ ﻻو ،ﻪﺗﺪﻋﺎﻗ ﻰﻟإ لﻮﺻﻮﻟا ﻦﻜﻤﻳ ﻻ جﺮﺑ سﺎﻴﻗ ﺔﻴﻔﻴآ ﺢﺿو .

ﺔﻠﺻﻮﺒﻟﺎﺑ ﺔﺣﺎﺴﻤﻟا

تﺎﻔﻳﺮﻌﺗ : 1 -ﻰﺴﻴﻃﺎﻨﻐﻤﻟا لﺎﻤﺸﻟا 2 -ﻰﻓاﺮﻐﺠﻟا لﺎﻤﺸﻟا 3 -فﻼﺘﺧﻹا ﺔﻳواز 4 -ﻂﺨﻟا فاﺮﺤﻧإ 5 -ىﺮﺋاﺪﻟا فاﺮﺤﻧﻹا 6 -ﺮﺼﺘﺨﻤﻟا فاﺮﺤﻧﻹا Summation of internal angles of a closed polygon = (n-2)*180 ………. (1)

Where:

n number of polygon’s sides.

Summation of internal angles of a closed polygon (with the help of member’s bearings) = Σ of back bearings - Σ of front bearings + L*360 ………. (2)

Where:

L number of points whose front bearing is greater than the back bearing of the previous member.

ﻦﻣ ﺮﺒآأ ﻖﺣﻼﻟا ﻂﺨﻠﻟ ﻰﻣﺎﻣﻷا فاﺮﺤﻧﻹا ﺎﻬﺑ ﻰﺘﻟا طﺎﻘﻨﻟا دﺪﻋ ﻖﺑﺎﺴﻟا ﻂﺨﻠﻟ ﻰﻔﻠﺨﻟا فاﺮﺤﻧﻹا

.

Error of internal angles (Δ) = (n-2)*180 –{ Σ of back bearings - Σ of front bearings + L*360 } Allowable error = K. 2.n

Where: K constant

The amount of correction for each direction (e) = n . 2 Δ Examples:

1- A closed polygon (abcd) was created. The bearings of all its members was measured by a compass, and they were as follows:

Member Ab bc Ca cd Da Front Bearing 1800 _{...` 260</sub>0 <sub>40` 61}0 _{50` 350</sub>0 <sub>30` 85}0 _{40`</sub> Back Bearing … 0 <sub>20` 81}0 _{… 240}0 _{40` 170</sub>0 <sub>40` 265}0 _30`

2- The following observations were taken by a compass for a closed traverse. It is required to adjust the bearings of that traverse by the exact method, and the sides of the traverse by “Bowditch” method (length of side’s method).

Side

Length

Front Bearing

Back Bearing

AB 96.23 64o 16` 244o 20` BC 121.15 128o 17` 307o 51` CD 79.16 201o 03` 22o 36` DE 102.77 288o 46` 107o 59` EA 94.75 324o _{16` 144</sub>o <sub>12`}

Solution

ﻞﺤﻟا تاﻮﻄﺧ : 1 -ىواﺰﻟا ﺄﻄﺨﻟا ﺢﻴﺤﺼﺗ : ﻦﻣ ىوﺎﺴﺗ ﻖﻠﻐﻣ ﻊﻠﻀﻣ ىﻷ ﺔﻴﻠﺧاﺪﻟا ﺎﻳاوﺰﻟا عﻮﻤﺠﻣ نأ فوﺮﻌﻤﻟا :

= 180 (n-2)

………. (1)

where n = total number of angles, or sides of the polygon

ﻦﻴ ﺒﻣ ﻮ ه ﺎ ﻤآ ﻖﺑﺎﺴ ﻟا ﻂ ﺨﻠﻟ ﻰ ﻔﻠﺨﻟا فاﺮﺤﻧﻹاو ﻂﺨﻟ ﻰﻣﺎﻣﻷا فاﺮﺤﻧﻹا ﺔﻴﻣﻮﻠﻌﻤﺑ ﻊﻠﻀﻤﻠﻟ ﺔﻴﻠﺧاﺪﻟا ﺎﻳاوﺰﻟا ﺔﻤﻴﻗ ﺔﻓﺮﻌﻣ ﻦﻜﻤﻳو ﻟﺎﺑ ﻢﺳﺮ :

A

B

C

D

E

64°16'0" 128°17'0" 244°20'0" 201°03'0" 307°51'0" 22°36'0" 288°46'0" 107°59'0" 324°16'00" 144°12'00" Angle (abc) = 244o 20` - 128o 17` Angle (cde) = 360 – (288o _{46` - 22</sub>o <sub>36`)}

ﺢﺒﺼﻳ ﻰﻟﺎﺘﻟﺎﺑو : ﻞﻜﺸﻠﻟ ﺔﻴﻠﺧاﺪﻟا ﺎﻳاوﺰﻟا عﻮﻤﺠﻣ = ﺔﻴﻔﻠﺨﻟا تﺎﻓاﺮﺤﻧﻹا عﻮﻤﺠﻣ – ﺔﻴﻣﺎﻣﻷا تﺎﻓاﺮﺤﻧﻹا عﻮﻤﺠﻣ + ل × 360 ………… ) 2 ( نأ ﺚﻴﺣ ) ل = ( ﻣﺎﻣﻷا فاﺮﺤﻧﻹا ﺎﻬﺑ ﻰﺘﻟا ﻂﻘﻨﻟا دﺪﻋ ﻖﺑﺎﺴﻟا ﻂﺨﻠﻟ ﻰﻔﻠﺨﻟا فاﺮﺤﻧﻹا ﻦﻣ ﺮﺒآأ ﻖﺣﻼﻟا ﻂﺨﻠﻟ ﻰ ﺔﻄﻘﻧ ًﻼﺜﻣ ، (d) ﻮه ﺎﻬﻟ ﻖﺣﻼﻟا ﻂﺨﻟا (de) ﻮ ه ﺎ ﻬﻟ ﻖﺑﺎﺴ ﻟا ﻂ ﺨﻟاو (cd) ﻂ ﺨﻠﻟ ﻰﻣﺎ ﻣﻷا فاﺮ ﺤﻧﻹا نﺎ آ اذﺈ ﻓ ، (de) ﻦ ﻣ ﺮ ﺒآأ ﻂﺨﻠﻟ ﻰﻔﻠﺨﻟا فاﺮﺤﻧﻹا (cd) ﺔﻄﻘﻧ نﺈﻓ (d) ﺔﻤﻴﻗ دﺎﺠﻳﻹ ﺔﻣﺪﺨﺘﺴﻤﻟا ﻂﻘﻨﻟا ﻰﻟإ فﺎﻀﺗ ) ل .( ﺔ ﻤﻴﻗ بﺎﺴ ﺤﻟو ) ل ( ﻂ ﺧ ﻰﻧﺎ ﺜﻟ ﻰﻣﺎ ﻣﻷا فاﺮﺤﻧﻹﺎ ﺑ أﺪ ﺒﻧ ﻖﺑﺎﺴ ﻟا لﺎ ﺜﻤﻟا ﻰ ﻓ (bc) ﻂ ﺨﻠﻟ ﻰ ﻔﻠﺨﻟا فاﺮﺤﻧﻹﺎ ﺑ ﻪ ﻧرﺎﻘﻧو ﻖﺑﺎﺴﻟا (ab) نرﺎﻘﻧ ىأ 1280 _{17`</sub> ﻊﻣ 2440 <sub>20`} ﻰﻟﺎﺘﻟا لوﺪﺠﻟا ﻰﻓ تﺎهﺎﺠﺗﻹا ﻦﻣ ﻦﻴﺒﻣ ﻮه ﺎﻤآ اﺬﻜهو ، :

Point

Line

Front Bearing

Back Bearing

A AB 64o 16` 244o 20` B BC 128o 17` 307o 51` C CD 201o 03` 22o 36` D DE 288o 46` 107o 59` E EA 324o 16` 144o 12` Summation 1006o _{38` 826</sub>o <sub>58`} ىوﺎﺴ ﻳ ﻖﺑﺎﺴ ﻟا ﻂ ﺨﻠﻟ ﻰ ﻔﻠﺨﻟا فاﺮ ﺤﻧﻹا ﻦ ﻋ ﻖﺣﻼﻟا ﻂﺨﻠﻟ ﻰﻣﺎﻣﻷا فاﺮﺤﻧﻹا ﺎﻬﻴﻓ ﺪﻳﺰﻳ ﻰﺘﻟا ﻂﻘﻨﻟا دﺪﻋ نأ ﺪﺠﻧ اﺬﻜهو (2) ﺎ ﻤه ﻰﺘﻄﻘﻧ (d, e) ﻹا نأ ﺔﻈﺣﻼﻣ ﻊﻣ ، ﺮﻴﺧﻷا ﻊﻠﻀﻠﻟ ﻰﻔﻠﺨﻟا فاﺮﺤﻧﻹﺎﺑ نرﺎﻘﻳ لوﻷا ﻊﻠﻀﻠﻟ ﻰﻣﺎﻣﻷا فاﺮﺤﻧ . ل = 2 ىواﺰﻟا ﺄﻄﺨﻟا ﺔﻤﻴﻗ دﺎﺠﻳإ ﻰه ﺔﻴﻟﺎﺘﻟا ةﻮﻄﺨﻟا (Δ) ﺔﻟدﺎﻌﻤﻟا ﻦﻣ : ﺔﻟدﺎﻌﻤﻟا ﻦﻣ ﺔﺑﻮﺴﺤﻤﻟا ﺎﻳاوﺰﻟا عﻮﻤﺠﻣ ) 1 ( – ﺔﻟدﺎﻌﻤﻟا ﻦﻣ ﺔﺑﻮﺴﺤﻤﻟا ﺎﻳاوﺰﻟا عﻮﻤﺠﻣ ) 2 ( Δ = 180 (5-2) – { (826o 58` - 1006o 38`) + 2 * 360 } = 540.0 – 540o 20` = -00o 20` ﺄﻄﺨﻟا نﺎآ اذإ ) ﺐﻟﺎﺳ ( ﺐﺠﻳ ﻪﻧﺈﻓ ةدﺎﻳز و ﺔﻴﻣﺎﻣﻷا تﺎﻓاﺮﺤﻧﻹا ﻞﻴﻠﻘﺗ ﺔﻴﻔﻠﺨﻟا تﺎﻓاﺮﺤﻧﻹا ﺄﻄﺨﻟا نﺎآ اذإ ) ﺐﺟﻮﻣ ( ﺐﺠﻳ ﻪﻧﺈﻓ ﻞﻴﻠﻘﺗ و ﺔﻴﻣﺎﻣﻷا تﺎﻓاﺮﺤﻧﻹا ةدﺎﻳز ﺔﻴﻔﻠﺨﻟا تﺎﻓاﺮﺤﻧﻹا ﺎﻌﻤﻟا ﻦﻣ ﻪﺑﺎﺴﺣ ﻦﻜﻤﻳ ﻪﺑ حﻮﻤﺴﻤﻟا ﺄﻄﺨﻟاو ﺔﻟد :

Allowable error = k. 2.n, k is constant, n: number of internal angles.

ىوﺎﺴﺗ ﻰهو ةدﻮﺻﺮﻤﻟا تﺎهﺎﺠﺗﻹا دﺪﻋ ﻰﻠﻋ عزﻮﻳ ﺄﻄﺨﻟا اﺬه ) عﻼﺿﻷا دﺪﻋ × 2 = ( 5 × 2 = 10 تﺎهﺎﺠﺗا ﺢﻴﺤﺼﺘﻟا ﺔﻤﻴﻗ = ﺄﻄﺨﻟا ﺔﻤﻴﻗ ÷ ﻞ ﻜﻟ ﺢﻴﺤﺼ ﺘﻟا نﻮ ﻜﻳ نأ ﻞﻀﻔﻳو ، تﺎهﺎﺠﺗﻹا دﺪﻋ ﺔ ﻤﻴﻗ ﺖ ﻧﺎآ اذإ ﻚﻟﺬ ﻟ ، ﺢﻴﺤ ﺻ راﺪ ﻘﻣ ﻩﺎ ﺠﺗإ ﻰﺗﻵا ﻢﺘﻳ ﻪﻧﺈﻓ ﺮﺴآ ﺪﺣاﻮﻟا ةﺎﺠﺗﻺﻟ ﺢﻴﺤﺼﺘﻟا :

error for each direction (e) = n . 2

Δ

, (fraction) ex: assume that n = 5, Δ = 46,

e = 5 * 2 46 = 4.6 1 ﺄﻄﺨﻠﻟ ﻦﻴﺘﺤﻴﺤﺻ ﻦﻴﺘﻤﻴﻗ ﻦﻴﻌﻧ (e) ﺎﺒﻣ ﺮﻐﺻأ ﺔﺤﻴﺤﺼﻟا ﺔﻤﻴﻘﻟا ﻚﻠﺗ ﻢﺛ ﺮﺴﻜﻟا ﺔﻤﻴﻗ ﻦﻣ ةﺮﺷ + 1 ﺎﻤه ﺢﻴﺤﺼﺘﻠﻟ ﻦﻴﺘﺤﻴﺤﺼﻟا ﻦﻴﺘﻤﻴﻘﻟا (4, 4+1) . e1 = 4, and e2 = 5 2 ﺢﻴﺤﺼﺘﻟا ﺬﺧﺄﺗ ﻰﺘﻟا تﺎهﺎﺠﺗﻹا دﺪﻋ ﻦﻴﻌﻧ (e2) d1 = Δ - 2.n.e1

= 46 – 2*5*4 = 6 directions (takes the correction 5)

3 -ﻴﺤﺼﺘﻟا ﺬﺧﺄﺗ ﻰﺘﻟا تﺎهﺎﺠﺗﻹا دﺪﻋ ﺢ (e1) d2 = 2.n – d1

d2 = 10 – 6 = 4 directions (takes the correction 4)

ﻰﻠﺻﻷا لﺎﺜﻤﻟا ﻰﻟإ ةدﻮﻌﻟﺎﺑ

:

Δ = -20.0`, e = n . 2 Δ = -20.0 / 10 = -2.0` راﺪﻘﻤﺑ ﺔﻴﻣﺎﻣﻷا تﺎﻓاﺮﺤﻧﻹا ﻊﻴﻤﺟ ةدﺎﻳز ﻢﺘﻳ (2.0`) راﺪﻘﻤﺑ ﺔﻴﻔﻠﺨﻟا تﺎﻓاﺮﺤﻧﻹا ﻊﻴﻤﺟ ﻞﻴﻠﻘﺗو (2.0`) .

Line Front

Bearing Back

Bearing Corrected

Front Bearing

Corrected

Back Bearing

AB 64o 16` 244o 20` 64o 18` 244o 18` BC 128o 17` 307o 51` 128o 19` 307o 49` CD 201o 03` 22o 36` 201o 05` 22o 34` DE 288o 46` 107o 59` 288o 48` 107o 57` EA 324o 16` 144o 12` 324o 18` 144o 10` Summation 1006o 48` 826o 48` ﺢﻴﺤﺼﺘﻟا ﺪﻌﺑ ﺔﻴﻠﺧاﺪﻟا ﺎﻳاوﺰﻟا عﻮﻤﺠﻣ = = 826o 48` - 1006o 48` + 2 * 360 = 540o o.k.

2 -ﺔﻴﻠﺤﻤﻟا ﺔﻴﺑذﺎﺠﻠﻟ ﺔﻘﻴﻗﺪﻟا ﺔﻘﻳﺮﻄﻟﺎﺑ تﺎﻓاﺮﺤﻧﻹا ﺢﻴﺤﺼﺗ :

δ

A

F

a

1

F

₂

b

δ

B

ﻦﻴﺒﻤﻟا ﻞﻜﺸﻟا ﻰﻓ (δa , δb) ﻦﻴﺘﻄﻘﻨﻟا ﺪﻨﻋ ﻰﺴﻴﻃﺎﻨﻐﻤﻟا لﺎﻤﺸﻟاو ﻰﻓاﺮﻐﺠﻟا لﺎﻤﺸﻟا ﻦﻴﺑ قﺮﻔﻟا (a , b) ﺎﺣ ﻰﻓ ، مﺪﻋ ﺔﻟ ىوﺎﺴ ﻳ ﻂ ﺨﻠﻟ ﻰﻣﺎ ﻣﻷاو ﻰ ﻔﻠﺨﻟا فاﺮ ﺤﻧﻹا ﻦﻴ ﺑ قﺮﻔﻟا نﺈﻓ ﺔﻴﻠﺤﻣ ﺔﻴﺑذﺎﺟ دﻮﺟو 180 ﺎ ﻤآ ﺔ ﻴﻠﺤﻣ ﺔ ﻴﺑذﺎﺟ دﻮ ﺟو ﺔ ﻟﺎﺣ ﻰ ﻓ ﺎ ﻣأ ، ﻞﻜﺸﻟﺎﺑ : F2 + δb – 180 = F1 + δa δb - δa = 180 – (F2 – F1) ………… (3) ﻢﻴﻗ دﺎﺠﻳﻹو (δa , δb , δc , …….) ﺔﻴﺗﻵا تاﻮﻄﺨﻟا ﻊﺒﺘﻧ : 1 -ﺑ قﺮﻔﻟا ﻦﻌﻴﻧ ﻂﺧ ﻞﻜﻟ ﻰﻣﺎﻣﻷاو ﻰﻔﻠﺨﻟا فاﺮﺤﻧﻹا ﻦﻴ ) ىواﺰﻟا ﺢﻴﺤﺼﺘﻟا ﺪﻌﺑ ( 2 قﺮﻔﻟا حﺮﻄﻧ ) ةرﺎﺷإ نوﺪﺑ ( ﻦﻣ 180 ةرﻮﺼﻟا ﻰﻠﻋ تﻻدﺎﻌﻤﻟا ﻦﻣ دﺪﻋ ﻰﻠﻋ ﻞﺼﺤﻨﻓ : δb - δa = k1 δc - δb = k2 δd - δc = k3 .. ……… تﻻدﺎﻌﻤﻟا ﻩﺬه عﻮﻤﺠﻣ نﻮﻜﻳ نأ ﺐﺠﻳو = نأ ىأ ،ﺮﻔﺻ : k1 + k2 + k3 + ………. = 0.0

Line

Corrected

Front Bearing

Corrected

Back Bearing

Difference

Back - Front

180 - d

AB 64o 18` 244o 18` 180 0.0 BC 128o 19` 307o 49` 179o 30` 0o 30` CD 201o 05` 22o 34` -178o 31` 1o 29` DE 288o 48` 107o 57` -180o 51` -00 51` EA 324o 18` 144o 10` -180o 8` -00 8` δb - δa = 0.0 δc - δb = 0o _{30`</sub> δd - δc = -1o <sub>29`} δe - δd = 0o _{51`</sub> δa - δe = 00 <sub>8`} for check 0o 30` - 1o 29` + 0o 51` + 00 8` = 0.0 δb = δa = 0.0 δc = 00 _{30`</sub> δd = -1o <sub>29` + δc = -1}o _{29` + 0</sub>o <sub>30` = -0}o _{59`</sub> δe = 0o <sub>51` + δd = 0}o _{51` - 0</sub>o <sub>59` = -0}o _8` the last equation is used for check

δa - δe = 0.0 – (-0o _{8`) = 0</sub>o <sub>8` o.k.}

Line

δfront δBack

Corrected

Front Bearing

Corrected

Back Bearing

AB 0.0 0.0 64o 18` + 0.0 = 64o 18` 244 o <sub>18` + 0.0</sub> = 244o 18` BC 0.0 0o 30` 128o 19` + 0.0 = 128o _{19`</sub> 307 o <sub>49` + 0}o _{30`</sub> = 308o <sub>19`} CD 0o 30` -0o 59` 201o 05` + 00 30` = 201o _{35`</sub> 22 o <sub>34` - 0}o _{59`</sub> = 21o <sub>35`} DE -0o 59` -0o 8` 288o 48` - 0o 59` = 287o 49` 107o 57` - 0o 8` = 107o 49` EA -0o 8` 0.0 324o 18` - 0o 8` = 324o 10` 144o 10` + 0.0 = 144o 10`

ﻞﻔﻘﻟا ﺄﻄﺧ ﺢﻴﺤﺼﺗ ) تﺎﺒآﺮﻤﻟا ﻖﻳﺮﻃ ﻦﻋ :(

W

es

t

E

as

t

+

_

+

_

+

+

_

_

ﻪﻌﻗﻮﻣ فﺮﻌﻧ ﻰﻟﺎﺘﻟﺎﺑو ،ﻊﻠﺿ ﻞﻜﻟ ﺮﺼﺘﺨﻤﻟا فاﺮﺤﻧﻹا دﺪﺤﻧ :

N.E (North-East) the Vertical component is (+) and Horizontal component is (+) N.W (North-West) the Vertical component is (+) and Horizontal component is (-) S.E (South-East) the Vertical component is (-) and Horizontal component is (+) S.W (South-West) the Vertical component is (-) and Horizontal component is (-) Then,

The vertical component (V) = L * cos (θ) The horizontal component (H) = L * sin (θ),

Where θ is the reduced bearing (ﺮﺼﺘﺨﻤﻟا فاﺮﺤﻧﻹا)

Line Length Corrected

Front Bearing

Reduced

Bearing

Vertical

Component

Horizontal

Component

AB 96.23 64o 18` N.E 64o 18` +41.731 +86.711 BC 121.15 128o 19` S.E 51o 41` -75.114 +95.054 CD 79.16 201o 35` S.W 21o 35` -73.61 -29.119 DE 102.77 287o _{49` N.W 72</sub>o <sub>11` +31.445 -97.841} EA 94.75 324o 10` N.W 35o 50` +76.816 -55.469 494.06 +1.2684 -0.6655 Length of error =

∑

(

)

2 +

∑

(

)

2 Components Horizontal Components Vertical = _(1.2684)2 _+(−0.6655)2 _{= 1.4324}

% of error = 06 . 494 4324 . 1 = lines traverse of length Total error of Length = 2.8992*10-3_. ﻞ ﻔﻘﻟا ﺄ ﻄﺧ ﺔﺒﺴﻧ زوﺎﺠﺘﻳ ﻻأ ﻰﻠﻋ تﺎﻔﺻاﻮﻤﻟا ﺺﻨﺗ (1:100) ﻊ ﻣ ،ةﺪﻳﺪﺸ ﻟا ﺔ ﻴﻓاﺮﻏﻮﺒﻄﻟا تاذ ةﺮﻋﻮ ﻟا ﻰ ﺿارﻷا ﻰ ﻓ زوﺎﺠﺘﻳ ﻻأو ،ﺮﻳﺰﻨﺠﻟﺎﺑ سﺎﻴﻘﻟا (1:500) نﺪﻤﻟا ﻰﻓ . ﻰﺗﻵﺎآ ﻂﺧ ﻞﻜﻟ ﺔﻴﺳأﺮﻟا ﺔﺒآﺮﻤﻟا ﺢﻴﺤﺼﺗ ﻢﺘﻳ : ﻂﺨﻟا لﻮﻃ ﺔﻴ ﺳأﺮﻟا ﺔ ﺒآﺮﻤﻟا ﺢﻴﺤﺼ ﺗ ﻂﺨﻠﻟ = طﻮﻄﺧ لاﻮﻃأ عﻮﻤﺠﻣ ﻊﻠﻀﻤﻟا × ﺄﻄﺨﻟ ﺔﻴﺳأﺮﻟا ﺔﺒآﺮﻤﻟا ﻞﻔﻘﻟا ﻰﺗﻵﺎآ ﻂﺧ ﻞﻜﻟ ﺔﻴﻘﻓﻷا ﺔﺒآﺮﻤﻟا ﺢﻴﺤﺼﺗ ﻢﺘﻳو : ﻂﺨﻟا لﻮﻃ ﻂ ﺨﻠﻟ ﺔ ﻴﻘﻓﻷا ﺔﺒآﺮﻤﻟا ﺢﻴﺤﺼﺗ = طﻮﻄﺧ لاﻮﻃأ عﻮﻤﺠﻣ ﻊﻠﻀﻤﻟا × ﺄﻄﺨﻟ ﺔﻴﻘﻓﻷا ﺔﺒآﺮﻤﻟا ﻞﻔﻘﻟا ﻄﺨﻟ ﺔﻴ ﺳأﺮﻟا تﺎ ﺒآﺮﻤﻟا ﺢﻴﺤﺼ ﺗ نﺈ ﻓ ،ﺔ ﺒﺟﻮﻣ ﻞ ﻔﻘﻟا ﺄ ﻄﺨﻟ ﺔﻴ ﺳأﺮﻟا ﺔ ﺒآﺮﻤﻟا ﺖ ﻧﺎآ اذإ ﻪ ﻧأ ﻆ ﺣﻼﻧو نﻮ ﻜﺗ ﻊ ﺑﺮﻤﻟا طﻮ ﺄﻄﺨﻟا ةرﺎﺷإ ﺲﻜﻋ نﻮﻜﺗ ﺢﻴﺤﺼﺘﻟا ةرﺎﺷإ نﺈﻓ اﺬﻜهو ،ﺔﺒﻟﺎﺳ .

Line Length Vertical Component Horizontal Component Vl. Corr. Hl. Corr. Corrected Vl. Comp. Corrected Hl. Comp. AB 96.23 +41.731 +86.711 -0.2471 +0.1296 41.484 86.841 BC 121.15 -75.114 +95.054 -0.3110 +0.1632 -75.425 95.217 CD 79.16 -73.61 -29.119 -0.2032 +0.1066 -73.813 -29.012 DE 102.77 +31.445 -97.841 -0.2638 +0.1384 31.181 -97.703 EA 94.75 +76.816 -55.469 -0.2433 +0.1276 76.573 -55.341 494.06 +1.2684 -0.6655 0.0 0.0 ﺢﻴﺤﺼﺗ ﺔﻴﻧﺎﻴﺒﻟا ﺔﻘﻳﺮﻄﻟﺎﺑ ﺄﻄﺨﻟا :

A

A`

B

C

D

A

B`

C`

D`

A

_B

_C

_D

Area:

1- It is required to find the area of the closed traverse, if the coordinates of its corners are as follows: Point A B C D E F X-Coordinate 10 45 87 138 104 36 Y-Coordinate 40 77 65 83 3 14

A

B

C

D

E

F

_X

Area = 0.50 { (X1.Y2 + X2.Y3 + X3.Y4 + X4.Y5 + X5.Y6 + X6.Y1) – (Y1.X2 + Y2.X3 + Y3.X4 + Y4.X5 + Y5.X6 + Y6.X1) }

= 0.50 { (10 * 77 + 45 * 65 + 87 * 83 + 138 * 3 + 104 * 14 + 36 * 40) – (40 * 45 + 77 * 87 + 65 * 138 + 83 * 104 + 3 * 36 + 14 * 10) = 0.50 * 12123 = 6061.50

2- The area by the lines’ components: A B C D E

Line X component Y component Vertical 2*Area

AB 150 -275 Y1 = -275 -41250 BC 175 -135 2Y1+Y2 = -685 -119875 CD -65 -130 2(Y1+Y2)+Y3 = -950 61750 DE -160 300 2(Y1+Y2+Y3)+Y4 = -780 124800 EA -100 240 2(Y1+Y2+Y3+Y4)+Y5 = -240 24000 49425 Area = 0.50 * 49425 = 24712.50

ﺔﻳﺮﺘﻣﻮﻴآﺎﺘﻟا ﺔﺣﺎﺴﻤﻟا

ﺔ ﺻﺎﺨﻟا تﺎﻓﺎ ﺿﻹا ﺾﻌﺒ ﺑ ﺰ ﻬﺠﻣ ﺖ ﻴﻠﻳدﻮﻴﺗ ماﺪﺨﺘ ﺳﺎﺑ ﻢﺘ ﺗو ،ﺔﻌﻳﺮﺴ ﻟا ﺔﺣﺎﺴﻤﻟا ﻰه . ﻊ ﻣو تﺎﻓﺎ ﺿﻹا ﻩﺬ ه ﺔﻄ ﺳاﻮﺑو ﻜﻤﻳ ةﺪﻋﺎﺴﻤﻟا ةﺰﻬﺟﻷا ﺾﻌﺑ ماﺪﺨﺘﺳا ﺔﻟﻮﻘﻌﻣ ﺔﻗﺪﺑو ﺔﻋﺮﺴﺑ ﺔﻴﺳأﺮﻟاو ، ﺔﻴﻘﻓﻷا تﺎﻓﺎﺴﻤﻟا دﺎﺠﻳإ ﻦ .

1

-ﺎﻳدﺎﺘﺳﻹا تاﺮﻌﺷ ﺔﻘﻳﺮﻃ

:

ﺔﻴﺴﻴﺋﺮﻟا ةﺮﻌﺸﻟا ﻞﻔﺳأو ﻰﻠﻋأ ﻦﻴﺘﻴﻓﺎﺿإ ﻦﻴﺘﻴﻘﻓأ ﻦﻴﺗﺮﻌﺸﺑ ﻪﻠﻴﻟد دوﺰﻳ ﺖﻴﻠﻳدﻮﻴﺗ ماﺪﺨﺘﺳﺎﺑ ﻢﺘﺗو : X X X t k

D

H

a b c a b c 1 1 1 1 2 m o m ﺔﻴﺌﻴﺸﻟا ﺔﺳﺪﻌﻠﻟ ىﺮﺼﺒﻟا ﺰآﺮﻤﻟا a , c ﺎﻳدﺎﺘﺳﻹا ﺎﺗﺮﻌﺷ b ﻰﻄﺳﻮﻟا ﺔﻴﻘﻓﻷا ةﺮﻌﺸﻟا a1 , b1 , c1 ﺔﻣﺎﻘﻟا ﻊﻣ ﻰﻄﺳﻮﻟا ةﺮﻌﺸﻟاو ، ﺎﻳدﺎﺘﺳﻹا ﻰﺗﺮﻌﺷ ﻊﻃﺎﻘﺗ X ﺔﻴﺌﻴﺸﻟا ﺔﺳﺪﻌﻠﻟ ىرﺆﺒﻟا ﺪﻌﺒﻟا X1 ﺔﻴﺌﻴﺸﻟا ﺔﺳﺪﻌﻠﻟ ىﺮﺼﺒﻟا ﺰآﺮﻤﻟاو ﺔﻣﺎﻘﻟا ﻦﻴﺑ ﺔﻴﻘﻓﻷا ﺔﻓﺎﺴﻤﻟا X2 تاﺮﻌﺸﻟا ﻞﻣﺎﺣ ىﻮﺘﺴﻣو ، ﺔﻴﺌﻴﺸﻟا ﺔﺳﺪﻌﻟا ﺰآﺮﻣ ﻦﻴﺑ ﻰﻘﻓﻷا ﺪﻌﺒﻟا t ناروﺪ ﻟ ﻰ ﺳأﺮﻟا رﻮﺤﻤﻟاو ، ﺔﻴﺌﻴﺸﻟا ﺔ ﺳﺪﻌﻠﻟ ىﺮﺼﺒﻟا ﺰآﺮﻤﻟا ﻦﻴﺑ ﻰﻘﻓﻷا ﺪﻌﺒﻟا رﺎﻈﻨﻤﻟا . H a1.b1 = ، ﺎﻳدﺎﺘﺳﻹا ﻰﺗﺮﻌﺷ ﻦﻴﺑ ﺔﻣﺎﻘﻟا ﻰﻠﻋ ﺔﻋﻮﻄﻘﻤﻟا ﺔﻓﺎﺴﻤﻟا o ﺔﻴﺌﻴﺸﻟا ﺔﺳﺪﻌﻟا ةرﺆﺑ ) ( . X t ab X H D= + + ﺚﻴﺣ (ab) = تاﺮﻌﺸﻟا ﻞﻣﺎﺣ لﻮﻃ ﺔﺳﺪﻌﻠﻟ ىرﺆﺒﻟا ﺪﻌﺒﻟا ﻟا ﺔﻴﺌﻴﺸ K1 abX = ىﺮﺘﻣﻮﻴآﺎ ﺘﻟا ﺖ ﺑﺎﺜﻟا ﻰﻤﺴ ﻳ = تاﺮﻌﺸﻟا ﻞﻣﺎﺣ لﻮﻃ

(X + t ) = K ﻰﻓﺎﺿﻹا ﺖﺑﺎﺜﻟا ﻰﻤﺴﻳ = ﺔﻴﺌﻴﺸﻟا ﺔﺳﺪﻌﻠﻟ ىرﺆﺒﻟا ﺪﻌﺒﻟا + ﻰ ﻟإ ﺔﻴﺌﻴﺸ ﻟا ﺔ ﺳﺪﻌﻟا ىﻮﺘﺴﻣ ﻦﻣ ﺔﻓﺎﺴﻤﻟا ﻰﺳأﺮﻟا رﻮﺤﻤﻟا ﻦﻴﺑ ﻰﻓﺎﺿﻹا ﺖﺑﺎﺜﻟا ﺔﻤﻴﻗ حواﺮﺘﻳو 30 ، 60 ﻢﺳ . D = H . K1 + K ………. ( ﻟا ﺔ ﻣﺎﻘﻟا ﻰ ﻟإ ﺪ ﺻﺮﻟا ﺔ ﻄﻘﻧ ﻦﻴﺑ ﺔﻴﻘﻓﻷا ﺔﻓﺎﺴﻤ = ﻰﻠﻔﺴ ﻟاو ﺎ ﻴﻠﻌﻟا ﻦﻴﺗﺮﻌﺸ ﻟا ةءاﺮ ﻗ قﺮ ﻓ × ىﺮﺘﻣﻮﻴآﺎ ﺘﻟا ﺖ ﺑﺎﺜﻟا + ﺖ ﺑﺎﺜﻟا ﻰﻓﺎﺿﻹا . ﺔﻠﺋﺎﻣ ةﺪﺻر ﺬﺧأ ﺔﻟﺎﺣ ﻰﻓو ) ﺔﻌﻔﺗﺮﻣ ﺮﻈﻨﻟا ﺔﻳواز :( H X X t D k b c a m c₁ 1 b X 2 a1 1 θ Y Y1 θ θ ( ).cos cos . . 2 t X ab X H D= + + D = H . K1 . cos2 θ + K . cos θ ………. ( عﺎﻔﺗرإ دﺎﺠﻳإ ﻦﻜﻤﻳو زﺎﻬﺠﻟﺎﺑ رﺎﻤﻟا ﻰﻘﻓﻻا ىﻮﺘﺴﻤﻟا ﻦﻋ ةدﻮﺻﺮﻤﻟا ﺔﻄﻘﻨﻟا ضﺎﻔﺨﻧإ وأ (Y) Y = D . tan θ ………. (

لﺎﺜﻣ : 1 تاﺮﻌﺸ ﻟا ةءاﺮ ﻗ ﺖ ﻧﺎﻜﻓ ،ﺔ ﻴﻘﻓأ ﺮﻈﻨﻟا ﺔﻳواز ﺖﻧﺎآو ،ﺎﻳدﺎﺘﺳﻹا تاﺮﻌﺸﺑ دوﺰﻣ ﺖﻴﻟدﻮﻴﺗ زﺎﻬﺠﺑ ﺔﻄﻘﻧ تﺪﺻر 2.613 – 4.106 ﺖ ﻧﺎﻜﻓ ﻂﻳﺮ ﺷ ﺔﻄ ﺳاﻮﺑ ﺔﻓﺎﺴ ﻤﻟا ﺖﺴ ﻴﻗ ﻢ ﺛ ، 150 م . ﺔ ﻳواﺰﺑ ًﻼﺋﺎ ﻣ ﺮ ﻈﻨﻟا ﻂ ﺧ نﺎ آو ، ىﺮ ﺧأ ﺔ ﻄﻘﻧ تﺪ ﺻر 7 تﺎ ﺟرد تاﺮﻌﺸ ﻟا ةءاﺮ ﻗ ﺖ ﻧﺎآو 1.146 – 3.154 ﺖ ﻧﺎﻜﻓ ﻂﻳﺮﺸ ﺑ ﺔﻓﺎﺴ ﻤﻟا ﺖﺴ ﻴﻗو 200 ﺖ ﺑﺎﺜﻟاو ،ىﺮﺘﻣﻮﻴآﺎ ﺘﻟا ﺖ ﺑﺎﺜﻟا ﺪ ﺟوأ ،ﺮ ﺘﻣ ﻰﻓﺎﺿﻹا . Solution ﻰﻘﻓأ ﺮﻈﻨﻟا ﻂﺧ ﺔﻟﺎﺣ ﻰﻓ : D = H . K1 + K 150 = (4.106-2.613) . K1 + K ………. (1) ﺔﻳواﺰﺑ ﻞﻴﻤﻳ ﺮﻈﻨﻟا ﻂﺧ ﺔﻟﺎﺣ ﻰﻓ 7 تﺎﺟرد : D = H . K1 . cos2 θ + K . cos θ 200 = (3.154 – 1.146) . K1.cos2 _{( 7 ) + K . cos ( 7 ) ………. (2)} ﻦﻴﺘﻟدﺎﻌﻤﻟا ﻞﺤﺑ 1 ، 2 نأ ﺞﺘﻨﻳ ًﺎﻴﻧﺁ : K1 = 103 (ىﺮﺘﻣﻮﻴآﺎﺘﻟا ﺖﺑﺎﺜﻟا) K = 3.785 (ﻰﻓﺎﺿﻹا ﺖﺑﺎﺜﻟا) 2 ﺟ ﻰﻓ تاﺮﻌﺸ ﻟا ﻦ ﻣ جوز ﻞ آ ﻦﻴ ﺑ ﺔﻓﺎﺴ ﻤﻟا ﺖ ﻧﺎآ تاﺮﻌ ﺷ ﻞ ﻣﺎﺤﺑ دوﺰﻣ ﺖﻴﻟدﻮﻴﺗ زﺎﻬ 1 / 40 ﺔ ﺳﺪﻌﻠﻟ ىرﺆ ﺒﻟا ﺪ ﻌﺒﻟاو ،ﺔ ﺻﻮﺑ ﺔﻴﺌﻴﺸﻟا (X) = 9 ﺔﻴﺌﻴﺸﻟا ﺔﺳﺪﻌﻟا ىﻮﺘﺴﻣ ﻰﻟإ زﺎﻬﺠﻠﻟ ﻰﺳأﺮﻟا رﻮﺤﻤﻟا ﻦﻴﺑ ﺔﻓﺎﺴﻤﻟاو ،ﺔﺻﻮﺑ (t) = 4.5 ﺔﺻﻮﺑ . ﺔﻄﻘﻧ تﺪﺻر ﺔ ﻳواﺰﺑ ﻞ ﻴﻤﻳ ﺮ ﻈﻨﻟا ﻂ ﺧ نﺎ آو زﺎ ﻬﺠﻟا اﺬ ه ﺔﻄ ﺳاﻮﺑ 9 ،تﺎ ﺟرد تاﺮﻌﺸ ﻟا ةءاﺮ ﻗ ﺖ ﻧﺎآو 1.63 – 2.14 2.65 ﺔ ﻣﺎﻗ ﻰ ﻠﻋ ﺔﺻﻮﺒﻟاو مﺪﻘﻟﺎﺑ ﺔﻤﺴﻘﻣ . ةدﻮﺻﺮﻤﻟا ﺔﻄﻘﻨﻟا بﻮﺴﻨﻣ نﺎآ اذإو ،ةدﻮﺻﺮﻤﻟا ﺔﻄﻘﻨﻟاو زﺎﻬﺠﻟا ﻦﻴﺑ ﺔﻴﻘﻓﻷا ﺔﻓﺎﺴﻤﻟا ﺪﺟوأ = 80 مﺪﻗ ﺮﻈﻨﻟا ﻂﺧ عﺎﻔﺗرإو، 4.50 ﺪﺻﺮﻟا ﺔﻄﻘﻧ بﻮﺴﻨﻣ ﺪﺟوأ ،مﺪﻗ ) زﺎﻬﺠﻟا ﺎﻬﺑ ﻰﺘﻟا ﺔﻄﻘﻨﻟا .( Solution ab = 2 * 40 1 = 20 1 inch X = 9.0 inch , then K1 = =9.0*20.0=180 ab X (dimensionless value) K = X + t = 9.0 + 4.50 = 13.50 inch = 13.50 / 12.0 = 1.125 Ft H = 2.65 – 1.63 = 1.02 Ft D = H . K1 . cos2 θ + K . cos θ = 1.02 * 180 * cos2 _{(9) + 1.125 * cos (9) = 182.451 Ft} ةدﻮﺻﺮﻤﻟا ﺔﻄﻘﻨﻟا ﻦﻴﺑو ، ﺮﻈﻨﻟا ﻂﺧ ﻦﻴﺑ ﺔﻴﺳأﺮﻟا ﺔﻓﺎﺴﻤﻟا : Y = D . tan θ = 182.451 * tan ( 9 ) = 28.879 Ft

V Y Y θ D a c b 1 b H 1 c 1 1 a . ةدﻮﺻﺮﻤﻟا ﺔﻄﻘﻨﻟاو ﺪﺻﺮﻟا ﺔﻄﻘﻧ ﻦﻴﺑ بﻮﺴﻨﻤﻟا قﺮﻓ : = Y – Y1 + V = 28.879 – 2.14 + 4.5 = 31.239 Ft ﺪﺻﺮﻟا ﺔﻄﻘﻧ بﻮﺴﻨﻣ : = 80 – 31.239 = 48.761 Ft

H b a θ φ D c

2

-لﻼﻈﻟا ﺔﻘﻳﺮﻃ

:

مﺎ ﻌﻟا ﻰ ﻟإ ﺔ ﻘﻳﺮﻄﻟا ﻩﺬه روﺬﺟ ﻊﺟﺮﺗو 1850 ﻰ ﺳأﺮﻟا ﺪ ﻌﺒﻟاو ﺔ ﻴﻘﻓﻷا ﺔﻓﺎﺴ ﻤﻟا ﻦﻴ ﻴﻌﺗ ﻦ ﻜﻤﻳ ﺔ ﻘﻳﺮﻄﻟا ﻩﺬ ه ﺔﻄ ﺳاﻮﺑو ، ﺪﺻﺮﻠﻟ نﺎﺘﻟﺎﺣ ﺪﺟﻮﺗو ، ىدﺎﻋ ﺖﻴﻟدﻮﻴﺗ ماﺪﺨﺘﺳﺎﺑ : أ -ﺔﻴﻘﻓأ ةﺪﺻر ﺬﺧﺄﺑ ﺢﻤﺴﺗ ضرﻷا ﺔﻌﻴﺒﻃ ﺖﻧﺎآ اذإ : ﻧ ﺪ ﻨﻋ ﺖ ﻴﻟدﻮﻴﺘﻟا ﻊﻀ ﻧ ﺪ ﺻﺮﻟا ﺔ ﻄﻘ ) ﺔ ﻠﺘﺤﻤﻟا ﺔ ﻄﻘﻨﻟا ةدﺎ ﻋ ﺎ ﻬﻴﻠﻋ ﻖ ﻠﻄﻳ ( ﺬ ﺣﺄﻧ ﻢ ﺛ ،ةدﻮ ﺻﺮﻤﻟا ﺔ ﻄﻘﻨﻟا ﺪ ﻨﻋ ﺔ ﻣﺎﻘﻟاو ، ﻦﻜﺘﻟو ،ًﺎﻴﻘﻓأ ﺮﻈﻨﻟا ﻂﺧ نﻮﻜﻳ ﺎﻣﺪﻨﻋ ﺔﻣﺎﻘﻟا ةءاﺮﻗ (b) . ﺔ ﻨﻴﻌﻣ ﺔﻳواﺰﺑ رﺎﻈﻨﻤﻟا ﻞﻴﻤﻧ ﻢﺛ ) ﻞﻔ ﺳأ ﻰ ﻟإ وأ ﻰ ﻠﻋأ ﻰ ﻟإ ( ةءاﺮ ﻗ ﻦﻴ ﻌﻧو ﺔﻴﻧﺎﺜﻟا ﺔﻟﺎﺤﻟا ﻰﻓ ﺔﻣﺎﻘﻟا (a) :

θ

D

H

a

b

H = a – b D = θ tanH ب -ﺔﻴﻘﻓأ تاﺮﻈﻧ ﺬﺧﺄﺑ ضرﻷا ﺔﻌﻴﺒﻃ ﺢﻤﺴﺗ ﻻ ﺎﻣﺪﻨﻋ : ﻞ ﻴﻣ ﺔ ﻳواﺰﺑ ﺔ ﻣﺎﻘﻟا ﻰ ﻟإ رﺎ ﻈﻨﻤﻟا ﻪ ﺟﻮﻧ ﺔﻨﻴﻌﻣ (θ1) ﺔﻣﺎﻘﻟا ﻰﻠﻋ ةءاﺮﻘﻟا ﺬﺧﺄﻧو (b) ﺮ ﻴﻐﻧ ﻢﺛ ، ﻦﻜﺘ ﻟو رﺎ ﻈﻨﻤﻟا ﻞ ﻴﻣ ﺔ ﻳواز (φ) ةءاﺮ ﻘﻟا ﺬ ﺧﺄﻧو ، ﺔﻣﺎﻘﻟا ﻰﻠﻋ ﺔﻴﻧﺎﺜﻟا (a) .

ac = D . tan φ, bc = D . tan θ ac – bc = H H = D (tan φ - tan θ) D = θ φ tan tan − H

3

-تﺎﻓﺎﺴﻤﻟا ﺐﻴﻀﻗ ﺔﻘﻳﺮﻃ

:

a

b

θ

D

c

ﻦﻴﺑ ﺔﻴﻘﻓﻷا ﺔﻓﺎﺴﻤﻟا ﺔﻓﺮﻌﻣ ﻦﻜﻤﻳ ﻰﻟﺎﺘﻟﺎﺑو ،ةﺪﻳﺪﺷ ﺔﻗﺪﺑ ﻪﻟﻮﻃ فوﺮﻌﻣ ﺐﻴﻀﻗ ﻰﻓﺮﻃ ﻰﻠﻋ ﺔﻴﻘﻓأ ﺔﻳواز سﺎﻴﻗ ﻖﻳﺮﻃ ﻦﻋ ﻢﺘﺗو ﻰﺗﻵﺎآ ةدﻮﺻﺮﻤﻟا ﺔﻄﻘﻨﻟاو ﺔﻠﺘﺤﻤﻟا ﺔﻄﻘﻨﻟا : D = 2 tan . . 2 1 θ ab

ىﺮﺘﻣﻮﻴآﺎﺘﻟا سﺎﻴﻘﻟا ﻰﻠﻋ ﻞﺋﺎﺴﻣ

1 ﺖ ﺑﺎﺜﻟا ﻦﻴ ﻴﻌﺗ ضﺮ ﻐﺑ ىﺮﺘﻣﻮﻴآﺎ ﺗ زﺎ ﻬﺟ ﺔﻄ ﺳاﻮﺑ ﻦﻴ ﺘﻄﻘﻧ ﺪ ﻨﻋ ﺔﻋﻮ ﺿﻮﻣ ﺔﻴ ﺳأر ﺔ ﻣﺎﻗ ﻰ ﻠﻋ ﺔ ﻴﺗﻵا تاءاﺮ ﻘﻟا تﺬ ﺧأ ﻰﻓﺎﺿﻹاو ،ىﺮﺘﻣﻮﻴآﺎﺘﻟا . ﻦﻴﺘﺑﺎﺜﻟا ﻦﻴﻋ : ﺔﻣﺎﻘﻟا تاءاﺮﻗ ﺔﻳواز عﺎﻔﺗرﻹا ﺔﻓﺎﺴﻤﻟا ﺔﻴﻘﻓﻷا 2.613 3.359 4.106 ﺮﻔﺻ 150 1.146 2.150 3.154 7 تﺎﺟرد 200 2 ﺎﻬﻨﻣ جوز ﻞآ ﻦﻴﺑ ﺔﻓﺎﺴﻤﻟا تاﺮﻌﺷ ثﻼﺜﺑ دوﺰﻣ ﺮﺘﻣﻮآﺎﺗ

40

1

ﺔﻴﺌﻴﺸ ﻠﻟ ىرﺆ ﺒﻟا ﺪ ﻌﺒﻟاو ،ﺔﺻﻮﺒﻟا ﻦﻣ 9.0 ﺔﻓﺎﺴ ﻤﻟاو ﺔ ﺻﻮﺑ ﻰ ﺳأﺮﻟا رﻮ ﺤﻤﻠﻟ ﺔﻴﺌﻴﺸ ﻟا ﻦ ﻣ

2

1

4

ﺔ ﺻﻮﺑ . ﻄﻘﻧ ﺪ ﻨﻋ ﺔﻴ ﺳأر ﺔ ﻣﺎﻗ ﺖﻌ ﺿو ﺎﻬﺑﻮﺴ ﻨﻣ ﺔ 80.0 رﺎ ﻈﻨﻤﻟا ﻞ ﻴﻣأ ، مﺪ ﻗ 9.00 ﻰ ﻠﻋ ﺔﻣﺎﻘﻟا تاءاﺮﻗ ﺖﻧﺎآو ،ﻰﻘﻓﻷا : (1.63 – 2.14 – 2.65 ) feet زﺎﻬﺠﻟا عﺎﻔﺗرا نﺄﺑ ًﺎﻤﻠﻋ ﺮﻈﻨﻟا ﻂﺧ بﻮﺴﻨﻣ ﻚﻟﺬآو ،ﺔﻣﺎﻘﻟاو زﺎﻬﺠﻟا ﻦﻴﺑ ﺔﻴﻘﻓﻷا ﺔﻓﺎﺴﻤﻟا ﺪﺟوأ 4.50 مﺪﻗ . 3 ﻦﻴﺘﻄﻘﻧ ﻦﻴﺑ راﺪﺤﻧﻹا لﺪﻌﻣ ﻦﻴﻋ (A, B) ﺔﻴﺗﻵا دﺎﺻرﻷا ﻦﻣ ﺔﻴﻠﻴﻠﺤﺗ ﺔﺳﺪﻌﺑ ﺰﻬﺠﻣ ﺮﺘﻣﻮﻴآﺎﺘﺑ ةذﻮﺧﺄﻤﻟا ) ﻪ ﻟ ﻰﻓﺎ ﺿﻹا ﺖﺑﺎﺜﻟا = ﺮﻔﺻ ( ىﺮﺘﻣﻮﻴآﺎﺘﻟا ﻪﺘﺑﺎﺛو ، 100.0 . ﻦﻣ زﺎﻬﺠﻟا فاﺮﺤﻧﻹا تاءاﺮﻘﻟا ﺔﻳواﺰﻟا ﺔﻴﺳأﺮﻟا ﻰﻟإ A 75o 1.20 2.00 2.80 +10 o 12` ﻰﻟإ B 345o 2.90 2.10 1.30 - 4o <sub>48`</sub> 4 ﻂﺧ ﺲﻴﻗ (AB) رﺎﻔﻧﻹا ﺐﻴﻀﻗ ماﺪﺨﺘﺳﺎﺑ (Subtense Bar) ﻦ ﻣ ﺪﺣاو ﺐﻧﺎﺟ ﻰﻠﻋ ىدﻮﻤﻋ ﺪﻋﺎﺴﻣ ةﺪﻋﺎﻗ ﻂﺧو ، (AB) ، ﻂ ﺨﻟا لﻮ ﻃ نﺎ آ اذﺈ ﻓ ،ًﺎ ﻣﺎﻤﺗ ﻪﻔﺼ ﺘﻨﻣ ﻰﻓو (AB) ﻮ ه 684.0 ﻮ ه ﺪﻋﺎﺴ ﻤﻟا ةﺪ ﻋﺎﻘﻟا لﻮ ﻃو ، ﺮ ﺘﻣ 28.0 ﺔ ﻳواﺰﻟا ﻦﻴ ﻌﻓ ،ﺮ ﺘﻣ ﺐﻴﻀﻘﻟا ﻰﺘﻳﺎﻬﻧ ﻦﻴﺑ ﺔﻴﻘﻓﻷا ) ﺲﻜﻟاﺮﺒﻟا ﺔﻳواز ( ﻂﺨﻟا ﺔﻳﺎﻬﻧ ﺪﻨﻋ ةدﻮﺟﻮﻤﻟا ﺔﻳواﺰﻟاو ، . ﻋﺎﻘﻟا ﻂﺧ نﺎآ اذإو ﻒﺼﻨﻳ ﺪﻋﺎﺴﻤﻟا ةﺪ (AB) ﻂﺨﻟا ﺔﻳﺎﻬﻧ ﺪﻨﻋ ةدﻮﺻﺮﻤﻟا ﺔﻳواﺰﻟاو ؟ﺲﻜﻟاﺮﺒﻟا ﺔﻳواز ﻰه ﺎﻤﻓ ،ﻪﻴﺒﻧﺎﺟ ﻰﻠﻋو ، . 5 ﻪﻧﺄﺑ ﺎﻬﻋﺎﻔﺗرإ مﻮﻠﻌﻣ ﻞﺗ ﺔﻤﻗ 2055 ةﺮﻴﺤﺑ ﻰﻓ ﻩﺎﻴﻤﻟا ﺢﻄﺳ قﻮﻓ ﺮﺘﻣ . ﺖﻧﺎآو ،ةﺮﻴﺤﺒﻠﻟ ﺮﺧﻵا ﺐﻧﺎﺠﻟا ﻦﻣ ﺔﻤﻘﻟا ﻩﺬه تﺪﺻر ﺎﻬﻋﺎﻔﺗرإ ﺔﻳواز 5o 10`` ﻮﺻ ضﺎﻔﺨﻧإ ﺔﻳواز ﺖﻧﺎآ اذﺈﻓ ، ةﺮ ﻴﺤﺒﻟا ﻩﺎ ﻴﻣ ﻰ ﻓ ﺔ ﻤﻘﻟا ةر : 8o 40`` ﻦ ﻣ ﺔ ﻴﻘﻓﻷا ﺔﻓﺎﺴ ﻤﻟا ﺪ ﺟوأ ، ﻦﻴﺘﻄﻘﻨﻟا ﻰﺑﻮﺴﻨﻣ ﻦﻴﺑ قﺮﻔﻟا ﻚﻟﺬآ ﺪﺟوأو ،ﻞﺘﻟا ﻰﻟإ زﺎﻬﺠﻟا ) ءاﻮﻬﻠﻟ ﻪﺴﻔﻧ ﻮه ءﺎﻤﻠﻟ رﺎﺴﻜﻧﻹا ﻞﻣﺎﻌﻣ نأ ﺮﺒﺘﻋا .( 6 ﺔﻄﻘﻨﻟا بﻮﺴﻨﻣ دﺎﺠﻳﻹ (A) ﺔﻄﻘﻨﻟا ﻦﻣ (B) ةﺪ ﻳﺪﺟ ﺔ ﻄﻘﻧ قﻮ ﻓ ﺖ ﻴﻟودﻮﻴﺘﻟا ﻊﺿو ﺎﻬﺑﻮﺴﻨﻣ مﻮﻠﻌﻤﻟا (C) تاءاﺮ ﻘﻟا تﺬ ﺧأو قﻮﻓ ًﺎﻴﺳأر ﻦﻴﺘﻋﻮﺿﻮﻤﻟا ﻦﻴﺘﻣﺎﻘﻟا ﻰﻠﻋ ﺔﻴﺗﻵا (A, B) ﺖﻧﺎﻜﻓ : ﺔﻣﺎﻘﻟا ﺔﻴﺳأﺮﻟا ﺔﻳواﺰﻟا تاﺮﻌﺸﻟا تاءاﺮﻗ ) ﺮﺘﻣ ( A -7o 35` 30`` 2.055 1.502 0.950 B +10o 50` 30`` 1.018 2.0 2.98 ىﺮﺘﻣﻮﻴآﺎ ﺘﻟا ﺖ ﺑﺎﺜﻟاو ،ﺔﻴﻠﻴﻠﺤﺗ ﺔﺳﺪﻋ ﻪﺑ زﺎﻬﺠﻟا نأ ﻢﻠﻋ اذﺈﻓ = 100 ﻨﻣ نأو ﺔ ﻄﻘﻧ بﻮﺴ (B) = -42.03 ﺮ ﺘﻣ . بﻮﺴ ﻨﻣ ﺪ ﺟوأ ﺔﻄﻘﻧ (A) .

7 ﻻ ﺄ ﻄﺨﻟا نأ ﺚ ﻴﺤﺑ ﺮﺘﻣﻮﻴآﺎ ﺘﻟا سﺎ ﻴﻘﻟا ﻰ ﻓ ﺔﻴﻘﻓﻷا ﺔﻓﺎﺴﻤﻟا ىوﺎﺴﺗ ﺔﻠﺋﺎﻤﻟا ﺔﻓﺎﺴﻤﻟا رﺎﺒﺘﻋا ﻦﻜﻤﻳ ﺔﻴﺳأر ﺔﻳواز ىأ ﻰﻟإ ﺪﺟوأ ﻦﻋ ﺪﻳﺰﻳ

300

1

ﺔﻴﻠﻴﻠﺤﺗ ﺔﺳﺪﻋ زﺎﻬﺠﻟﺎﺑ ﺪﺟﻮﻳو ، ) ﻰﻓﺎﺿﻹا ﺖﺑﺎﺜﻟا نأ ىأ = ﺮﻔﺻ .( 8 ﻮ ه رﺎ ﻈﻨﻣ ﻰ ﻓ ﺔﻴﺌﻴﺸﻟا ﺔﺳﺪﻌﻠﻟ ىرﺆﺒﻟا ﺪﻌﺒﻟا 12.0 ،ةرﺆ ﺒﻟا ﻦﻴ ﺑ ﺔﻓﺎﺴ ﻤﻟا ﻒﺼ ﺘﻨﻣ ﻰ ﻓ ناروﺪ ﻠﻟ ﻰ ﺳأﺮﻟا رﻮ ﺤﻤﻟاو ،ﺔ ﺻﻮﺑ ﺔﻴﺌﻴﺸ ﻟا ﺔ ﺳﺪﻌﻟاو . ﺪ ﻌﺑ ﻰ ﻠﻋ ﺔ ﻣﺎﻗ ﺖﻌ ﺿو 301.50 ﻰﺗﺮﻌ ﺷ ﻦﻴ ﺑ عﻮ ﻄﻘﻤﻟا ءﺰ ﺠﻟا نﺎ آو ،زﺎ ﻬﺠﻠﻟ ﻰ ﺳأﺮﻟا رﻮ ﺤﻤﻟا ﻦ ﻣ مﺪ ﻗ ﺔﻣﺎﻘﻟا ﻰﻠﻋ ﺎﻳدﺎﺘﺳﻻا = 3.0 مﺪﻗ . ﻦﻴﺑ ﺔﻓﺎﺴﻤﻟا ﻰه ﺎﻣ ؟زﺎﻬﺠﻟا ﻰﻓ تاﺮﻌﺸﻟا ﻞﻣﺎﺣ ﻰﻠﻋ ﺎﻳدﺎﺘﺴﻟا ﻰﺗﺮﻌﺷ 10 ﺔﻓﺎﺴ ﻤﻟا ﺖ ﻧﺎآ اذﺈ ﻓ ،ﺔ ﻣﺎﻘﻟا ﻰ ﻠﻋ ﻦﻴﻓﺪ ﻬﻟ ﺔﻴ ﺳأﺮﻟا ﺎ ﻳاوﺰﻟا تﺪ ﺻرو ،ىدﺎ ﻋ ﺖ ﻴﻟودﻮﻴﺘﺑ تﺪ ﺻرو ﺔﻴ ﺳأر ﺔ ﻣﺎﻗ ﺖﻌ ﺿو ﺎ ﻤﻬﻨﻴﺑ ﺔﻴ ﺳأﺮﻟا 4.26 عﺎ ﻔﺗرﻹا ﻰ ﺘﻳواز ﻰ ﻠﻇ ﻦﻴ ﺑ قﺮ ﻔﻟاو ،ﺮ ﺘﻣ = 0.044 ﺔ ﻳواز ﻞ ﻇ نﺎ آ اذإ ﺔ ﻣﺎﻘﻟا ﺔ ﻄﻘﻧ بﻮﺴ ﻨﻣ ﻮهﺎ ﻣ ، ﻰﻠﻔﺴ ﻟا فﺪ ﻬﻟا = 0.161 ىﻮ ﻠﻌﻟا فﺪ ﻬﻠﻟ ضرﻷا ﻦ ﻣ عﺎ ﻔﺗرﻹاو = 1.75 ﺮ ﺤﺒﻟا ﺢﻄ ﺳ ﺖ ﺤﺗ زﺎ ﻬﺠﻟا ﺢﻄ ﺳ بﻮﺴ ﻨﻣو ،ﺮ ﺘﻣ راﺪﻘﻤﺑ 4.0 ﺮﺘﻣ . 12 ﻦﻴﺘﻄﻘﻨﻟا ﻦﻴﺑ ﺔﻓﺎﺴﻤﻟا دﺎﺠﻳإ ﺪﻳرأ ،سﺮﻓاﺮﺗ ﻰﻓ (A, B) ﺔﻄﻘﻨﻟا A ﺔ ﻄﻘﻧ ﻦ ﻣ ةﺮهﺎ ﻇ ﺖ ﻧﺎآ X ﺮﺘﻟا ﻂ ﻘﻧ ىﺪ ﺣا ﺎ ﻣأ ،سﺮ ﻓا ﺔﻄﻘﻧ B ىﺮﺧأ ﺔﻄﻘﻧ ﻦﻣ ةﺮهﺎﻇ ﺖﻧﺎﻜﻓ Y . ﻦﻴ ﺘﻄﻘﻨﻟا ﻦ ﻣ ﺔ ﻳﺮﺘﻣﻮﻴآﺎﺗ دﺎ ﺻرأ تﺬﺧأ (X, Y) قﻮ ﻓ ﻦﻴﺘﻋﻮ ﺿﻮﻣ ﻦﻴﺘﻤﺋﺎ ﻗ ﻰ ﻠﻋ ﻦﻴﺘﻄﻘﻨﻟا (A, B) ﻰﺗﻵﺎآ دﺎﺻرﻷا ﺖﻧﺎآو : Co-ordinates Occupied

Point Horizontal Vertical

Traverse Point Bearing Vertical angle Hair Readings X 3800 E 1000 N A 326o 42` 09o 22` 1.50 2.11 2.72 Y 3600 W 740 N B 10o 27` 12o 18` 1.80 2.25 2.70 ﺮﺘﻣﻮﻴآﺎﺘﻟا ﺖﺑﺎﺜﻟاو ،ﺔﻴﻠﻴﻠﺤﺗ ﺔﺳﺪﻌﺑ دوﺰﻣ زﺎﻬﺠﻟا = 100 . ﺔﻓﺎﺴﻤﻟا ﺐﺴﺣا AB ﻂﺨﻟا فاﺮﺤﻧاو ، AB . 13 ﺔﻄﻘﻧ ﺪﻨﻋ ﺮﺘﻣﻮﻴآﺎﺗ ىزﺎﻬﺟ ﻦﻣ تاءاﺮﻗ تﺬﺧأ A ﺎﻬﺑﻮﺴﻨﻣ ﻰﺘﻟا ، 15.05 قﻮﻓ ﺔﻋﻮﺿﻮﻣ ﺔﻣﺎﻗ ﻰﻟإ ،ﺮﺘﻣ B : لوﻷا زﺎﻬﺠﻟا ( ρ ) : ىﺮﺘﻣﻮﻴآﺎﺘﻟا ﻪﺘﺑﺎﺛ = 100 ﻰﻓﺎﺿﻹا ﻪﺘﺑﺎﺛو ، = 14.40 ﺔﺻﻮﺑ . ﻰﻧﺎﺜﻟا زﺎﻬﺠﻟا ( φ ) : ىﺮﺘﻣﻮﻴآﺎﺘﻟا ﻪﺘﺑﺎﺛ = 95 ﻰﻓﺎﺿﻹا ﻪﺘﺑﺎﺛو ، = 15.0 ﺔﺻﻮﺑ . زﺎﻬﺠﻟا عﺎﻔﺗرا نﺎآو ( ρ ) ﺔﻄﻘﻧ ﺪﻨﻋ A = 1.45 عﺎﻔﺗرﻹا ﺔﻳوازو ،ﺮﺘﻣ

Theory of Errors

ﻰ ﺘﻟا ﺔ ﻟﻵا وأ ﺪ ﺻاﺮﻟا ﺺﺨﺸ ﻟا ﻰ ﻟإ دﻮﻌﺗ ﺪﻗو ،ءﺎﻄﺧﻷا ﻩﺬه ﺔﻌﻴﺒﻃ ﻒﻠﺘﺨﺗو ،ءﺎﻄﺧأ تﺪﺟو ﺎﻤﻠآ ،دﺎﺻرأ تﺪﺟو ﺎﻤﻠآ

ﺔﻴﺟرﺎﺧ ﺔﻴﻌﻴﺒﻃ ﻞﻣاﻮﻋ ﻰﻟإ وأ ﺎﻬﺑ ﺪﺻﺮﻳ :

1- Types of Errors:

1-1- According to the sources: a- Personal Errors

b- Instrumental Errors c- Natural Errors 2-1 According to kind

a- Systematic Error

b- Un-systematic (random) Error c- Gross error 2- propagation of Errors سﺎﻴﻘﻟا تاﺮﻣ دﺪﻋ نﺈﻓ ،تاﺮﻣ ةﺪﻋ ﻂﻳﺮﺸﻟا لﻮﻃ ﻦﻋ ﺪﻳﺰﻳ ﻪﻟﻮﻃ ﻂﺧ سﺎﻴﻗ ﺪﻨﻋ (n) ﻪ ﻧﺈﻓ سﺎﻴﻘﻟا ﻰﻓ ﺄﻄﺧ ﺪﺟو اذإو ، رﺮﻜﺘﻳ ةﺪﺣاﻮﻟا ةﺮﻤﻟا ﻰﻓ ﺄﻄﺨﻟا اﺬه ﺔﻤﻴﻗ ﺖﻧﺎآ اذﺈﻓ ،تاﺮﻤﻟا ﻦﻣ دﺪﻌﻟا ﺲﻔﻨﺑ (E) ﻪﻠآ ﻂﺨﻟا ﻰﻓ ﺞﺗﺎﻨﻟا ﺄﻄﺨﻟا نﺈﻓ ، = (n.E) . ﻂﺨﻟا لﻮﻃ ةدﺎﻳﺰﺑ ﻪﺘﺒﺴﻧ ﺮﻴﻐﺘﺗ ﻻو ﻊﻤﺠﺘﻳ ﺄﻄﺨﻟا نأ ﻆﺣﻼﻧو . 3- Random Errors: ﺠﻟا ﻦﻣ وأ ﻰﺼﺨﺷ نﺎآ ءاﻮﺳ ﻩرﺪﺼﻣ ﻰﻟإ ﺄﻄﺨﻟا عﺎﺟرﻹو ،ةدﺪﻌﺘﻣ ردﺎﺼﻣ ﺎﻬﻟ ءﺎﻄﺧﻷا ﻰﺗﻵا ﻢﺘﻳ مﺪﺨﺘﺴﻤﻟا زﺎﻬ : -ﻰﺼﺨﺷ نﻮﻜﻳ ءﺎﻄﺧﻷا رﺪﺼﻣ نﺈﻓ زﺎﻬﺠﻟا ﺲﻔﻧ ﻰﻠﻋ صﺎﺨﺷﻷا ﻦﻣ ﺔﻋﻮﻤﺠﻣ ﻞﻤﻋ ﺪﻨﻋ . -،زﺎﻬﺠﻟا ﻦﻣ نﻮﻜﻳ ﺄﻄﺨﻟا رﺪﺼﻣ نﺈﻓ ،ةﺰﻬﺟﻷا ﻦﻣ ﺔﻋﻮﻤﺠﻣ ﺪﺣاو ﺺﺨﺷ ماﺪﺨﺘﺳا ﺪﻨﻋ ًاﺪﺟ ةﺮﻴﻐﺻ ﺎﻬﻧﺄﺑ ﺰﻴﻤﺘﺗو ، ﺔﺿرﺎﻌﻟا ءﺎﻄﺧﻷا ﻰﻘﺒﺗ ﺔﺘﺑﺎﺜﻟاو ،ﺔﻤﻈﺘﻨﻤﻟا ءﺎﻄﺧﻷا ﻦﻣ ﺺﻠﺨﺘﻟا ﺪﻌﺑ ةرﺎﺷﻹا ﺔﺘﺑﺎﺛ ﺮﻴﻏو ، ﺎﻬﻨﻣ ﺺﻠﺨﺘﻟا وأ ﺎﻬﺑﺎﺒﺳأ ﺔﻓﺮﻌﻣ ﻦﻜﻤﻳ ﻻو ،ﺔﻤﻴﻘﻟا وأ . ﺔﻣﺎﻬﻟا ﻒﻳرﺎﻌﺘﻟا ﺾﻌﺑ 1 -ﺮﺷﺎﺒﻤﻟا سﺎﻴﻘﻟا : Direct Measure ﺔ ﻤﻴﻗ سﺎ ﻴﻗ وأ ،ﻂﻳﺮﺸ ﻟا ﺔﻄ ﺳاﻮﺑ ﻂﺧ لﻮﻃ سﺎﻴﻗ ﻞﺜﻣ ،ةﺮﺷﺎﺒﻣ ﺔﺳﺎﻘﻤﻟا ﺔﻴﻤﻜﻟا ﺪﻳﺪﺤﺗ ﻪﺘﻄﺳاﻮﺑ ﻢﺘﻳ ىﺬﻟا سﺎﻴﻘﻟا ﻮه ﺖﻴﻟودﻮﻴﺘﻟا ﺔﻄﺳاﻮﺑ ﺔﻳواز . 2 -ﺮﺷﺎﺒﻤﻟا ﺮﻴﻏ سﺎﻴﻘﻟا وأ ﺪﺻﺮﻟا : Indirect Measure ﺎﻬﻟ ﺮ ﺷﺎﺒﻣ سﺎ ﻴﻗ ﻞ ﻤﻋ ﻢﺘ ﻳ ﻢ ﻟ ، ﺔ ﻴﻤآ ﻰ ﻠﻋ لﻮﺼ ﺤﻟا ﻰ ﻨﻌﻳ ﺮﺷﺎﺒﻣ ﺮﻴﻐﻟا سﺎﻴﻘﻟا . ﺚ ﻠﺜﻣ ﻰ ﻓ ﻊﻠ ﺿ لﻮ ﻃ بﺎﺴ ﺣ ﻞ ﺜﻣ ﺎﻤﻧإو ،ﻪﺳﺎﻴﻗ ﻰﻓ ﻂﻳﺮﺸﻟا مﺪﺨﺘﺴﻳ ﻢﻟ ﻪﺳﺎﻴﻗ ﺔﻤﻴﻗ ﻰﻠﻋ لﻮﺼﺤﻟا ﻢﺗ ىﺬﻟا ﻊﻠﻀﻟﺎﻓ ،ﺎﻤﻬﻨﻴﺑ ﺔﻳواﺰﻟاو ،ﻦﻳﺮﺧﻵا ﻦﻴﻌﻠﻀﻟا ﺔﻴﻣﻮﻠﻌﻤﺑ ﺚﻠﺜﻤﻟا عﻼﺿأ ﻂﺑﺮﺗ ﺔﺻﺎﺧ ﺔﻗﻼﻋ ﺖﻣﺪﺨﺘﺳا . 3 -ًﻻﺎﻤﺘﺣا ﺮﺜﻛﻷا ﺔﻤﻴﻘﻟا :

The most probable value

ﻰ ﻠﻋ لﻮﺼ ﺤﻠﻟ دﺎ ﺻرﻷا ﻦ ﻣ ﻰﺋﺎ ﻬﻧ ﻻ دﺪﻋ ﺬﺧأ ًﺎﻀﻳأ رﺬﻌﺘﻳ ﺎﻤآ ،ﺔﻤﻈﺘﻨﻤﻟاو ﺔﻳدﺎﻌﻟا ءﺎﻄﺧﻷا ﻦﻣ ًﺎﻣﺎﻤﺗ ﺺﻠﺨﺘﻟا رﺬﻌﺘﻳ ﺔﺳﺎﻘﻤﻟا ﺔﻴﻤﻜﻠﻟ ﺔﻴﻘﻴﻘﺤﻟا ﺔﻤﻴﻘﻟا . ﺔﻳدﺎﻌﻟا ءﺎﻄﺧﻷا نأ ضﺮﻔﺑو نﺈﻓ ،ﺎﻬﻟﺎﻤها ﻦﻜﻤﻳ ﺚﻴﺤﺑ ﻦﻜﻤﻣ ﺪﺣ ﻰﺼﻗأ ﻰﻟإ ﺖﺷﻼﺗ ﺔﻤﻈﺘﻨﻤﻟاو ﺢﻴﺤﺼﺘﻟا تﺎﻌﺑﺮﻤﻟ عﻮﻤﺠﻣ ﻞﻗأ ﺔﻳﺮﻈﻨﻟ ًﺎﻌﺒﺗ ﺎهﺮﻴﻏ ﻦﻣ ﺮﺜآأ ﺎﻬﺘﺤﺻ ﻞﻤﺘﺤﻳ ﻰﺘﻟا ﺔﻤﻴﻘﻟا ﻰه نﻮﻜﺗ ﺔﻴﻟﺎﻤﺘﺣا ﺔﻤﻴﻗ ﻞﻀﻓأ .

ﻘﻳو ،ًﻻﺎ ﻤﺘﺣا ﺮ ﺜآﻷا ﺔ ﻤﻴﻘﻟا ﺎ ﻬﻴﻠﻋ ﻖ ﻠﻄﻳ ﺔ ﺑﻮﻠﻄﻤﻟا ﺔﻴﻤﻜﻠﻟ ةﺪﻳﺪﻌﻟا دﺎﺻرﻸﻟ ﺔﻄﺳﻮﺘﻤﻟا ﺔﻤﻴﻘﻟا نﺈﻓ ﻚﻟﺬﻟ ﺎ ﻣ بﺮ ﻗأ ﺎ ﻬﻧأ ﻚﻟﺬ ﺑ ﺪﺼ ﺔﻴﻘﻴﻘﺤﻟا ﺔﻴﻤﻜﻠﻟ ﻦﻜﻤﻳ . 4 -ﻰﻘﻴﻘﺤﻟا ﺄﻄﺨﻟا : True error ةدﻮﺻﺮﻤﻟا ﺔﻴﻤﻜﻠﻟ ﺔﻴﻘﻴﻘﺤﻟا ﺔﻤﻴﻘﻟا ﻦﻴﺑ قﺮﻔﻟا ﻮه ) ﺎﻬﻴﻠﻋ لﻮﺼ ﺤﻟا ﻞﻴﺤﺘﺴﻳ وا رﺬﻌﺘﻳ ﻰهو ( ﺎﻬﻟ ةدﻮ ﺻﺮﻤﻟا ﺔ ﻤﻴﻘﻟا ﻦﻴ ﺑو . ﺎ ﻤﻟو ﺘﺴﻣ ﻰﻘﻴﻘﺤﻟا ﺄﻄﺨﻟا ﻰﻠﻋ لﻮﺼﺤﻟا نﺈﻓ ،ﻞﻴﺤﺘﺴﻣ ﺔﻴﻘﻴﻘﺤﻟا ﺔﻤﻴﻘﻟا ﻰﻠﻋ لﻮﺼﺤﻟا نﺎآ ًﺎﻀﻳأ ﻞﻴﺤ . ﻰﻘﻴﻘﺤﻟا ﺄﻄﺨﻟا = ةدﻮﺻﺮﻤﻟا ﺔﻤﻴﻘﻟا – ﺔﻴﻘﻴﻘﺤﻟا ﺔﻤﻴﻘﻟا 5 -ﺔﻴﻗﺎﺒﻟا ءﺎﻄﺧﻷا : Residual error ﺎﻬﻟ ًﻻﺎﻤﺘﺣا ﺮﺜآﻷا ﺔﻤﻴﻘﻟا ﻦﻴﺑو ،ةدﻮﺻﺮﻤﻟا ﺔﻤﻴﻘﻟا ﻦﻴﺑ قﺮﻔﻟا ﻮه ﻰﻗﺎﺒﻟا ﺄﻄﺨﻟا = ةدﻮﺻﺮﻤﻟا ﺔﻤﻴﻘﻟا – ًﻻﺎﻤﺘﺣا ﺮﺜآﻷا ﺔﻤﻴﻘﻟا 6 -ﻰﺑﺎﺴﺤﻟا ﻂﺳﻮﺘﻤﻟا : Arithmetic mean تاﺮ ﻤﻟا ﻦ ﻣ دﺪﻋ ﺎﻣ ﺔﻴﻤآ ﺪﺻﺮﺑ ﺪﺻاﺮﻟا مﻮﻘﻳ ﺎﻣﺪﻨﻋ = n دﺎ ﺻرﻷا ﻩﺬ ه ﻂ ﺳﻮﺘﻣو ،ﺔ ﻔﻠﺘﺨﻣ ﺞﺋﺎ ﺘﻧ ﻰ ﻠﻋ ﻞﺼ ﺤﻳ ﻪ ﻧﺈﻓ ، دﺎﺻرﻸﻟ ﻰﻠﻜﻟا عﻮﻤﺠﻤﻟا ىوﺎﺴﻳ ÷ ﺪﺻﺮﻟا تاﺮﻣ دﺪﻋ . ﺎﻬﻤهأ صاﻮﺧ ةﺪﻋ ﻰﺑﺎﺴﺤﻟا ﻂﺳﻮﺘﻤﻠﻟو : أ ﺮﻔﺻ ىوﺎﺴﻳ ﺔﻴﻗﺎﺒﻟا ءﺎﻄﺧﻷا عﻮﻤﺠﻣ ) ﻰﻗﺎ ﺒﻟا ﺄﻄﺨﻟا نﻷ = ةدﻮ ﺻﺮﻤﻟا ﺔ ﻤﻴﻘﻟا – ﺔ ﻤﻴﻘﻟا ًﻻﺎ ﻤﺘﺣا ﺮ ﺜآﻷا ( ﺔ ﻤﻴﻘﻟا ﺖ ﻧﺎآ ﺎ ﻤﻟو ، دﺎﺻرﻸﻟ ﺔﻴﻗﺎﺒﻟا ءﺎﻄﺧﻷا عﻮﻤﺠﻣ نﺈﻓ ﻰﺑﺎﺴﺤﻟا ﻂﺳﻮﺘﻤﻟا ﻞﺜﻤﺗ ًﻻﺎﻤﺘﺣا ﺮﺜآﻷا = ﺮﻔﺻ . ب -ﻦﻜﻤﻳ ﺎﻣ ﻞﻗأ ﺔﻴﻗﺎﺒﻟا ءﺎﻄﺧﻷا تﺎﻌﺑﺮﻣ عﻮﻤﺠﻣ 7 -ﻂﺳﻮﺘﻤﻟا ﻰﻌﻴﺑﺮﺘﻟا ﺄﻄﺨﻟا :

Mean square error (MSE)

ﻘﺤﻟا ﺎ ﻬﺘﻤﻴﻗ مﻮﻠﻌﻣ ﺎﻣ ﺔﻴﻤﻜﻟ دﺎﺻرﻷا ﺔﻗد ﺮﻳﺪﻘﺘﻟ ءﺎ ﻄﺧﻷا ﺔ ﻳﺮﻈﻧ ﻰ ﻓ ﻰﻤﺴ ﻳ ﺎ ﻣ وأ ىرﺎ ﻴﻌﻤﻟا فاﺮ ﺤﻧﻹا مﺪﺨﺘﺴ ﻳ ﺔ ﻴﻘﻴ

ﻂﺳﻮﺘﻤﻟا ﻰﻌﻴﺑﺮﺘﻟا ﺄﻄﺨﻟﺎﺑ Mean Square Error (M.S.E)

.

-ةﺪﺣاﻮﻟا ةﺪﺻﺮﻠﻟ ﻂﺳﻮﺘﻤﻟا ﻰﻌﻴﺑﺮﺘﻟا ﺄﻄﺨﻟا :

MSE of a single observation

(

)

1

1 2

−

−

=

∑

=

n

X

X

n i i

σ

-ﻰﺑﺎﺴﺤﻟا ﻂﺳﻮﺘﻤﻠﻟ ﻂﺳﻮﺘﻤﻟا ﻰﻌﻴﺑﺮﺘﻟا ﺄﻄﺨﻟا :

MSE of the arithmetic mean

ىﺮ ﺧأ ةﺪﺻر ىأ ماﺪﺨﺘﺳﺎﺑ ﺎﻬﻨﻴﻴﻌﺗ ﻦﻣ ﺮﺒآأ ﻰﺑﺎﺴﺤﻟا ﻂﺳﻮﺘﻤﻟا ماﺪﺨﺘﺳﺎﺑ ﺔﻟﻮﻬﺠﻤﻟا ﺔﻴﻤﻜﻟا ﻦﻴﻴﻌﺗ ﺔﻗد نأ مﻮﻠﻌﻤﻟا ﻦﻣ . ﻦﻴ ﺑ ﺔ ﻗﻼﻌﻟاو ،ةﺪ ﺣاﻮﻟا ةﺪ ﺻﺮﻠﻟ ﻂ ﺳﻮﺘﻤﻟا ﻰ ﻌﻴﺑﺮﺘﻟا ﺄ ﻄﺨﻟا ﻦ ﻣ ﻞ ﻗأ ﻰﺑﺎﺴ ﺤﻟا ﻂ ﺳﻮﺘﻤﻠﻟ ﻂ ﺳﻮﺘﻤﻟا ﻰﻌﻴﺑﺮﺘﻟا ﺄﻄﺨﻟا نﺈﻓ ﻰﻟﺎﺘﻟﺎﺑو ﻮﺘﻤﻟا ﻰﻌﻴﺑﺮﺘﻟا ﺄﻄﺨﻟا ﻰه ﻰﺑﺎﺴﺤﻟا ﻂﺳﻮﺘﻤﻠﻟ ﻂﺳﻮﺘﻤﻟا ﻰﻌﻴﺑﺮﺘﻟا ﺄﻄﺨﻟا ﻦﻴﺑو ،ةﺪﺣاﻮﻟا ةﺪﺻﺮﻠﻟ ﻂﺳ :

n

o 2 2

σ

σ

=

where σo the mean square error of the arithmetic mean

n

o

σ

(

)

)

1

.(

1 2

−

−

=

∑

=

n

n

X

X

n i i o

σ

8 -ﻰﺒﺴﻨﻟا ﺄﻄﺨﻟا : Relative error ﺔﺳﺎﻘﻤﻟا تﺎﻴﻤﻜﻟا فﻼﺘﺧا ﺪﻨﻋ تاﺪﺣﻮﻟا ﻰﻓ ) ﺔ ﻳوازو لﻮ ﻃ ًﻼﺜﻣ ( ﻦ ﻣ ﻪ ﻧﺈﻓ ،سﺎ ﻴﻘﻟا ﻰ ﻓ قدأ ﺎ ﻤﻬﻳأ ﺔ ﻓﺮﻌﻣ بﻮ ﻠﻄﻤﻟاو قدأ دﺎﺻرﻷا ىأ ﺪﻳﺪﺤﺘﻟ ﺔﻧرﺎﻘﻤﻠﻟ ﻰﺒﺴﻨﻟا ﺄﻄﺨﻟا ﻰﻤﺴﺗ ﺔﻴﻤآ ﻦﻴﻌﺗ ﻚﻟﺬﻟ ،ﻦﻴﺘﻟﺎﺤﻟا ﻰﻓ ءﺎﻄﺧﻷا ﺔﻧرﺎﻘﻣ ﺄﻄﺨﻟا .

X

Error

lative

σ

Re

=

9 -ﻂﺳﻮﺘﻤﻟا ﺄﻄﺨﻟا : Mean error ﻰﺑﺎﺴﺤﻟا ﻂﺳﻮﺘﻤﻟا ىوﺎﺴﻳ ﻮهو ﺰﻣﺮﻟﺎﺑ ﻪﻟ ﺰﻣﺮﻳو ،ةرﺎﺷﻹا ﻦﻣ ةدﺮﺠﻣ ءﺎﻄﺧﻸﻟ (η) نأ تﺎﺒﺛا ﻦﻜﻣأ ﺪﻗو ، :

σ

π

η

=

2

.

= 0.7979 σ 10 -ﻞﻤﺘﺤﻤﻟا ﺄﻄﺨﻟا : Probable error ﻪﻋﻮﻗو لﺎﻤﺘﺣا نﻮﻜﻳ ىﺬﻟا ﺄﻄﺨﻟا ﻮه = 50% ﺐ ﻴﺗﺮﺗ ﻰ ﻓ ﻂ ﺳوﻷا ﻢﻗﺮ ﻟا ﻞ ﺜﻤﻳ ﻮ ﻬﻓ ﻚﻟﺬ ﻟ ،ءﺎ ﻄﺧﻷا ﻊ ﻴﻤﺟ عﻮﻗو ﻦﻣ ﻗو ،ﺔﻴﻗﺎﺒﻟا ءﺎﻄﺧﻷا ﻦﻣ ﻪﻨﻴﻌﺗ ﻦﻜﻤﻳ ﻞﻤﺘﺤﻤﻟا ﺄﻄﺨﻟا نأ تﺎﺒﺛا ﻦﻜﻣأ ﺪ : r = 0.6745 σ example:

3- A distance was measured eight times and the observations were as follows:

118.167 m 118.750 m 118.273 m 118.266 m 118.165 m 118.167 m 118.760 m 118.280 m

a- calculate the mean value (

X

) of this distance, and the mean square error (MSE) of the observations (σx) of a single measurement.

b- Calculate the (MSE) of the arithmatic mean (σx). c- Calculate the relative error.

d- Find the mean error (a) and the probable error (r) from the above observations, then check their relations with (σ)

e- Supposing that if the true value of that distance was 118.248 m, what would you comment upon these observations?

Solution

No. Observation Residual error

_X

_X

i

−

error

Sorting True error X_i −X_t

error Sorting 1 118.167 -0.1865 0.0735 -0.081 0.018 2 118.750 0.3965 0.0805 0.502 0.025 3 118.273 -0.0805 0.0875 0.025 0.032 4 118.266 -0.0875 0.1865 0.018 0.081 5 118.165 -0.1885 0.1865 -0.083 0.081 6 118.167 -0.1865 0.1885 -0.081 0.083 7 118.760 0.4065 0.3965 0.512 0.502 8 118.280 -0.0735 0.4065 0.032 0.512 Σ = 946.828 a- mean value =

8

828

.

946

1

₌

∑

=

n

X

n i i = 118.3535 m

- mean square error of a single observation (σn-1) =

(

)

1

1 2

−

−

∑

=

n

X

X

n i i = 0.252725

b- MSE of the arithmetic mean =

(

)

)

1

.(

1 2

−

−

=

∑

=

n

n

X

X

n i i o

σ

=

8

252725

.

0

1

₌

−

n

n

σ

= 0.08935 c- relative error : Re =

3535

.

118

252725

.

0

1

₌

−

X

n

σ

= 0.002135

d- Mean error (a) =

8

606

.

1

1

₌

−

∑

=

n

X

X

n i i = 0.20075 check of mean error value with (σn-1)

a = 0.7979 σ = 0.7979 * 0.2527 = 0.2016 d- Probable error: ﻢ ﻗر ﺄ ﻄﺨﻟا نﺈ ﻓ ﺔ ﻳدﺮﻓ دﺎ ﺻرﻷا دﺪ ﻋ نﺎ آ اذإ ،ةرﺎ ﺷﻹا ﻰ ﻟإ ﺮ ﻈﻨﻟا نوﺪ ﺑ ،ﺮ ﺒآﻷا ﻰ ﻟإ ﺮﻐﺻﻷا ﻦﻣ ءﺎﻄﺧﻷا ﺐﻴﺗﺮﺗ ﻢﺘﻳ

2

1

+

n

ًﻻﺎﻤﺘﺣا ﺮﺜآﻷا ﺄﻄﺨﻟا ﻞﺜﻤﻳ . ﺎﺻرﻷا دﺪﻋ ﺖﻧﺎآ اذإو ًﻻﺎﻤﺘﺣا ﺮﺜآﻷا ﺄﻄﺨﻟا نﺈﻓ ﺔﻴﺟوز د = ﻰﻤﻗر ﻦﻳﺄﻄﺨﻟا ﻂﺳﻮﺘﻣ (

1

,

n

+

n

) .

r =

2

1865

.

0

1865

.

0

+

= 0.1865 r = 0.6745 σ = 0.6745 * 0.2527 = 0.1704

e- if the true value = 118.248, then, it can be concluded that both observations (118.750, and 118.760) contains random errors, and must be excluded.

11 -نازوﻷا : نزو ﺎ ﻬﻟ ﻰ ﺘﻟا دﺎ ﺻرﻷا نﺈ ﻓ ﻰﻟﺎ ﺘﻟﺎﺑو ،نزو ﺎ ﻬﻟ دﺎ ﺻرأ ﻪﻟ ﻖﻴﻗﺪﻟا ﺺﺨﺸﻟا ًﻼﺜﻣ ،ةﺪﺻﺮﻟﺎﺑ داﺪﺘﻋﻹا ىﺪﻣ ﺎﻬﺑ ﺪﺼﻘﻳو ﺔﻴﺒﺴﻧ ﺔﻘﺛ ﺎﻬﻟ دﺎﺻرأ ﻦﻋ ةرﺎﺒﻋ . weight =

1

₂

σ

Where: σ Standard deviation

(

)

1

1 2

−

−

=

∑

=

n

X

X

n i i

σ

ﺎﻬﻠآ دﺎﺻرﻸﻟ ﺔﻄﺳﻮﺘﻤﻟا ﺔﻤﻴﻘﻟا ﻦﻋ ةﺪﺣاﻮﻟا ةﺪﺻﺮﻟا فاﺮﺤﻧإ ﻞﺜﻤﻳ ىرﺎﻴﻌﻤﻟا فاﺮﺤﻧﻹاو . فاﺮﺤﻧﻹا ﻞﻗ ﺎﻤﻠآ ﻰﻟﺎﺘﻟﺎﺑو ﺎﻬﻧزو دادز ﺎﻤﻠآ ةﺪﺻﺮﻠﻟ ىرﺎﻴﻌﻤﻟا . 12 ءﺎﻄﺧﻷا ﻰﻨﺤﻨﻣ صاﻮﺧ ) سوﺎﺟ ﻰﻨﺤﻨﻣ :( ﻰﻘﻓﻷا رﻮﺤﻤﻟا ﻪﻴﻓ ﻞﺜﻤﻳ ﻰﻨﺤﻨﻣ ﻦﻋ ةرﺎﺒﻋ عﻮﻗو لﺎﻤﺘﺣا ﻰﺳأﺮﻟا رﻮﺤﻤﻟا ﻞﺜﻤﻳو ،ءﺎﻄﺧﻸﻟ ﺔﻳﺮﺒﺠﻟاو ﺔﻳدﺪﻌﻟا ﺔﻤﻴﻘﻟا ﻪﺛوﺪﺣ تاﺮﻣ دﺪﻋ وأ ﺄﻄﺨﻟا . ﺔﻤﻴﻘﻟا ﻦﻴﺑو ةدﻮﺻﺮﻤﻟا ﻢﻴﻘﻟا ﻦﻴﺑ قﺮﻔﻟا ىوﺎﺴﻳو ﻰﻗﺎﺒﻟا ﺄﻄﺨﻟا ﺎﻧدﺪﺣ ﻢﺛ ،تاﺮﻤﻟا ﻦﻣ ًاﺪﺟ ﺮﻴﺒآ دﺪﻋ ﺎﻣ ﺔﻴﻤآ ﺎﻨﺴﻗ اذإ ًﻼﺜﻤﻓ ﻟﺎﺘﻟﺎآ لوﺪﺟ ﻰﻓ ﺞﺋﺎﺘﻨﻟا ﺐﻴﺗﺮﺗ ﻢﺗ ﻢﺛ ،ًﻻﺎﻤﺘﺣا ﺮﺜآﻷا ﻰ :

لﺎﺜﻣ : 1- the angle (ABC) was measured 25 times, with equally accuracy. By taking the arithmetic mean of the observations, the most probable value of the angle was (94o 04` 20.55``). The values of observations are listed in the following table:

No. Observations Residual Error

X

X

_i

−

+V -V 1 94o 04` 20.85`` 0.30 1.80 1.80 2 94o 04` 20.50`` -0.05 1.45 1.50 3 94o 04` 20.60`` 0.05 1.45 1.50 4 94o 04` 20.50 -0.05 1.20 1.05 5 94o <sub>04` 20.70`` 0.15 0.95 1.00</sub> 6 94o 04` 19.05`` -1.50 0.70 0.80 7 94o 04` 20.50 -0.05 0.70 0.55 8 94o 04` 22.00`` 1.45 0.70 0.55 9 94o 04` 19.55`` -1.0 0.40 0.30 10 94o 04` 19.75`` -0.80 0.30 0.05 11 94o 04` 20.50`` -0.05 0.15 0.05 12 94o 04` 19.05`` -1.50 0.05 0.05 13 94o 04` 18.75`` -1.80 0.05 14 94o 04` 20.00`` -0.55 15 94o <sub>04` 20.00`` -0.55</sub> 16 94o 04` 21.25`` 0.70 17 94o <sub>04` 21.25`` 0.70</sub> 18 94o 04` 22.00`` 1.45 19 94o 04` 21.50`` 0.95 20 94o 04` 19.50`` -1.05 21 94o 04` 21.75`` 1.20 22 94o 04` 22.35`` 1.80 23 94o 04` 20.95`` 0.40 24 94o 04` 20.35`` -0.20 25 94o 04` 21.25`` 0.70 ةﺮﺘﻔﻟا ﺔﻤﻴﻗ ﻦﻜﺘﻟو تاﺮﺘﻓ ﻰﻟإ ءﺎﻄﺧﻷا ﻢﺴﻘﻧ (0.50``) ﻰﻟﺎﺘﻟﺎآ ةﺮﺘﻓ ﻞآ ﻰﻓ ءﺎﻄﺧﻷا دﺪﻋ دﺪﺤﻧ ﻢﺛ : Period No. of Errors 2.0-1.5 1 1.5-1.0 3 1.0-0.5 4 0.5-0.0 4 0.0-(-0.5) 5 -0.5-(-1.0) 4 -1.0-(-1.5) 3 -1.5-(-2.0) 1 ،ﺄ ﻄﺨﻟا راﺮ ﻜﺗ تاﺮ ﻣ دﺪ ﻋ ﻪ ﻋﺎﻔﺗرا ﻞ ﺜﻤﻳو ،ﺄ ﻄﺨﻟا ةﺮ ﺘﻓ ﻞﻴﻄﺘﺴ ﻤﻟا ةﺪ ﻋﺎﻗ ﻞ ﺜﻤﺗ ﺚ ﻴﺤﺑ ،ةروﺎ ﺠﺘﻣ تﻼﻴﻄﺘﺴ ﻣ ةﺪ ﻋ ﻢﺳر ﻢﺘﻳ ﻢﺛ ﻵا ىراﺮﻜﺘﻟا ﻊﻴﻤﺠﺘﻟا ﻰﻨﺤﻨﻣ ﻰﻤﺴﻳ ىﺬﻟا ﻰﻨﺤﻨﻤﻟا اﺬه ﻰﻓ ﻆﺣﻼﻳو ﻰﺗ :

2 -ﺄﻄﺨﻟا ﺮﺒآ ﺎﻤﻠآ ﺎهدﺪﻋ ﻞﻘﻳو ،تﺮﻐﺻ ﺎﻤﻠآ ءﺎﻄﺧﻷا دﺪﻋ دادﺰﻳ . 3 -دﺎﺻرﻷا دﺪﻋ دادزا ﺎﻤﻠآ ﺔﻴﻘﻴﻘﺤﻟا ﺔﻤﻴﻘﻟا ﻦﻣ ًﺎﺑﺮﻗ ﻰﺑﺎﺴﺤﻟا ﻂﺳﻮﺘﻤﻟا دادﺰﻳ . (-2 t o -1.5 ) (-1 .5 t o -1 .0 ) (-1 .0 t o -0 .5 ) (-0 .5 t o 0 .0 ) (0 .0 t o 0 .5 ) (0 .5 to 1 .0 ) (1 .0 t o 1 .5 ) (1 .5 t o 2 .0 ) Period of Error 0 1 2 3 4 5 6 R e p e ta ti o n o f E rr o r ﻳ ﺎﻣ ﻞﻗأ ﻰﻟإ ةﺮﺘﻔﻟا ﺖﻠﻗ اذإ ،ﻖﺑﺎﺴﻟا ﻞﻜﺸﻟا ﻰﻓو ﻰ ﻨﺤﻨﻣ ﻰ ﻠﻋ ﻞﺼ ﺤﻧ ﺎ ﻨﻧﺈﻓ ﻦﻜﻤﻳ ﺎﻣ ﺮﺒآأ ﻰﻟإ دﺎﺻرﻷا دﺪﻋ دازو ،ﻦﻜﻤ ﻪﻴﻠﻋ ﻖﻠﻄﻳ ىﺬﻟاو ،ءﺎﻄﺧﻷا ) سوﺎﺟ ﻰﻨﺤﻨﻣ ( ﻰﺛاﺪ ﺣﻹا لاﺪﺒﺘ ﺳا ﺪ ﻌﺑ ﻚ ﻟذو تﻻﺎ ﻤﺘﺣﻹا ﻰ ﻨﺤﻨﻣ ﻩﻮ ﺟﻮﻟا ﻊ ﻴﻤﺟ ﻦ ﻣ ﺔﺒﺸﻳ ﻮهو ، ﻦﻣ ًﻻﺪﺒﻓ ، ﻰﺳأﺮﻟا (n) ﻊﻀﻧ ﺄﻄﺨﻟا ثوﺪﺣ تاﺮﻣ دﺪﻋ ﻰﻠﻋ لﺪﺗ ﻰﺘﻟا (y) ثوﺪ ﺣ لﺎ ﻤﺘﺣا ﻰ ﻠﻋ لﺪ ﺗ ﻰ ﺘﻟا ،ﻖﺑﺎﺴ ﻟا ﺄ ﻄﺨﻟا ﺲ ﻔﻧ ﺔﻟﺎﺤﻟا ﻩﺬه ﻰﻓ ﻪﻴﻠﻋ ﻖﻠﻄﻳو ) ﺄﻄﺨﻟا ثوﺪﺣ لﺎﻤﺘﺣا ﻰﻨﺤﻨﻣ ( ﺔﻴﻟﺎﺘﻟا ﻖﺋﺎﻘﺤﻟا ﻰﻨﺤﻨﻤﻟا اﺬه ﻦﻣ ﺢﻀﺘﺗو ، : 1 ﺔﻳدﺪﻌﻟا ﺔﻤﻴﻘﻟا ﺲﻔﻧ ﺎﻬﻟ ﻰﺘﻟا ﺔﺒﻟﺎﺴﻟا ءﺎﻄﺧﻷا ثوﺪﺣ لﺎﻤﺘﺣا ﻞﺜﻣ ﺔﺒﺟﻮﻤﻟا ءﺎﻄﺧﻷا ثوﺪﺣ لﺎﻤﺘﺣا . 2 -ةﺮﻴﺒﻜﻟا ءﺎﻄﺧﻷا ﻦﻣ ًﺎﺛوﺪﺣ ﺮﺜآأ ةﺮﻴﻐﺼﻟا ءﺎﻄﺧﻷا ىأ ﻪﺘﻤﻴﻗ تدادزا ﺎﻤﻠآ ﺄﻄﺨﻟا ثوﺪﺣ لﺎﻤﺘﺣا ﻞﻘﻳ . 3 رﻮﺤﻣ ﻞﺑﺎﻘﻳ ﻰﻨﺤﻨﻤﻟا نﺈﻓ اﺬﻬﻟو ،ثوﺪﺤﻟا ةردﺎﻧ ةﺮﻴﺒﻜﻟا ءﺎﻄﺧﻷا (X) ﺔﻳﺎﻬﻧﻻﺎﻣ ﻰﻓ . Error G(e)

Error

G(e)

ﺔﻘﻴﻗد دﺎﺻرأ ﺔﻘﻴﻗد ﺮﻴﻏ دﺎﺻرأ

لﺎﻤﺘﺣﻹا ﻰﻨﺤﻨﻤﻟ ﺔﻣﺎﻌﻟا ﺔﻟدﺎﻌﻤﻟا

:

لﺎﻤﺘﺣﻹا ﻰﻨﺤﻨﻤﻟ ﺔﻴﺗﻵا ﺔﻗﻼﻌﻟا سوﺎﺟ ﺪﺟوأ : ) . ( 2 2

.

)

(

h X

e

h

G

=

−

π

ε

Where:

G(ε) Probability of error X (for the Y-axis) e natural logarithmic base (2.71828)

X the error h is a constant (h =

2

.

1

σ

)

σ the standard deviation (MSE)

ﺖ ﺑﺎﺛ راﺪﻘﻣ ﻂﺳﻮﺘﻤﻟا ﻰﻌﻴﺑﺮﺘﻟا ﺄﻄﺨﻟاو (σ) ﺧاو ،دﺎ ﺻرﻷا ﺔ ﻗد ﻰ ﻠﻋ ﻪ ﺘﻤﻴﻗ ﻒ ﻗﻮﺘﻳو ،ﺪ ﺣاﻮﻟا ﻰ ﻨﺤﻨﻤﻠﻟ ﺔﺒﺴ ﻨﻟﺎﺑ فﻼﺘ ﻰﻟﺎﺘﻟا ﻞﻜﺸﻟا ﻰﻓ ﺢﺿﻮﻣ ﻮه ﺎﻤآ ﻰﻨﺤﻨﻤﻟا ﻞﻜﺷ ﻰﻓ ﺮﺛﺆﻳ ﻪﺘﻤﻴﻗ : ﺔﻤﻴﻗ ﺖﻠﻗ ﺎﻤﻠآ ، دﺎﺻرﻷا ﺔﻗد تدادزا ﺎﻤﻠآ (σ) ﻰﻨﺤﻨﻤﻟا بﺪﺤﺗ دادزا ﻰﻟﺎﺘﻟﺎﺑو ، . و ،ﺢﻴﺤﺼﻟا ﺪﺣاﻮﻟاو ﺮﻔﺼﻟا ﻦﻴﺑ ﻪﺘﻤﻴﻗ ىﺮﺸﻋ ﺮﺴﻜﺑ ﻪﻨﻋ ﺮﺒﻌﻳو ﻪﺛوﺪﺣ ﺔﺒﺴﻧ ﻮه ﺎﻣ ءﻲﺷ ثوﺪﺣ لﺎﻤﺘﺣا ﻰﻨﻌﻣ نإ ضﺮﻌﺑ ﻰﻟﺎﺘﻟا لﺎﺜﻤﻟا ضﺮﻌﻧ تﻻﺎﻤﺘﺣﻹا ﺔﻳﺮﻈﻧ ﻦﻋ ﺔﻄﺴﺒﻣ ةﺮﻜﻓ : ثﺪﺤﻟا راﺮﻜﺗ تاﺮﻣ دﺪﻋ ﺎﻣ ﺄﻄﺧ وأ ﺎﻣ ءﻲﺷ ثوﺪﺣ لﺎﻤﺘﺣإ = ثاﺪﺣﻸﻟ ﻰﻠﻜﻟا دﺪﻌﻟا

ﻪﺑ قوﺪﻨﺻ ﻦﻣ ءاﺮﻤﺣ ةﺮآ جاﺮﺧإ لﺎﻤﺘﺣا ًﻼﺜﻤﻓ 10 و ،ءاﺮﻤﺣ تاﺮآ 15 و ،ءادﻮﺳ ةﺮآ 25 ﻮه ءﺎﻀﻴﺑ ةﺮآ :

%

20

5

1

15

25

10

10

=

=

+

+

ﺑو سوﺎﺟ ﻰﻨﺤﻨﻣ ﺖﺤﺗ ﺔﺣﺎﺴﻤﻟا نﺈﻓ ﻰﻟﺎﺘﻟﺎ = 1 ءﺎﻄﺧﻷا ﻊﻴﻤﺟ ثوﺪﺣ تﻻﺎﻤﺘﺣا عﻮﻤﺠﻣ ﻞﺜﻤﺗ ﺎﻬﻧﻷ ، .

Area under the (Gauss) curve =

∫

+∞ ∞ − −

dx

e

h

_.

h2.X2

_.

π

= 1 ﻦﻴﺘﻤﻴﻗ ﻦﻴﺑ ﺄﻄﺧ عﻮﻗو لﺎﻤﺘﺣا ﺐﺴﺤﻳ ﺎﻤﻧإو ،ﻦﻴﻌﻣ ﺄﻄﺧ عﻮﻗو ﺐﺴﺤﻳ نأ دﺎﺘﻌﻤﻟا ﻦﻣ ﺲﻴﻟو (X1 , X2)

X

1

X

2

Error

X

G(e)

1 X 2 X 2 X 1 ﺄﻄﺧ عﻮﻗو لﺎﻤﺘﺣا ﻦﻴﺘﻤﻴﻗ ﻦﻴﺑ (X1 , X2) = =

∫

− 2 1 2 2_.

.

X X X h

e

h

π

دﺎﺻرﻷا ﺔﻗد ﺮﻴﻳﺎﻌﻣ ﺔﻧرﺎﻘﻣ : ﺄﻄﺨﻟا ﻰﻨﻌﻣ ) ﺮﺜآأ ﻪﺛوﺪﺣ ﻞﻤﺘﺤﻤﻟا ( ﺮﻐ ﺻأ ءﺎ ﻄﺧأ ﺎ ﻬﺑ ﻰ ﺘﻟا دﺎ ﺻرﻷا دﺪ ﻋ نﻮ ﻜﻳ دﺎ ﺻرﻷا ﻦﻣ ﺔﻋﻮﻤﺠﻣ ىأ ﻰﻓ ﻪﻧأ ﻪﻨﻣ ﺮﺒآأ ءﺎﻄﺧأ ﺎﻬﺑ ﻰﺘﻟا دﺎﺻرﻷا دﺪﻋ ىوﺎﺴﺗ ﻞﻤﺘﺤﻤﻟا ﺄﻄﺨﻟا ﻦﻣ – ﺿو ﺎﻤآ ﻒﻳﺮﻌﺗ ﻰﻓ ﻚﻟذ ﺎﻨﺤ (Probable Error) . r = 0.6745 σ ﻂﺳﻮﺘﻤﻟا ﺄﻄﺨﻟا ﻚﻟﺬآو (η) : η = 0.7979 σ ﻂﺳﻮﺘﻤﻟا ﻰﻌﻴﺑﺮﺘﻟا ﺄﻄﺨﻟا ﻦﻣ ﻞآ ﺰﻴﻤﺘﻳو (σ) ﻰﺗﻵﺎﺑ ﻞﻤﺘﺤﻤﻟا ﺄﻄﺨﻟاو ، : 1 ﺔﺒﺟﻮﻣ وأ ﺔﺒﻟﺎﺳ ﺖﻧﺎآ ﻮﻟ ﺎﻤﻴﻓ ﺄﻄﺨﻟا ةرﺎﺷﺈﺑ ﺎﻤﻬﻨﻣ ﻞﻜﻟ ﺔﻗﻼﻋ ﻻ ) ﻊﻴﺑﺮﺘﻟا دﻮﺟﻮﻟ .( 2 ﺮﻴﺛﺄﺗ ﺎﻤﻬﻴﻓ ﺮﻬﻈﻳ ﻦﻣ ًﻼآ مﺪﺨﺘﺴﻳ ﻚﻟﺬﻟ ، ﺔﻤﻴﻘﻟا ﺪﻳﺰﻳ ﻊﻴﺑﺮﺘﻟا نﻷ ﻚﻟذو ، ًاﺮﻜﺒﻣ ءﺎﻄﺧﻷا (σ) و (r) دﺎ ﺻرﻷا ﺔ ﻗد سﺎﻴﻗ ﻰﻓ ﻂﺳﻮﺘﻤﻟا ﻰﻌﻴﺑﺮﺘﻟا ﺄﻄﺨﻟا ماﺪﺨﺘﺳا ﻞﻀﻔﻳو ﺎﻬﻨﻣ ﺢﻟﺎﺻ ﺮﻴﻐﻟا دﺎﻌﺒﺘﺳاو MSE عﻮ ﻗﻮﻟا ﺔ ﻠﻤﺘﺤﻣ ﺮﻴﻐﻟا دﺎﺻرﻷا ﺾﻌﺑ ﺾﻓر ﺪﻨﻋ ءﺎﻄﺧﻷا ﻰﻨﺤﻨﻣ ﺖﺤﺗ . ﺎﻬﻴﻓ ﻊﻘﻳ ﻰﺘﻟا ءﺎﻄﺧﻷا ﺪﺣ (50%) دﺎﺻرﻷا ﻦﻣ : r = ± 0.6745 σ

Error

G(e)

0.

67

45

σ

r

r

0.

67

45

σ

ﻦﻴﺘﻤﻴﻗ ﻦﻴﺑ ﺄﻄﺧ عﻮﻗو لﺎﻤﺘﺣا نأ ﺎﻧﺮﺷأ نأ ﻖﺒﺳ ﺎﻤآو (X1 , X2) = =

∫

− 2 1 2 2_.

.

X X X h

e

h

π