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ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡
7.1 °ÂÓÈο
7.2 ƒÔ‹ ·‰Ú¿ÓÂÈ·˜ ÔÚıÔÁˆÓÈ΋˜ ‰È·ÙÔÌ‹˜
7.3 ™Ù·ÙÈ΋ ÚÔ‹7.4 ƒÔ‹ ·ÓÙ›ÛÙ·Û˘. ¶ÔÏÈ΋ ÚÔ‹
·‰Ú¿ÓÂÈ·˜–·ÓÙ›ÛÙ·Û˘
7.5 ∞ÎÙ›Ó· ·‰Ú¿ÓÂÈ·˜
7.6 £ÂÒÚËÌ· Steiner
7.7 ∂Ê·ÚÌÔÁ¤˜
133
7K E º A § A I O
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134 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡
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∂ȉȈÎfiÌÂÓÔÈ ÛÙfi¯ÔÈ:∂ȉȈÎfiÌÂÓÔÈ ÛÙfi¯ÔÈ:
™ÎÔfi˜ ÙÔ˘ ÎÂÊ·Ï·›Ô˘ ·˘ÙÔ‡ ›ӷÈ, ÔÈ Ì·ıËÙ¤˜ Ó· ηٷÓÔ‹ÛÔ˘Ó ÙËÓ
·‰Ú¿-
ÓÂÈ· Ù˘ ‡Ï˘, ÙË ÚÔ‹ ·‰Ú¿ÓÂÈ·˜ ÙÔ˘ ÛÒÌ·ÙÔ˜ Û·Ó ¯·Ú·ÎÙËÚÈÛÙÈÎfi
̤ÁÂ-
ıÔ˜ Ù˘ ‰˘Ûη̄›·˜ ÌÈ·˜ ηÌÙfiÌÂÓ˘ ηٷÛ΢‹˜.
¡· ˘ÔÏÔÁ›˙Ô˘Ó ÙȘ ÚÔ¤˜ ·‰Ú¿ÓÂÈ·˜–·ÓÙ›ÛÙ·Û˘–ÙËÓ ÔÏÈ΋ ÚÔ‹
·‰Ú¿ÓÂÈ·˜ Î·È ·ÓÙ›ÛÙ·Û˘–ÙȘ ·ÎÙ›Ó˜ ·‰Ú¿ÓÂÈ·˜ ÌÈ·˜ ‰È·ÙÔÌ‹˜
ˆ˜
ÚÔ˜ ÙÔÓ ¿ÍÔÓ·.
¡· ‰È·Ù˘ÒÓÔ˘Ó ÙÔ £ÂÒÚËÌ· Steiner Î·È Ó· ÙÔ ÂÊ·ÚÌfi˙Ô˘Ó ÛÂ
Ú·-
ÎÙÈΤ˜ ÂÊ·ÚÌÔÁ¤ .̃
£ÂˆÚ‹Û·Ì ÛÎfiÈÌÔ Ó· ·Ú·ı¤ÛÔ˘Ì ·Ú·‰Â›ÁÌ·Ù· Ï˘Ì¤Ó· Ô˘ ·Ó·-
ʤÚÔÓÙ·È ÛÙÔ ·ÏÔ‡ Ù·˘ –ÛÙÔ ‰ÈÏfi Ù·˘– ÛÙËÓ ÎÔ›ÏË ‰È·ÙÔÌ‹. ªÂ
‚¿ÛË
·˘Ù¿, ÊÚÔÓԇ̠fiÙÈ ÔÈ Ì·ıËÙ¤˜ ı· Â›Ó·È Û ı¤ÛË Ó· ÂÈÏ‡Ô˘Ó ÌÂ
¢¯¤ÚÂÈ·
‰È¿ÊÔÚ· Ú·ÎÙÈο ÚÔ‚Ï‹Ì·Ù·.
7.1 °∂¡π∫∞
°ÓˆÚ›˙Ô˘Ì ·fi ÙË Ê˘ÛÈ΋, fiÙÈ Ë ‡ÏË ·ÚÔ˘ÛÈ¿˙ÂÈ ·ÓÙ›ÛÙ·ÛË ÛÂ
οı ÌÂ-
Ù·‚ÔÏ‹ Ù˘ ÎÈÓËÙÈ΋˜ Ù˘ ηٿÛÙ·Û˘. ∆ËÓ ·ÓÙ›ÛÙ·ÛË ·˘Ù‹˜ Ù˘
ÌÂÙ·‚Ô-
Ï‹˜ ·fi ÙËÓ ‡ÏË, ÙËÓ ÔÓÔÌ¿˙Ô˘Ì ·‰Ú¿ÓÂÈ· Ù˘ ‡Ï˘. ŸÛÔ ÈÔ ÌÂÁ¿ÏË
Ì¿-
˙· ¤¯Ô˘Ó Ù· ÛÒÌ·Ù·, Â›Ó·È Ê·ÓÂÚfi, ÙfiÛÔ ÈÔ ÌÂÁ¿ÏË ·‰Ú¿ÓÂÈ·
ÂÌÊ·Ó›-
˙Ô˘Ó.
∂›Û˘ ÁÓˆÚ›˙Ô˘Ì fiÙÈ ÁÈ· ¤Ó· ÛÒÌ· Ô˘ ÂÚÈÛÙÚ¤ÊÂÙ·È (Û¯. 7.1.·)
Á‡-
Úˆ ·fi ¤Ó· ÌfiÓÈÌÔ ¿ÍÔÓ· Ì ÛÙ·ıÂÚ¿ ÁˆÓȷ΋ Ù·¯‡ÙËÙ· ˆ, Ë
ÎÈÓËÙÈ΋ ÙÔ˘
ÂÓ¤ÚÁÂÈ· Â›Ó·È ›ÛË ÌÂ:
(7.1.1)
™ÙÔÓ Ù‡Ô 7.1.1 ›ӷÈ: ˆ = ÛÙ·ıÂÚ‹ ÁˆÓȷ΋ Ù·¯‡ÙËÙ· ÙÔ˘
ÛÒÌ·ÙÔ˜
Ô˘ ÂÚÈÛÙÚ¤ÊÂÙ·È Á‡Úˆ ·fi ÙÔ ÌfiÓÈÌÔ ¿ÍÔÓ· y - yã, m i = ÔÈ Ôχ
ÌÈÎÚ¤˜
(ÛÙÔȯÂÈÒ‰ÂȘ) Ì¿˙˜ ·fi ÙȘ Ôԛ˜ ·ÔÙÂÏÂ›Ù·È ÙÔ ÛÒÌ· ™, Î·È yi
= oÈ ·-
ÔÛÙ¿ÛÂȘ ·fi ÙÔÓ ¿ÍÔÓ· y - yã ÙˆÓ Ì·˙ÒÓ ÙÔ˘ ÛÒÌ·ÙÔ˜.
™ÙËÓ ÂχıÂÚË ÂÚÈÛÙÚÔÊÈ΋ ΛÓËÛË ÙÔ˘ ÛÒÌ·ÙÔ˜ ÔÈ ÂχıÂÚÔÈ
¿ÍÔÓ˜
‰ÂÓ ·Ú·Ì¤ÓÔ˘Ó ÌfiÓÈÌÔÈØ ¤¯Ô˘Ì ÙÚÂȘ ¿ÍÔÓ˜ ÂÚÈÛÙÚÔÊ‹˜ Ô˘ ÙÔ˘˜
ÔÓÔ-
Ì¿˙Ô˘Ì ·ÚÈÔ˘˜ ¿ÍÔÓ˜ ·‰Ú·Ó›·˜. ∞˘ÙÔ› Â›Ó·È ÔÈ ∞, µ, ° Û¯.
7.1.‚.
E = 12
ˆ ™ m i r i2
ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 135
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™¯‹Ì· 7.1· ¶ÂÚÈÛÙÚÔÊ‹ ÛÒÌ·ÙÔ˜ ™¯‹Ì· 7.1‚ √È ÙÚÂȘ ·ÚÈÔÈ ¿ÍÔÓ˜
∞-µ-°
·‰Ú¿ÓÂÈ·˜
7.2 ƒ√¶∏ ∞¢ƒ∞¡∂π∞™ √ƒ£√°ø¡π∫∏™ ¢π∞∆√ª∏™
√ÓÔÌ¿˙Ô˘Ì ÚÔ‹ ·‰Ú¿ÓÂÈ·˜ ÙÔ˘ ÛÒÌ·ÙÔ˜ ˆ˜ ÚÔ˜ ¿ÍÔÓ· ÂÚÈÛÙÚÔÊ‹˜
y - yã, ÙÔ ¿ıÚÔÈÛÌ· Ù̂ Ó ÁÈÓÔÌ¤ÓˆÓ fiÏˆÓ ÙˆÓ ÌÈÎÚÒÓ Ì·˙ÒÓ mi ›
ÙÔ ÙÂ-
ÙÚ¿ÁˆÓÔ ÙˆÓ ·ÔÛÙ¿ÛÂˆÓ ÙÔ˘ ΤÓÙÚÔ˘ ‚¿ÚÔ˘˜ ÙÔ˘˜ ·fi ÙÔÓ ¿ÍÔÓ·.°È·
ÌÈ· ÂÈÊ¿ÓÂÈ· F, Î·È ÙȘ ÂÈÊ¿ÓÂȘ f i ·˘Ù‹˜, Ë ÚÔ‹ ·‰Ú¿ÓÂÈ·˜
Ù˘
ÂÈÊ¿ÓÂÈ·˜ ·˘Ù‹˜ ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· y - yã Û˘Ì‚ÔÏ›˙ÂÙ·È Ì Jy ηÈ
ÌÂÙÚ¿-
Ù·È ÛÂ cm4.
ŸÌÔÈ·, ÁÈ· ÙËÓ ›‰È· ÂÈÊ¿ÓÂÈ· F Î·È ÙȘ ÂÈÊ¿ÓÂȘ fi ·˘Ù‹˜, Ë ÚÔ‹
·-
‰Ú¿ÓÂÈ·˜ Ù˘ ÂÈÊ¿ÓÂÈ·˜ ·˘Ù‹˜ ˆ˜ ÚÔ˜ ¿ÍÔÓ· x-xãÛ˘Ì‚ÔÏ›˙ÂÙ·È Ì Jx
ηÈ
ÌÂÙÚ¿Ù·È ÛÂ cm4.
∂›Ó·È Ê·ÓÂÚfi, fiÙÈ ÙÔ ÁÈÓfiÌÂÓÔ ™miri2
ÂÎÊÚ¿˙ÂÈ ÙÔ Ì¤ÙÚÔ Ù˘ ·‰Ú¿ÓÂÈ·˜
ÙÔ˘ ÛÒÌ·ÙÔ˜, ‰ËÏ·‰‹ ÙËÓ ·ÓÙ›ÛÙ·ÛË Ô˘ ÚÔ‚¿ÏÏÂÙ·È ·fi ÙÔ Û‡ÛÙËÌ·,
fi-
Ù·Ó ·fi ÙËÓ Î·Ù¿ÛÙ·ÛË ËÚÂÌ›·˜, ÙÔ ı¤ÙÔ˘Ì Û ΛÓËÛË.
∆Ô Ì¤ÁÂıÔ˜ Ù˘ ÚÔ‹˜ ·‰Ú¿ÓÂÈ·˜ ÌÈ·˜ ‰È·ÙÔÌ‹˜, ÌÔÚԇ̠̠·Ï¿
Ì·ıËÌ·ÙÈο, ηٿ ÚÔÛ¤ÁÁÈÛË, Ó· ̆ ÔÏÔÁ›˙Ô˘Ì ÂÚÁ·˙fiÌÂÓÔÈ ̂ ˜
ÂÍ‹˜:
1. Èڛ˙Ô˘ÌÂ, Û fiÛÔ ÙÔ ‰˘Ó·ÙfiÓ ÌÂÁ·Ï‡ÙÂÚÔ ·ÚÈıÌfi ÂÈÊ·ÓÂÈÒÓ
fi(ÛÙÔȯÂÈÒ‰Ë ÂÌ‚·‰¿), ÙËÓ ‰È·ÙÔÌ‹.
2. ∫¿ı ÂÈÊ¿ÓÂÈ· fi ÙËÓ ÔÏÏ·Ï·ÛÈ¿˙Ô˘Ì › ÙÔ ÙÂÙÚ¿ÁˆÓÔ Ù˘
·fi-
ÛÙ·Û˘ Ù˘ xi, yi ·fi ÙÔ˘˜ ¿ÍÔÓ˜ y - yãÎ·È x-xã(ÔÈ ¿ÍÔÓ˜ x-xã
Î·È y -
yã ÂÚÓÔ‡Ó ·fi ÙÔ Î¤ÓÙÚÔ ‚¿ÚÔ˘˜ G Ù˘ ‰È·ÙÔÌ‹˜
m2
y
B
°
A
x
r2
m1
™
r1r3
m3
136 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡
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™¯‹Ì· 7.2· ∂‡ÚÂÛË ÚÔ‹˜ ·‰Ú¿ÓÂÈ·˜ ‰È·ÙÔÌ‹˜
‰ËÏ·‰‹ ı· ¤¯Ô˘ÌÂ
(7.2.1)
‰ËÏ·‰‹ ÙȘ Û¯¤ÛÂȘ:
Jx = f1y1 + f2y2 +… + fiy1
2Î·È Jy = f1x
2
1 + f2x2
2 +… + fixi
2
°È· ÙȘ ηٷÔÓ‹ÛÂȘ Ô˘ ı· ÌÂÏÂÙ‹ÛÔ˘Ì ÈÔ Î¿Ùˆ (ο̄Ë,
‰È¿ÙÌËÛË,ÛÙÚ¤„Ë), Â›Ó·È Ê·ÓÂÚfi fiÙÈ ÚÔÛ‰ÈÔÚÈÛÙÈÎfi ̤ÁÂıÔ˜ ÙˆÓ
‰È·ÊfiÚˆÓ ‰È·ÙÔ-
ÌÒÓ (ÔÚıÔÁˆÓÈÎÒÓ, ΢ÎÏÈÎÒÓ, Û˘Óı¤ÙˆÓ) Â›Ó·È ·˘Ùfi ÙÔ ¿ıÚÔÈÛÌ·
ÙˆÓ ·Ô-
ÛÙ¿ÛÂÒÓ ÙÔ˘˜ ·fi ÙÔ˘˜ ¿ÍÔÓ˜ x-xã Î·È y - yã.
∏ ‰È·ÙÔÌ‹ ÂÓfi˜ ÛÒÌ·ÙÔ˜ ¤¯ÂÈ –ÚÔÊ·ÓÒ˜– ‰È·ÊÔÚÂÙÈΤ˜ ÚÔ¤˜
·‰Ú¿-
ÓÂÈ·˜ ̂ ˜ ÚÔ˜ ‰È·ÊÔÚÂÙÈÎÔ‡˜ ¿ÍÔÓ˜.
∞˘Ùfi Û˘Ó¿ÁÂÙ·È fiÙÈ, fiÛÔ ÈÔ ÌÂÁ¿ÏË Â›Ó·È Ë ÚÔ‹ ·‰Ú¿ÓÂÈ·˜ Ù˘
‰È·-
ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ¿ÍÔÓ·, ÙfiÛÔ ÈÔ ÌÂÁ¿ÏË Â›Ó·È Î·È Ë ·ÓÙ›ÛÙ·ÛË (Ù˘
‰È·ÙÔ-
Ì‹˜ ÛÙËÓ ÂÚÈÛÙÚÔÊ‹ ˆ˜ ÚÔ˜ ·˘ÙfiÓ ÙÔÓ ¿ÍÔÓ·Ø Â›Ó·È ‰ËÏ·‰‹ Ë ÚÔ‹
·-
‰Ú·Ó›·˜ ¯·Ú·ÎÙËÚÈÛÙÈÎfi ̤ÁÂıÔ˜ Ù˘ ‰˘Ûη̄›·˜ ÌÈ·˜ ηÌÙfiÌÂÓ˘
η-
Ù·Û΢‹˜.
∂›Ó·È ÚÔÊ·Ó¤˜ fiÙÈ ·Ó ı¤ÏÔ˘Ì ӷ ˘ÔÏÔÁ›ÛÔ˘Ì ÙË ÚÔ‹
·‰Ú¿ÓÂÈ·˜ÌÈ·˜ ÔÚıÔÁˆÓÈ΋˜ ‰È·ÙÔÌ‹˜ ‰È·ÛÙ¿ÛÂˆÓ h, b ‰ÂÓ ¤¯Ô˘Ì ·Ú¿
Ó· ıˆڋ-
ÛÔ˘Ì ÙÔ ÔÚıÔÁÒÓÈÔ ¯ˆÚÈṲ̂ÓÔ Û i ÙÔ Ï‹ıÔ˜ ÂÈÊ¿ÓÂȘ (ψڛ‰Â˜)
fi, ›-
Û˜ Î·È ·Ú¿ÏÏËϘ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· .¯. x-xã Ï¿ÙÔ˘˜:
, Ì‹ÎÔ˘˜ b Î·È ÂÌ‚·‰Ô‡
°È· οı ٤ÙÔÈ· ψڛ‰· ı· ¤¯Ô˘Ì ·Ó¿ÏÔÁ· yi Î·È fi, ‰ËÏ·‰‹ ¤Ó·
¿ıÚÔÈ-
ÛÌ· yi2 . fi Ô˘ ÙÔ ¿ıÚÔÈÛÌ¿ ÙÔ˘˜ ı· ‰ÒÛÂÈ ÁÈ· i → ∞.
fi = ⋅
2iy
i=
h2i
= h2i
Jx = ™fi yi2 Î·È Jy = ™fi xi
2
y
yã
xã x
xi
yi
fi
G
ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 137
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(7.2.2)
∞Ó¿ÏÔÁ· ÚÔ·ÙÂÈ Î·È Ô ¯Ú‹ÛÈÌÔ˜ Ù‡Ô˜
(7.2.3)
∂¿Ó ÁÈ· ·Ú¿‰ÂÈÁÌ· Ï¿‚Ô˘Ì ÌÈ· ‰È·ÙÔÌ‹ ÔÚıÔÁˆÓÈ΋ ‰È·ÛÙ¿ÛˆÓ
12 cm Î·È 4 cm, ‰ËÏ·‰‹ h = 12 cm Î·È b = 4 cm, ÙfiÙ ÁÈ’ ·˘Ù‹Ó
ı· ¤¯Ô˘ÌÂ,
∞fi ÙË Û‡ÁÎÚÈÛË ·˘ÙÒÓ ÙˆÓ ·ÔÙÂÏÂÛÌ¿Ù̂ Ó, ÚÔ·ÙÂÈ fiÙÈ: Ix =
9Iy.
7.3 ™∆∞∆π∫∏ ƒ√¶∏ ø™ ¶ƒ√™ ∞•√¡∞ ∂¶π¶∂¢√À ™Ã∏ª∞∆√™
°È· Ó· ‚Úԇ̠ÙË ÛÙ·ÙÈ΋ ÚÔ‹ ÂÓfi˜ ÔÚıÔÁˆÓ›Ô˘, Ï¢ÚÒÓ · Î·È h
ˆ˜
ÚÔ˜ ¿ÍÔÓ· Ô˘ Û˘Ì›ÙÂÈ Ì ÙËÓ ÏÂ˘Ú¿ ÙÔ˘ –¤ÛÙˆ ÙËÓ ∞µ–
¯ˆÚ›˙Ô˘ÌÂ
ÙËÓ ÂÈÊ¿ÓÂÈ· ÛÂ Ó ÙÔ Ï‹ıÔ˜ ÛÙÔȯÂÈÒ‰ÂȘ ÂÈÊ¿ÓÂȘ (Û¯. 7.3·)
·Ú¿Ï-
ÏËϘ ÚÔ˜ ÙËÓ ÏÂ˘Ú¿ ∞µ, ›Ûˆ˜ ÂÈÊ¿ÓÂÈ·˜ Â. ∆fiÙ ı· ¤¯Ô˘ÌÂ
Ó . Â = · . h (1)
™¯‹Ì· 7.3· ∂‡ÚÂÛË ÛÙ·ÙÈ΋˜ ÚÔ‹˜ ›‰Ԣ Û¯‹Ì·ÙÔ˜
y7
y7
A B
y1
Â
h
Â
Â
R=7Â
Â
Â
Â
·
xxã
I y =b ⋅ h
3
12=
12 ⋅ 43
12= 64 cm 4
Ix
=b ⋅ h
3
12=
4 ⋅ 123
cm4
12= 576 cm 4
Iy–y′ =b3 ⋅ h
12
ÚÔ‹ ·‰Ú¿ÓÂÈ·˜ ÔÚıoÁˆÓÈ΋˜
‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ y-yã
Ix–x′ =b ⋅ h3
12
ÚÔ‹ ·‰Ú¿ÓÂÈ·˜ ÔÚı.
‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ x –xã
138 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡
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fiÔ˘ ∂ ÙÔ ÂÌ‚·‰fiÓ ÙÔ˘ ÔÚıÔÁˆÓ›Ô˘ (∂ = · . h)
°È· ÙËÓ ÂÈÊ¿ÓÂÈ· ÙÔ˘ Û¯‹Ì·ÙÔ˜ Ì·˜ Ë ÛÙ·ÙÈ΋ ÚÔ‹ ÙÔ˘ ˆ˜ ÚÔ˜
ÙÔÓ
¿ÍÔÓ· x-xã ı· ›ӷÈ:
ÕÚ·:
¶∞ƒ∞¢∂π°ª∞ 1Ô
¡· ‚ÚÂı› Ë ÛÙ·ÙÈ΋ ÚÔ‹ Ù˘ ‰È·ÙÔÌ‹˜, ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· x-xã
§‡ÛË
∏ ÛÙ·ÙÈ΋ ÚÔ‹ ª1 Ù˘ ‰È·ÙÔÌ‹˜ 1 ̂ ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· x-xã›ӷÈ:
ª1 = 18 cm2 . 6 cm = 108 cm3
H ÛÙ·È΋ ÚÔ‹ ª2 Ù˘ ‰È·ÙÔÌ‹˜ 2 ̂ ˜ ÚÔ˜ ¿ÍÔÓ· x-xã›ӷÈ:
ª2 = 27 cm2 . 1,5 cm = 40,5 cm3
3 cm
3 cm
6 cm
9 cm
x
2
1
G2
G1
M™ = · ⋅ h ⋅ h 1 – h2
M™
= · ⋅ 2h1
⋅ h1 – · ⋅ 2h
2 ⋅ h
2= 2· ⋅ h
2h
1 – h
2
™=
1 1 –
2 ⋅
2
M™
=· h
2
2= E ⋅
h
2
140 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡
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∂Ô̤ӈ˜ Ë ÛÙ·ÙÈ΋ ÚÔ‹ ̂ ˜ ÚÔ˜ ¿ÍÔÓ· x-xãÙ˘ ‰È·ÙÔÌ‹˜ ›ӷÈ:
ª = ª1 + ª2 = 148,5 cm3
¶∞ƒ∞¢∂π°ª∞ 2Ô
¡· ‚ÚÂı› Ë ÛÙ·ÙÈ΋ ÚÔ‹ ÌÈ·˜ ‰È·ÙÔÌ‹˜ ·ÏÔ‡ Ù·˘ ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ·
x-
xã
§‡ÛË
∏ ÛÙ·ÙÈ΋ ÚÔ‹ ª1 Ù˘ ‰È·ÙÔÌ‹˜ 1 ̂ ˜ ÚÔ˜ ¿ÍÔÓ· x-xã›ӷÈ:
ª1 = 24 cm2 . 2,5 cm = 60 cm3
∏ ÛÙ·ÙÈ΋ ÚÔ‹ ª2 Ù˘ ‰È·ÙÔÌ‹˜ 2 ̂ ˜ ÚÔ˜ ¿ÍÔÓ· x-xã›ӷÈ:
ª2 = 3 cm2 . 0,5 cm @ 1,5 cm3
∏ ÛÙ·ÙÈ΋ ÚÔ‹ ª3 Ù˘ ‰È·ÙÔÌ‹˜ 3 ÚÔ˜ ¿ÍÔÓ· x-xã ›ӷÈ:
ª3 = 12 cm2 . 2 cm = 24 cm3
∂Ô̤ӈ˜ Ë ÛÙ·ÙÈ΋ ÚÔ‹ ˆ˜ ÚÔ˜ ¿ÍÔÓ· x-xãÙÔ˘ ·ÏÔ‡ Ù·˘ ›ӷÈ
ª = ª1 + ª2 - ª3 = 37,5 cm3
3 cm
3 cm 2.5 cm
1 cm
4 cm
8 cm
xx’2
2
1
G1
G1
G3
ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 141
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7.4.1 ƒÔ‹ ·ÓÙ›ÛÙ·Û˘
£ÂˆÚԇ̠ÌÈ· ÔÚıÔÁˆÓÈ΋ ‰È·ÙÔÌ‹ ‰È·ÛÙ¿ÛÂˆÓ h, b. (Û¯.
7.4.1·)
√ÓÔÌ¿˙Ô˘Ì ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wx Ù˘ ‰È·ÙÔÌ‹˜ ∞µ°¢ ÙÔ ËÏ›ÎÔ Ù˘
‰È-
·›ÚÂÛ˘ Ù˘ ÚÔ‹˜ ·‰Ú·Ó›·˜ Ix Ù˘ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ÙÔÓ
ÎÂÓÙÚÔ‚·ÚÈÎfi
¿ÍÔÓ· x, ‰È· Ù˘ ·ÔÛÙ¿Ûˆ˜ ÙÔ˘ ÈÔ ·ÔÌ·ÎÚ˘Ṳ̂ÓÔ˘ ÛËÌ›Ԣ Ù˘
‰È·-
ÙÔÌ‹˜ ·fi ÙÔÓ ¿ÍÔÓ· x. Œ¯Ô˘Ì ‰ËÏ·‰‹:
™¯‹Ì· 7.4.1· ƒÔ‹ ·ÓÙ›ÛÙ·Û˘
(7.4.1.1)
¶·Ú·Ù‹ÚËÛË
∞Ó Ô Î‡ÚÈÔ˜ ¿ÍÔÓ·˜ ·‰Ú·Ó›·˜ x-xã Â›Ó·È Î·È ¿ÍÔÓ·˜ Û˘ÌÌÂÙÚ›·˜
Ù˘ ‰È·-
ÙÔÌ‹˜ ÙfiÙ ı· ¤¯Ô˘ÌÂ:
Î·È ı· ¤¯Ô˘ÌÂ:
(7.4.1.2)
∞Ó¿ÏÔÁ· ÚÔ·ÙÂÈ:
Wx =
bh 3
12h2
=bh 3
6h=
bh2
6‰ËÏ. Wx =
bh2
6
Wx =I xh2
fiÔ˘ · = h2
Wx
=Ix
„
y2
y1
x1 x2
xxã
yã
y
G
142 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡
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§‡ÛË
°È· ÙËÓ ÂÚ›ÙˆÛË (·) ı· ¤¯Ô˘ÌÂ:
°È· ÙËÓ ÂÚ›ÙˆÛË (‚) ı· ¤¯Ô˘ÌÂ
∞fi ÙË Û‡ÁÎÚÈÛË ÙˆÓ ‰‡Ô ·˘ÙÒÓ ÚÔÒÓ ·ÓÙ›ÛÙ·Û˘ ÚÔ·ÙÂÈ:
W· = 3 . Wb
‰ËÏ·‰‹ ‰˘ÛÌÂÓ¤ÛÙÂÚË Â›Ó·È Ë ‰Â‡ÙÂÚË ÙÔÔı¤ÙËÛË ÁÈ·Ù› ¤¯ÂÈ
ÌÈÎÚfiÙÂ-
ÚË ·ÓÙÔ¯‹.
7.4.2 ¶ÔÏÈ΋ ÚÔ‹ ·‰Ú¿ÓÂÈ·˜
∞˜ ıˆڋÛÔ˘Ì ÌÈ· ÔÚıÔÁˆÓÈ΋ ‰È·ÙÔÌ‹ ∞µ°¢ Ì ÙÔ˘˜ ·ÚÈÔ˘˜ ¿ÍÔÓ˜
·-
‰Ú·Ó›·˜ Ù˘ x-xã Î·È y-yã.
°È· ÙÔ ÛÙÔȯÂÈ҉˜ ÂÌ‚·‰fiÓ fi Î·È ·fi ÙË ÁˆÌÂÙÚ›· ÙÔ˘
Û¯‹Ì·ÙÔ˜, ¤-
¯Ô˘ÌÂ
Î·È Û˘ÓÔÏÈο ı· ÚÔ·„ÂÈ:
(7.4.2.1)Ip = Ix + Iy
™f iÚ2 = I x + I y ‰ËÏ.
Ú 2 = x 2 + y 2 ‰ËÏ. fiÚ 2 = f
ix
2+ f
iy
2
Wb
=h ⋅b
2
6=
3 ⋅ b ⋅ b2
6=
3b3
6
W·
=b ⋅h
2
6=
b ⋅ 9b2
6=
9b3
6
144 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡
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™¯‹Ì· 7.4.2· ¶ÔÏÈ΋ ÚÔ‹ ·‰Ú¿ÓÂÈ·˜ ‰È·ÙÔÌ‹˜
√ÓÔÌ¿˙Ô˘Ì ÔÏÈ΋ ÚÔ‹ ·‰Ú¿ÓÂÈ·˜ πp
Ù˘ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ÙËÓ ·Ú¯‹
G ÙˆÓ Î˘Ú›ˆÓ ÎÂÓÙÚÔ‚·ÚÈÎÒÓ ·ÍfiÓˆÓ ·‰Ú·Ó›·˜ x - xã Î·È y - yã
ÙÔ ¿ıÚÔÈ-
ÛÌ· ÙˆÓ ‰‡Ô ÚÔÒÓ ·‰Ú·Ó›·˜ Ù˘ ‰È·ÙÔÌ‹˜ Î·È ÂÎÊÚ¿˙ÂÙ·È ÛÂ
cm4.
ŒÙÛÈ ÁÈ· ÙËÓ ÔÚıÔÁˆÓÈ΋ ‰È·ÙÔÌ‹ ÙÔ˘ Û¯‹Ì·ÙÔ˜ ı· ¤¯Ô˘ÌÂ:
ÕÚ·
°È· ÔÚıÔÁˆÓÈ΋ ‰È·ÙÔÌ‹ Ì b = 4 cm Î·È h = 6 cm ı· ¤¯Ô˘ÌÂ:
7.4.3 ¶ÔÏÈ΋ ÚÔ‹ ·ÓÙ›ÛÙ·Û˘
¶ÚÔËÁÔ‡ÌÂÓ· ·Ó·ÊÂÚı‹Î·Ì ÛÙË ÚÔ‹ ·‰Ú·Ó›·˜ ÌÈ·˜ ‰È·ÙÔÌ‹˜ ˆ˜
ÚÔ˜
ÙÔ˘˜ ·ÚÈÔ˘˜ ¿ÍÔÓ˜ x-xã Î·È y-yã, ÛÙË ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Î·È ÛÙËÓ
ÔÏÈ΋
ÚÔ‹ ·‰Ú·Ó›·˜ Ù˘.™˘Ó·Ê‹˜ Â›Ó·È Î·È Ë ¤ÓÓÔÈ· Ù˘ ÔÏÈ΋˜ ÚÔ‹˜
·ÓÙ›ÛÙ·Û˘ W
pÌÈ·˜
‰È·ÙÔÌ‹˜ Î·È ¯·Ú·ÎÙËÚ›˙ÂÙ·È ˆ˜ Ô ÏfiÁÔ˜ Ù˘ ÔÏÈ΋˜ ÚÔ‹˜
·‰Ú·Ó›·˜
Ù˘ ‰È·ÙÔÌ‹˜ ·˘Ù‹˜ ÚÔ˜ ÙËÓ ·fiÛÙ·ÛË ÙÔ˘ ÈÔ ·ÎÚ·›Ô˘ ÛËÌ›Ԣ Ù˘
(·˘-
ÙÔ‡ ‰ËÏ·‰‹ Ô˘ ‚Ú›ÛÎÂÙ·È ÈÔ Ì·ÎÚÈ¿ ·fi ÙÔ Î¤ÓÙÚÔ ‚¿ÚÔ˘˜ G Ù˘
‰È·ÙÔ-
Ì‹˜.
∂›Ó·È ‰ËÏ·‰‹: (7.4.3.1)
Î·È ÂÎÊÚ¿˙ÂÙ·È Û cm3.
Wp =Ip· =
Ix + Iy·
Ip
=6 ⋅ 412
62
+ 42
cm4 = 104 cm4
Ip
=hb
12= h
2+ b
2
I p =bh
3
12+
hb3
12= hb
12h
2+ b
2
G1
G2
G
y
y
f1
xxh
b
ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 145
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Afi ›Ó·Î˜ ·Ú¤¯ÔÓÙ·È ÔÈ ÚÔ¤˜ ·‰Ú·Ó›·˜ π ‰È·ÊfiÚˆÓ ‰È·ÙÔÌÒÓ,
ÔÈ
ÚÔ¤˜ ·ÓÙ›ÛÙ·Û˘ W·, oÈ ÔÏÈΤ˜ ÚÔ¤˜ ·‰Ú·Ó›·˜ πp, ˆ˜ Î·È ÔÈ
ÔÏÈΤ˜
ÚÔ¤˜ ·ÓÙ›ÛÙ·Û˘ Wp.
ŒÙÛÈ ÁÈ· ÙËÓ ÔÚıÔÁˆÓÈ΋ ‰È·ÙÔÌ‹ ‰È·ÛÙ¿ÛÂˆÓ h Î·È b ı·
¤¯Ô˘ÌÂ:
ÕÚ·:
°È· ÔÚıÔÁˆÓÈ΋ ‰È·ÙÔÌ‹ Ì b = 6 cm Î·È h = 4 cm, ı· ¤¯Ô˘ÌÂ:
7.5 AKTINA A¢PANEIA™
∏ ·ÎÙ›Ó· ·‰Ú·Ó›·˜ ix ÌÈ·˜ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ¿ÍÔÓ·, Â›Ó·È ›ÛË ÌÂ
ÙËÓ ÙÂ-ÙÚ·ÁˆÓÈ΋ Ú›˙· Ù˘ ÚÔ‹˜ ·‰Ú·Ó›·˜ Ù˘ ‰È·ÙÔÌ‹˜ ‰È· Ù˘
ÂÈÊ·Ó›·˜ F
Ù˘ ‰È·ÙÔÌ‹˜ ·˘Ù‹˜. ∂›Ó·È ‰ËÏ·‰‹:
(7.5.1)
∏ ·ÎÙ›Ó· ·‰Ú¿ÓÂÈ·˜ ·ÚÈÛÙ¿ÓÂÈ ÙËÓ ·fiÛÙ·ÛË ·fi ÙÔ˘˜ ¿ÍÔÓ˜
·Ó·ÊÔ-
Ú¿˜ fiÔ˘ ÔÏfiÎÏËÚË Ë ÂÈÊ¿ÓÂÈ· Ù˘ ‰È·ÙÔÌ‹˜ ÌÔÚÔ‡Û ӷ
ÙÔÔıÂÙËı›
“Û˘Ì˘Îӈ̤ÓË” Û ÛËÌ›Ô, ‰È·ÙËÚÒÓÙ·˜ ÙËÓ ›‰È· ÚÔ‹ ·‰Ú·Ó›·˜ ÌÂ
ÙËÓ
·Ú¯È΋ ‰È·ÙÔÌ‹.
ÕÚ· ÌÈ· ÂÈÊ¿ÓÂÈ· ¤¯ÂÈ ‰‡Ô ·ÎÙ›Ó˜ ·‰Ú·Ó›·˜ fiˆ˜ ÔÚ›˙ÔÓÙ·È ·fi
ÙÔÓ
Ù‡Ô 7.5.1.
°È· ÔÚıÔÁˆÓÈ΋ ‰È·ÙÔÌ‹ ‰È·ÛÙ¿ÛÂˆÓ b = 4 cm Î·È h = 6 cm ı·
¤¯Ô˘ÌÂ:
ix
=
bh3
12bh
=h
2
12= 0,289h ‰ËÏ·‰‹ i x = 1,734 cm
ix =IxF
Î·È iy =Iy
F
Wp
= 66
62
+ 42
cm3 = 52 cm 3
Wp =b h
2+ b
2
b
Wp =12
⋅ h2
+ b2
h2
=b h
2+ b
2
6
146 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡
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°È· ÙÂÙÚ·ÁˆÓÈ΋ ‰È·ÙÔÌ‹ Ì b = h = 8 cm, Â›Ó·È ÚÔÊ·Ó¤˜ fiÙÈ:
ix = iy = 0,289 b = 2,312 cm
7.6 £EøPHMA STEINER
∂‰Ò ı· ‚Úԇ̠ÔÈ· Û¯¤ÛË ˘¿Ú¯ÂÈ ÌÂٷ͇ Ù˘ ÚÔ‹˜ ·‰Ú·Ó›·˜ Ù˘
‰È·-
ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· .¯. y-yã(Ô˘ ‰È¤Ú¯ÂÙ·È ·fi ÙÔ Î¤ÓÙÚÔ
‚¿ÚÔ˘˜ G
Ù˘ ‰È·ÙÔÌ‹˜) Î·È Ù˘ ÚÔ‹˜ ·‰Ú·Ó›·˜ Ù˘ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜
¿ÍÔÓ·
Â-ÂãÔ˘ Â›Ó·È ·Ú¿ÏÏËÏÔ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· y-yã.
™¯‹Ì· 7.6· £ÂÒÚËÌ· Steiner
∞fi ÙË ÁˆÌÂÙÚ›· ÙÔ˘ Û¯‹Ì·ÙÔ˜ ¤¯Ô˘ÌÂ:
(1)
›Û˘ ¤¯Ô˘ÌÂ:
(2)I · = ™f i B°
B°2
= G°2
+ BG2 – 2 BG G° Û˘ÓBG°
Â
Â
Â'
fi
y'
y
°
AGB
iy
=
hb3
12bh
=b
2
12= 0,289b ‰ËÏ·‰‹ i y = 1,156 cm
ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 147
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(3)
∂¿Ó ÙË Û¯¤ÛË (1) ÙËÓ ÔÏÏ·Ï·ÛÈ¿ÛÔ˘Ì › ™fi ı· ¤¯Ô˘Ì fiÔ˘,
ÕÚ· ¤¯Ô˘ÌÂ:
(7.6.1)
‰ËÏ·‰‹ Ë ÚÔ‹ ·‰Ú·Ó›·˜ ÌÈ·˜ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ¿ÍÔÓ·
·-·ã·Ú·ÏÏ‹-
ÏÔ˘ ÙÔ˘ ·ÚÈÔ˘ ¿ÍÔÓ· y-yãÈÛÔ‡Ù·È Ì ÙËÓ Î‡ÚÈ· ÚÔ‹ ·‰Ú·Ó›·˜ ηÈ
ÙÔ ÁÈ-
ÓfiÌÂÓÔ Ù˘ ÂÈÊ·Ó›·˜ Ù˘ ‰È·ÙÔÌ‹˜ › ÙÔ ÙÂÙÚ¿ÁˆÓÔ Ù˘
·fiÛÙ·Û˘
ÙˆÓ ‰‡Ô ÚÔ·Ó·ÊÂÚÔÌ¤ÓˆÓ ·ÍfiÓˆÓ.
∞fi ÙÔÓ Ù‡Ô 7.6.1 Â›Ó·È ÚÔÊ·Ó¤˜ fiÙÈ π > πG.
1. °È· ÔÚıÔÁˆÓÈ΋ ‰È·ÙÔÌ‹ ÌÂ: h = 10 cm, b = 3 cm Î·È Â = 2 cm
ηٿ ÙÔ
£ÂÒÚËÌ· ·˘Ùfi ÙÔ˘ Steiner (Ù‡Ô˜ 7.6.1), ı· ¤¯Ô˘ÌÂ
ÕÚ·
2. °È· ÙÚÈÁˆÓÈ΋ ‰È·ÙÔÌ‹ ∞µ° ·Ó ı¤ÏÔ˘Ì ӷ ‚Úԇ̠ÙË ÚÔ‹
·‰Ú·Ó›-
·˜ Ù˘ ‰È·ÙÔÌ‹˜ ·˘Ù‹˜ ˆ˜ ÚÔ˜ ¿ÍÔÓ· Ô˘ ÂÚÓ¿ÂÈ ·fi ÙÔ Î¤ÓÙÚÔ
‚¿ÚÔ˘˜
·˘Ù‹˜ ı· ¤¯Ô˘ÌÂ
(1)
(2)
(3)Ix
1+ I
x2
=bh
12
I x1
= I G1
+ h3
2⋅
bh2
Ix2
= IG 2
+ 2
3h
2⋅
bh
2
IÂ
=3
3⋅ 10
3cm4 = 90 cm 4
IÂ =hb3
12+ h ⋅ b ⋅ b
4
2=
hb3
12+ 3 ⋅
h ⋅ b3
12=
4b3h
12=
b3h
3
IÂ = IG + F ⋅ Â
2
I·
= IG
+ F ⋅ Â2
ÁÈ·Ù› ™f iG A = 0
™fiB°
2= ™f
iG°
2+ ™f
iBG
2 – 2™ f
i –G°Û˘ÓÊ ⋅ G° Û˘ÓÊ
Ê = BG° Î·È GA = G° Û˘Ó 180° – Ê = – G° Û˘ÓÊ
IG
= ™fi
G°2
148 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡
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™¯‹Ì· 7.6‚ ƒÔ‹ ·‰Ú¿ÓÂÈ·˜ ÙÚÈÁˆÓÈ΋˜ ‰È·ÙÔÌ‹˜
∞fi ÙȘ Û¯¤ÛÂȘ (1), (2), (3) ÚÔ·ÙÂÈ:
7.6.2
7.7 ∂º∞ƒª√°∂™
1. ∆∂∆ƒ∞°ø¡π∫∏ ¢π∞∆√ª∏
¶∞ƒ∞¢∂π°ª∞
∆˘ ·Ú·Î¿Ùˆ ÙÂÙÚ·ÁˆÓÈ΋˜ ‰È·ÙÔÌ‹ ,̃ ÏÂ˘Ú¿˜ 4 cm, Ó· ‚ÚÂıÔ‡Ó:
·) √È ÚÔ¤˜ ·‰Ú·Ó›·˜ Ù˘ Ix Î·È Iy ˆ˜ ÚÔ˜ ÙÔ˘˜ ·ÚÈÔ˘˜ ¿ÍÔÓ˜
·‰Ú·-
Ó›·˜ Ù˘
‚) ∏ ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wx ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· ∞µ
Á) ∏ ÔÏÈ΋ ÚÔ‹ ·‰Ú·Ó›·˜ πp ˆ˜ ÚÔ˜ ÙÔ G
‰) ∏ ÔÏÈ΋ ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wp Ù˘ ‰È·ÙÔÌ‹˜ ·˘Ù‹˜
Â) ∏ ·ÎÙ›Ó· ·‰Ú·Ó›·˜ Ù˘ i¢° ˆ˜ ÚÔ˜ ÙË ‚¿ÛË ¢° Ù˘ ‰È·ÙÔÌ‹˜
I G =bh
3
36
h/3
h/3
h/3G1
x1
x2
x1ã
x2ã
Bã B
G2
xã xãA°
b
ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 149
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§‡ÛË
·) ∏ ÚÔ‹ ·‰Ú·Ó›·˜ Ù˘ ÙÂÙÚ·ÁˆÓÈ΋˜ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ¿ÍÔÓ· x-
xã›ӷÈ:
E›Ó·È ÚÔÊ·Ó¤˜ fiÙÈ (;) ηÈ
‚) ∏ ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wx Ù˘ ‰È·ÙÔÌ‹˜ ›ӷÈ:
Î·È ÂÂȉ‹ b = h ¤¯Ô˘ÌÂ
Î·È ÚÔÊ·ÓÒ˜
Á) ∏ ÔÏÈ΋ ÚÔ‹ ·‰Ú·Ó›˜ πp ˆ˜ ÚÔ˜ ÙÔ˘˜ ·ÚÈÔ˘˜ ¿ÍÔÓ˜
·‰Ú·Ó›·˜
Ù˘ ‰È·ÙÔÌ‹˜ ›ӷÈ:
Ip = Ix + Iy
Ip = 2. 21,333 cm4 = 42,666 cm4
Wy = Wx = 10,666 cm3
Wx =4
3
6cm3 = 10,666 cm 3
Wx
=bh
2
6
Iy
= 21,333 cm 4
Ix
=4
4
12cm4 = 21,333 cm 4. ÕÚ· I x = 21,333 cm
4
Ix
=bh
3
12fiÔ˘ b = h
xã
G
x
y
yã
4 cm
4 c m
A B
°¢
150 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡
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ÕÚ·: Ip = 42,666 cm4
‰) ∏ ÔÏÈ΋ ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wp Ù˘ ‰È·ÙÔÌ‹˜ ·˘Ù‹˜ ›ӷÈ:
ÕÚ·
Â) ∂¿Ó ÂÊ·ÚÌfiÛÔ˘Ì ÙÔ £ÂÒÚËÌ· Steiner ı· ¤¯Ô˘ÌÂ:
ÕÚ·:
ÕÚ· Ë ·ÎÙ›Ó· ·‰Ú·Ó›·˜ Ù˘ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ÙË ‚¿ÛË ÙÔ˘ ¢° ı·
›-
Ó·È:
ÕÚ·:
2. Oƒ£√°ø¡π∫∏ ¢π∞∆√ª∏
¶∞ƒ∞¢∂π°ª∞
∆˘ ·Ú·Î¿Ùˆ ÔÚıÔÁˆÓÈ΋˜ ‰È·ÙÔÌ‹˜ ‰È·ÛÙ¿ÛÂˆÓ 4cm Î·È 10 cm Ó·
‚ÚÂ-
ıÔ‡Ó
·) √È ÚÔ¤˜ ·‰Ú·Ó›·˜ Ix Î·È Iy Ù˘ ‰È·ÙÔÌ‹˜ ·˘Ù‹˜ ˆ˜ ÚÔ˜ ÙÔ˘˜
·ÚÈ-
Ô˘˜ ¿ÍÔÓ˜ ·‰Ú·Ó›·˜ Ù˘.
‚) ∏ ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wx Ù˘ ‰È·ÙÔÌ‹˜ ̂ ˜ ÚÔ˜ ¿ÍÔÓ· ∞µ
Á) ∏ ÔÏÈ΋ ÚÔ‹ ·‰Ú·Ó›·˜ Ip ˆ˜ ÚÔ˜ ÙÔ G
‰) ∏ ÔÏÈ΋ ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wp Ù˘ ‰È·ÙÔÌ‹˜
Â) ∏ ·ÎÙ›Ó· ·‰Ú·Ó›·˜ Ù˘ ˆ˜ ÚÔ˜ ÙË ‚¿ÛË Ù˘ ¢°,
i¢°
= 2,31 cm
i¢°
=I¢°
F
=85,333
16
⋅cm2
cm = 2,31 cm
I¢°
= 85,333 cm 4
I¢°
= 21,333 cm 4 + 16 ⋅ 22
cm4 = 21,333 cm 4 + 64 cm 4 = 85,333 cm 4
I¢°
= Ix
+ F ⋅ Â2
Wp
= 21,333 cm 3
Wp =Ip
· , Wp =42,666
2cm3 = 21,333 cm 3.
ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 151
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§‡ÛË
·) ∏ ÚÔ‹ ·‰Ú·Ó›·˜ Ù˘ ÔÚıÔÁˆÓÈ΋˜ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ¿ÍÔÓ· x-xã,
›-
Ó·È:
fiÔ˘ b = 6 cm, h = 10 cm
ÕÚ·:
ÕÚ·:
‚) ∏ ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wx Ù˘ ‰È·ÙÔÌ‹˜ ›ӷÈ:
ÕÚ·:
Wx =6 ⋅ 102
6cm3
Wx =b h2
6fiÔ˘ b = 6 cm, h = 10 cm
y= 180 cm 4
Iy
=10 ⋅ 6
3
12cm4 = 180 cm 4.
Ix
= 500 cm 4
Ix
=6 ⋅ 10
3
12cm4 =
103
2cm4 = 500 cm 4.
Ix =bh3
12
xã
G
x
y
yã
6 cm
c m
A
1 0
B
°¢
152 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡
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∞Ó¿ÏÔÁ· ¤¯Ô˘ÌÂ:
Á) ∏ ÔÏÈ΋ ÚÔ‹ ·‰Ú·Ó›·˜ Ip ˆ˜ ÚÔ˜ ÙÔ˘˜ ·ÚÈÔ˘˜ ¿ÍÔÓ˜
·‰Ú·Ó›˜
Ù˘ ‰È·ÙÔÌ‹˜ ›ӷÈ:
Ip = Ix + Iy Î·È ¿Ú· Ip = (500 + 180) cm4 = 680 cm4.
‰) ∏ ÔÏÈ΋ ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wp Ù˘ ‰È·ÙÔÌ‹˜ ·˘Ù‹˜ ›ӷÈ:
ÕÚ·:
Â) ∂¿Ó ÂÊ·ÚÌfiÛÔ˘Ì ÙÔ £ÂÒÚËÌ· Steiner ı· ¤¯Ô˘ÌÂ:
I¢° = πx + F . Â2
∂›Ó·È: π¢° = 500 cm4 + 60 . 52 cm4 = 500 cm4 + 1500 cm4 = 2000
cm4
ÕÚ·, Ë ·ÎÙ›Ó· ·‰Ú·Ó›·˜ Ù˘ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ÙË ‚¿ÛË ÙÔ˘ ¢° ı·
›-
Ó·È
ÕÚ·: i ¢° = 5,77 cm
i¢°
=I¢°
F= 2000
60cm = 5,77 cm
I¢°
= 2000 cm4
Wp
= 136 cm 3
Wp
=p
· , Wp =6805
cm3 = 136 cm 3
Wy
=hb
6= 60 cm 3
‰ËÏ·‰‹ Wx = 100 cm3
ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 153
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‚) ∏ ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wx Ù˘ ‰È·ÙÔÌ‹˜ ›ӷÈ:
ÕÚ·:
Á) ∏ ÔÏÈ΋ ÚÔ‹ ·‰Ú·Ó›·˜ Ip ˆ˜ ÚÔ˜ ·ÚÈÔ˘˜ ¿ÍÔÓ˜ ·‰Ú·Ó›·˜
Ù˘
‰È·ÙÔÌ‹˜ ›ӷÈ:
Ip = Ix + Iy Î·È ¿Ú· Ip = 2 . 63,585 cm4 = 127,17 cm4
‰) ∏ ÔÏÈ΋ ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wp Ù˘ ‰È·ÙÔÌ‹˜ ·˘Ù‹˜ ›ӷÈ:
ÕÚ·:
Â) ∂¿Ó ÂÊ·ÚÌfiÛÔ˘Ì ÙÔ £ÂÒÚËÌ· Steiner ı· ¤¯Ô˘ÌÂ:
π¢° = Ix + F. Â3
∂›Ó·È: π¢° = 63,585 cm4 + 28,26 . 32 cm4
fiÔ˘ R2 = 3,14 . 32 cm2 = 28,26 cm2
π¢° = 63,585 cm4 + 254,34 cm4
ÕÚ·:
ÕÚ·, Ë ·ÎÙ›Ó· ·‰Ú·Ó›·˜ Ù˘ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· ¢° ı·
›ӷÈ:
ÕÚ·
£ÂˆÚԇ̠¤Ó· ÛÙÔȯÂÈ҉˜ ÙÌ‹Ì· Ù˘ ‰È·ÙÔÌ‹˜, ÙÔ AGB, ·fi ÙÔ
Ï‹-
ıÔ˜ i Ù¤ÙÔÈˆÓ ÙÚÈÁÒÓˆÓ Ô˘ ··ÚÙ›˙Ô˘Ó ÙË ‰È·ÙÔÌ‹.
i¢°
= 3,354 cm
i¢°
=I¢°
F=
317,925
28,26cm
I¢°
= 317,925 cm4
Wp
= 42,39 cm 3
Wp
=127,17
3cm3 = 42,39 cm 3
Wp
=Ip
· fi Ô ˘ · = R =D
2
Wx
= 21,195 cm 3
Wx
= ⋅ D
3
32=
3,14 ⋅ 63
32cm3 = 21,195 cm 3
ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 155
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∏ ÚÔ‹ ·‰Ú·Ó›·˜ ˆ˜ ÚÔ˜ ¿ÍÔÓ· x-xã, fiÏˆÓ ·˘ÙÒÓ ÙˆÓ ÙÚÈÁÒÓˆÓ
Ô˘
‰È¤Ú¯ÔÓÙ·È ·fi ÙÔ G, ı· Â›Ó·È Ë ÔÏÈ΋ ÚÔ‹ ·‰Ú·Ó›·˜, Î·È ›ÛË
ÌÂ:
ÂÂȉ‹ πx = Iy ¤¯Ô˘ÌÂ:*
4. ∞¶§√ ∆∞À
¶∞ƒ∞¢∂π°ª∞
∆Ô˘ ·Ú·Î¿Ùˆ ·ÏÔ‡ Ù·˘, Ó· ‚ÚÂı› Ë ÚÔ‹ ·‰Ú·Ó›·˜ ÙÔ˘ π ˆ˜ ÚÔ˜
ÙÔÓ
·ÚÈÔ ¿ÍÔÓ· ·‰Ú·Ó›·˜ x-xã
Ix
= Iy
=D
4
64
Ip
= v ⋅ 14
⋅ bh3
= 14
vb h3
= 14
D ⋅D2
3=
D4
32
xã G xA
B
° bh
R
156 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡
* ™ËÌÂÈÒÓÂÙ·È fiÙÈ Ô Ù‡Ô˜ ‰›‰ÂÈ ÙË ÚÔ‹ ·‰Ú·Ó›·˜ ÙÚÈÁˆÓÈ΋˜
‰È·-
ÙÔÌ‹˜ ‚¿Ûˆ˜ b Î·È ‡„Ô˘˜ h ˆ˜ ÚÔ˜ ¿ÍÔÓ· Ô˘ ÂÚÓ¿ ·fi ÙËÓ ÎÔÚ˘Ê‹,
ÙËÓ ·¤-
Ó·ÓÙÈ ·fi ÙËÓ b Î·È Â›Ó·È ·Ú¿ÏÏËÏÔ˜ ÚÔ˜ ÙËÓ b. ¶·Ú·ÙËÚ›ÛÙ fiÙÈ
ÁÈ· ¿ÂÈÚÔ
Ï‹ıÔ˜ Ù¤ÙÔÈˆÓ ÙÚÈÁÒÓˆÓ ı· ›ӷÈ: vb = D ÔfiÙÂ Î·È h =D
—2
.
I = 14
bh3
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ÕÚ·, Ë ÚÔ‹ ·‰Ú·Ó›·˜ Ù˘ Û‡ÓıÂÙ˘ ·˘Ù‹˜ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ÙÔÓ
·-
ÚÈÔ ¿ÍÔÓ· ·‰Ú·Ó›·˜, ı· ›ӷÈ:
π = π1 + π2 ÕÚ·
5. ¢π¶§√ ∆∞À
¶∞ƒ∞¢∂π°ª∞
∆Ô˘ ·Ú·Î¿Ùˆ ‰ÈÏÔ‡ Ù·˘ Ó· ‚ÚÂı› Ë ÚÔ‹ ·‰Ú·Ó›·˜ ÙÔ˘ π ˆ˜ ÚÔ˜
ÙÔÓ
·ÚÈÔ ¿ÍÔÓ· ·‰Ú·Ó›·˜ x-xã, ηıÒ˜ Î·È Ë ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wy
·˘Ù‹˜.
§‡ÛË
£· ·Ú·ÙËÚ‹ÛÔ˘ÌÂ, fiÙÈ ÙÔ ‰ÈÏfi ·˘Ùfi Ù·˘, ‰ÂÓ Â›Ó·È ¿ÏÏÔ ·fi ÙÔ
ÙÂ-
ÙÚ¿ÁˆÓÔ ∞µ°¢, Â¿Ó ·fi ·˘Ùfi ·Ê·ÈÚ¤ÛÔ˘Ì ٷ ÂÌ‚·‰¿ ÙˆÓ
ÔÚıÔÁˆÓ›ˆÓ
∂∑∫π Î·È ∏£ª§, ÔfiÙÂ Ë ÚÔ‹ ·‰Ú·Ó›·˜ ÙÔ˘ ‰ÈÏÔ‡ ·˘ÙÔ‡ Ù·˘ ı·
ÈÛÔ‡Ù·È
xã x
y
yã
4cm
12cmG
8cm 8cm4cm
4cm
A B
° ¢
M§KI
E Z £H
I = 4693,332 cm3
= 4026,666 cm3
I 2 = 2666,666 + 1360 cm3 == 106,666 + 560 cm 3 = 666,666
cm3
I 2 =4 ⋅ 20
3
12cm3+ 4 ⋅ 20 ⋅ 17 cm 3I 1 =
20 ⋅ 43
12cm3 + 4 ⋅ 20 ⋅ 7 cm 3 =
I2
=b h
3
12+ E
2Â 22
I1
=bh
3
12+ E
1 Â 12
158 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡
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Ì ÙËÓ ÚÔ‹ ·‰Ú·Ó›·˜ ÙÔ˘ ÙÂÙÚ·ÁÒÓÔ˘ ∞µ°¢, ·Ê·ÈÚÔ˘Ì¤ÓˆÓ ÙˆÓ
ÚÔÒÓ
·‰Ú·ÓÂÈÒÓ ÙˆÓ ÔÚıÔÁˆÓ›ˆÓ ∂∑∫π Î·È ∏£ª§.
ÕÚ·: ππ = π∞µ°¢ – π∂∑∫π – π∏£ª§ (1)
∂›Ó·È:
ÕÚ· Ë (1) ‰›‰ÂÈ:
H ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wy Ù˘ ‰È·ÙÔÌ‹˜ ı· ›ӷÈ:
ÕÚ·:
AÓ¿ÏÔÁ· ‚Ú›Ù ÙË ÚÔ‹ ·‰Ú·Ó›·˜ Ù˘ ‰È·ÙÔÌ‹˜ ÙÔ˘ ·Ú·Î¿Ùˆ
Û¯‹-Ì·ÙÔ˜ ̂ ˜ ÚÔ˜ ¿ÍÔÓ· x-xã
£ÂˆÚÒÓÙ·˜ ÙË ‰È·ÙÔÌ‹ ·˘Ù‹ ̂ ˜ ‰È·ÊÔÚ¿ Ù˘ (2) ·fi ÙËÓ (1) ‚Ú›ÙÂ
fiÙÈ:
I1 = 4096 cm4 Î·È I2 = 1152 cm
4
¯' ¯
2cm
12cm
16cm 12cm
8cm
12cm
2cm
4cm8cm
12
Wy = 1102,933 cm3
Wy =Iyy
=11029,33 cm3
10= 1102,933 cm3
II = 11029,33 cm4
IH£M§ =8 ⋅ 123
12= 1152 cm4
IEZKI =8 ⋅ 123
12= 1152 cm4
IAB°¢ =20 ⋅ 20
12= 13333,33 cm4
ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 159
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6. ¢π∞∆√ª∏ [
¶∞ƒ∞¢∂π°ª∞
∆˘ ·Ú·Î¿Ùˆ ‰È·ÙÔÌ‹˜, Ó· ‚ÚÂı› Ë ÚÔ‹ ·‰Ú·Ó›·˜ Ù˘ ˆ˜ ÚÔ˜ ÙÔ˘˜
¿-
ÍÔÓ˜ x-xã Î·È y-yã, ηıÒ˜ Î·È ÔÈ ÚÔ¤˜ ·ÓÙ›ÛÙ·Û˘ ·˘Ù‹˜.
§‡ÛË
£· ˘ÔÏÔÁ›ÛÔ˘Ì ÙËÓ ·fiÛÙ·ÛË Â1. Œ¯Ô˘ÌÂ
ÕÚ·
Î·È Iy
=16 ⋅ 8
3
12 –
12 ⋅ 63
12cm4
Ix = 1866,7 cm4
Ix =8 ⋅ 163
12 –
6 ⋅ 123
12cm4
Â1 = 2,71 cm
Â1 =32 ⋅ 1 + 1 2 ⋅ 5 + 1 2 ⋅ 5
3 2 + 1 2 + 1 2cm
xã x
y
yã
G
2 cm
12 cm
2 cm
2cm 6 cm
Â2=5,29cm
G1
G2
G3
Â1
160 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡
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°È· ÙȘ ÚÔ¤˜ ·ÓÙ›ÛÙ·Û˘ Wx, Wy1, Wy2,
¤¯Ô˘ÌÂ:
ÕÚ·:
7. ∫√π§∏ ∫À∫§π∫∏ ¢π∞∆√ª∏
¶∞ƒ∞¢∂π°ª∞
∆˘ ÎÔ›Ï˘ ΢ÎÏÈ΋˜ ‰È·ÙÔÌ‹˜ Ì D = 8 cm Î·È d = 4 cm, Ó·
‚ÚÂıÔ‡Ó:
·) √È ÚÔ¤˜ ·‰Ú·Ó›·˜ Ix Î·È Iy ˆ˜ ÚÔ˜ ÙÔ˘˜ ·ÚÈÔ˘˜ ¿ÍÔÓ˜
·‰Ú·Ó›·˜
Ù˘.
‚) √È ÚÔ¤˜ ·ÓÙ›ÛÙ·Û˘ Wx Î·È Wy ·˘Ù‹˜
Á) √È ·ÎÙ›Ó˜ ·‰Ú·Ó›·˜ ix Î·È iy ·˘Ù‹˜
§‡ÛË
·) ™ÙÔ ·Ú¿‰ÂÈÁÌ· Ô˘ ·Ó·ÊÂÚfiÌ·ÛÙ·Ó ÁÈ· ΢ÎÏÈ΋ ‰È·ÙÔÌ‹
·Ô‰Â›Í·-
ÌÂ fiÙÈ:
Ix = Iy =D
64
xã xb
y
yã
G
D
Wy1= 172,2 cm3 Î·È Wy2
= 88,22 cm3
Wy1=
466,7
2,71cm3 = 172,2 cm3 Î·È Wy2
=466,7
5,29cm3 = 88,22 cm3
Wx = 197,4 cm3
Wx =8 ⋅ 162
6 –
6 ⋅ 122
6cm3 = 197,4 cm3
Iy = 466,7 cm4
ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 161
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∂‰Ò, ÁÈ· ÙËÓ ÎÔ›ÏË Î˘ÎÏÈ΋ ‰È·ÙÔÌ‹ Â›Ó·È ÚÔÊ·Ó¤˜ fiÙÈ ı·
¤¯Ô˘ÌÂ:
Î·È ÁÈ· D = 8 cm Î·È d = 4 cm, ı· ¤¯Ô˘ÌÂ:
ÕÚ·:
‚)∞Ó¿ÏÔÁ· ı· ¤¯Ô˘ÌÂ:
ÕÚ·:
Á) °È· ÙȘ ·ÎÙ›Ó˜ ·‰Ú·Ó›·˜ ix Î·È iy ı· ¤¯Ô˘ÌÂ:
∂›Ó·È
ÕÚ· ix = iy = 2,24 cm
ix = iy =1
48
2+ 42 = 5 cm
ix = iy =1
4D
2+ d2
·ÊÔ‡ F =
4D
2 – d
2
ix = iy =
64D
4 – d4
4
D2 – d2
=1
4
D2
+ d2 D2 – d2
D2
– d2
Wx = Wy = 376,8 cm3
Wx = Wy =
32D
4 – d4 = 376,8 cm3
Ix = Iy = 188,4 cm4
Ix
= Iy
=3,14
648
4 – 4
4cm4 = 188,4 cm4
Ix = Iy =
64D
4 – d4
162 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡
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¶∂ƒπ§∏æ∏
1. ∏ ÚÔ‹ ·‰Ú·Ó›·˜, Â›Ó·È ¤Ó· ¯·Ú·ÎÙËÚÈÛÙÈÎfi ̤ÁÂıÔ˜ Ù˘
‰˘Ûη̄›·˜ ÌÈ·˜ ηÌÙfiÌÂÓ˘ ηٷÛ΢‹˜.
ŸÛÔ ÈÔ ÌÂÁ¿ÏË Â›Ó·È Ë ÚÔ‹ ·‰Ú·Ó›·˜ Ù˘ ‰È·ÙÔÌ‹˜ ˆ˜
ÚÔ˜ ÙÔÓ ¿ÍÔÓ·, ÙfiÛÔ ÈÔ ÌÂÁ¿ÏË Â›Ó·È Î·È Ë ·ÓÙ›ÛÙ·ÛË Ù˘
‰È·ÙÔ-
Ì‹˜.
ø˜ ÚÔ˜ ÙÔ˘˜ ÎÂÓÙÚÔ‚·ÚÈÎÔ‡˜ ¿ÍÔÓ˜ ÙˆÓ ‰È·ÊfiÚˆÓ ‰È·ÙÔÌÒÓ ·fi
ÙÔ˘˜ ›Ó·Î˜ ÌÔÚԇ̠ӷ ‚Úԇ̠ÙȘ ÚÔ¤˜ ·‰Ú·Ó›·˜.
2. °È· Û‡ÓıÂÙ˜ ‰È·ÙÔ̤˜, ¯Ú‹ÛÈÌÔ Â›Ó·È ÙÔ £. Steiner, Ô˘ Ì·˜
·Ú¤¯ÂÈ
ÙË ÚÔ‹ ·‰Ú·Ó›·˜ Ù˘ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· Ô˘ ·¤¯ÂÈ ·fi
ÙÔÓ ÎÂÓÙÚÔ‚·ÚÈÎfi ¿ÍÔÓ·, ·fiÛÙ·ÛË Â. √ Ù‡Ô˜ ÙÔ˘ ıˆڋ̷ÙÔ˜
›-
Ó·È:
π = IG + F. Â2
fiÔ˘ F Ë ÂÈÊ¿ÓÂÈ· Ù˘ ‰È·ÙÔÌ‹˜.
3. ∆Ô ËÏ›ÎÔ Ù˘ ÚÔ‹˜ ·‰Ú·Ó›·˜ π· Ù˘ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ¿ÍÔÓ·
‰È·
Ù˘ ·ÔÛÙ¿Ûˆ˜ · ÙÔ˘ ¿ÍÔÓ· ·˘ÙÔ‡ ·fi ÙÔ Ï¤ÔÓ ·ÔÌ·ÎÚ˘Ṳ̂ÓÔÛËÌ›Ô
ÙÔ˘ ÙÌ‹Ì·ÙÔ˜ Ù˘ ‰È·ÙÔÌ‹˜ ÔÓÔÌ¿˙Ô˘Ì ÚÔ‹ ·ÓÙ›ÛÙ·Û˘
W·
Î·È Â›Ó·È:
4. ∆Ô ¿ıÚÔÈÛÌ· ÙˆÓ ÚÔÒÓ ·‰Ú·Ó›·˜ Ù˘ ‰È·ÙÔÌ‹˜ Ip ˆ˜ ÚÔ˜ ÙÔ˘˜
·-
ÚÈÔ˘˜ ¿ÍÔÓ˜ ·‰Ú·Ó›·˜ Ù˘ ‰È·ÙÔÌ‹˜ ÔÓÔÌ¿˙Ô˘Ì ÔÏÈ΋ ÚÔ‹ ·-
‰Ú·Ó›·˜ πp Ù˘ ‰È·ÙÔÌ‹˜ Î·È Â›Ó·È Ip = Ix + Iy.
5. To ËÏ›ÎÔ Ù˘ ÔÏÈ΋˜ ÚÔ‹˜ ·‰Ú·Ó›·˜ Ip Ù˘ ‰È·ÙÔÌ‹˜ ‰È· Ù˘
·Ô-
ÛÙ¿Ûˆ˜ · ÙˆÓ Ï¤ÔÓ ·ÔÌ·ÎÚ˘ÛÌ¤ÓˆÓ ÛËÌ›ˆÓ Ù˘ ·fi ÙÔ Î¤ÓÙÚÔ
‚¿ÚÔ˘˜ ·˘Ù‹˜, ÔÓÔÌ¿˙Ô˘Ì ÔÏÈ΋ ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wp
Î·È Â›Ó·È
6. ∆ËÓ ÙÂÙÚ·ÁˆÓÈ΋ Ú›˙· Ù˘ ÚÔ‹˜ ·‰Ú·Ó›·˜ Ix Ù˘ ‰È·ÙÔÌ‹˜ ‰È·
Ù˘
ÂÈÊ·Ó›·˜ F Ù˘ ‰È·ÙÔÌ‹˜ ·˘Ù‹˜, ÔÓÔÌ¿˙Ô˘Ì ·ÎÙ›Ó· ·‰Ú·Ó›·˜
i
Ù˘ ‰È·ÙÔÌ‹˜
Wp =Ip·
W· =·
·
ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 163
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∞¿ÓÙËÛË:
3. ¢›‰ÔÓÙ·È ‰‡Ô ÔÚıÔÁˆÓÈΤ˜ ‰È·ÙÔ̤˜ ÛÙ·ıÂÚÔ‡ ÂÌ‚·‰Ô‡ 2b2.
µÚ›-
Ù ÔÈ· ·fi ÙȘ ‰‡Ô ‰È·ÙÔ̤˜ Â›Ó·È Ë ÈÔ ÛÙ·ıÂÚ‹ ˆ˜ ÚÔ˜ ¿ÍÔÓ· x-xã
ηÈ
ÔÈ· Â›Ó·È Ë ÈÔ ÛÙ·ıÂÚ‹ ˆ˜ ÚÔ˜ ¿ÍÔÓ· y-yã.
A¿ÓÙËÛË: ∏ ÚÒÙË ‰È·ÙÔÌ‹ Â›Ó·È Ë ÈÔ
ÛÙ·ıÂÚ‹ (;) ˆ˜ ÚÔ˜ x-xã Î·È Ë
‰Â‡ÙÂÚË Â›Ó·È Ë ÈÔ ÛÙ·ıÂÚ‹
ˆ˜ ÚÔ˜ ¿ÍÔÓ· y-yã.
4. √ÚıÔÁˆÓÈ΋˜ ‰È·ÙÔÌ‹˜ ‰È·ÛÙ¿ÛÂˆÓ b = 4 cm Î·È h = 6 cm,
‚Ú›ÙÂ
ÙËÓ ÔÏÈ΋ ÚÔ‹ ·‰Ú·Ó›·˜ Ù˘.
∞¿ÓÙËÛË: 104 cm4.
5. BÚ›Ù ÙȘ ÚÔ¤˜ ·‰Ú·Ó›·˜ Ix, Iy Ù˘ ‰È·ÙÔÌ‹˜:
G Gxã xxã x
y
yã
y
yã
b
2b
2b b
iAB
= 28824
cm = 3,46 cm
xã x
y
yãAB
° ¢
6 cm
4 cm
ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 165
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34/35
∞¿ÓÙËÛË: Ix = 1459,58 cm4, Iy = 4958,33 cm
4
6. ¢›‰ÂÙ·È ‰È·ÙÔÌ‹ ‰ÈÏÔ‡ Ù·˘ Ì ÂÁÎÔ‹. µÚ›Ù ÙË ÚÔ‹
·‰Ú·Ó›·˜
Î·È ÙË ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Ù˘ ‰È·ÙÔÌ‹˜ ·˘Ù‹˜.
∞¿ÓÙËÛË: πx = 41962,7 cm4, Wx = 2331,26 cm
3
7. µÚ›Ù ÙË ÚÔ‹ ·‰Ú·Ó›·˜ Ù˘ ÎÔ›Ï˘ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ÙÔ˘˜
¿ÍÔ-Ó˜ x-xã Î·È y-yã.
2 cm
2 cm
8 cm
8 cm
16 cm
12 cm 12 cm4cm
5.5 cm
15 cm
10 cm5 cm
4 cm
166 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡
-
8/20/2019 Ropes adraneias
35/35
∞¿ÓÙËÛË: πx = 9583,34 cm4, πy = 5520,83 cm
4
8. ¢›‰ÂÙ·È ·ÌÊÈ·ÚıÚˆÙfi˜ ÛÙ‡ÏÔ˜, Û‡ÓıÂÙ˘ ‰È·ÙÔÌ‹˜.
¡· ‚ÚÂı›:
·) Ë ı¤ÛË ÙÔ˘ ∫µ Ù˘ ‰È·ÙÔÌ‹˜
‚) Ë ÚÔ‹ ·‰Ú·Ó›·˜ ̂ ˜ ÚÔ˜ ÙÔÓ ÎÂÓÙÚÔ‚·ÚÈÎfi ¿ÍÔÓ· y-yã
Á) Ë ·ÎÙ›Ó· ·‰Ú·Ó›·˜ iy-yã
∞¿ÓÙËÛË: (yG = 8 cm, Iy-yã=504 cm4, iy-yã = 2,6 cm)
xã x
y
yã
6 cm
14 cm
2 cm
2 cm
2 cm
16 cm
xã x
y
yã
5 cm
5 cm
5 cm 5 cm 5 cm
10 cm
ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 167
