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

134 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡

∂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 Û ڷ-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 Ì ¢¯¤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 Û 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, mi = 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

ˆ2

™ m i r i2

ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 135

™¯‹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· ÙˆÓ 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 fi ·˘Ù‹˜, Ë ÚÔ‹ ·‰Ú¿ÓÂÈ·˜ Ù˘ÂÈÊ¿ÓÂÈ·˜ ·˘Ù‹˜ ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· y - yã Û˘Ì‚ÔÏ›˙ÂÙ·È Ì Jy Î·È ÌÂÙÚ¿-Ù·È Û 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 ηÈÌÂÙÚ¿Ù·È Û 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 ˆ˜ 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

°

Ax

r2

m1

™

r1r3

m3

136 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡

™¯‹Ì· 7.2· ∂‡ÚÂÛË ÚÔ‹˜ ·‰Ú¿ÓÂÈ·˜ ‰È·ÙÔÌ‹˜

‰ËÏ·‰‹ ı· ¤¯Ô˘ÌÂ

(7.2.1)

‰ËÏ·‰‹ ÙȘ Û¯¤ÛÂȘ:

Jx = f1y1 + f2y2 + … + fiy12 Î·È Jy = f1x

21 + f2x

22 + … + 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·˜ ˆ˜ 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· yi

2 . fi Ô˘ ÙÔ ¿ıÚÔÈÛÌ¿ ÙÔ˘˜ ı· ‰ÒÛÂÈ ÁÈ· i → ∞.

fi = b ⋅ h2i

y 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

(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¿ÛÂˆÓ 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¿ÙˆÓ, 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˘ÌÂÙËÓ 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 ⋅ 4 3

12= 64 cm 4

I x =b ⋅ h 3

12=

4 ⋅ 12 3 cm 4

12= 576 cm 4

Iy–y′ =b3 ⋅ h

12ÚÔ‹ ·‰Ú¿ÓÂÈ·˜ ÔÚıoÁˆÓÈ΋˜

‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ y-yã

Ix–x′ =b ⋅ h3

12ÚÔ‹ ·‰Ú¿ÓÂÈ·˜ ÔÚı.

‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ x–xã

138 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡

µÚ›ÛÎÔ˘Ì ÙȘ ÛÙ·ÙÈΤ˜ ÚÔ¤˜ ÁÈ· οı ÂÈÊ¿ÓÂÈ· ¯ˆÚÈÛÙ¿, ˆ˜ ÚÔ˜ÙËÓ ÏÂ˘Ú¿ ∞µ.

£· ¤¯Ô˘ÌÂ:

(2)

fiÔ˘: y1, y2, …, yÓ Â›Ó·È ÔÈ ·ÔÛÙ¿ÛÂȘ ÙˆÓ ÎÂÓÙÚÔÂȉÒÓ Î¿ı ÂÈÊ¿ÓÂÈ-·˜ ·fi ÙËÓ ÏÂ˘Ú¿ ∞µ.

∂›Ó·È ‰Â:

(3)

∏ Û¯¤ÛË (2), ÙfiÙ ı· Á›ÓÂÈ:

™¯‹Ì· 7.3‚ ™Ù·ÙÈ΋ ÚÔ‹ ÔÚıÔÁˆÓÈ΋˜ ‰È·ÙÔÌ‹˜

¶·Ú·ÙËÚԇ̠fiÙÈ ÛÙÔ ›‰ÈÔ ·ÔÙ¤ÏÂÛÌ· ÌÔÚԇ̠ӷ Êı¿ÛÔ˘Ì ÂÊ·Ú-Ìfi˙ÔÓÙ·˜ ÙÔ £ÂÒÚËÌ· Varignon.

£· ¤¯Ô˘ÌÂ:

M ™ = R ⋅h

2= 7 ⋅ Â ⋅

h

2= 7 ⋅ · ⋅

h

7⋅

h

2

E1

G1

G2

h1

h2

h

xxã

·

E2

M ™ =· h

2

2

M ™ = Â ⋅ 3,5 ⋅ h =· ⋅ h

7⋅ 3,5 h = · ⋅ h ⋅

h

2

y1 = h14

, y5 = h14

+ 4h7

y2 = h14

+ h7

, y6 = h14

+ 5h7

y3 = h14

+ 2h7

, y7 = h14

+ 6h7

y4 = h14

+ 3h7

M™ = Ây1 + Ây2 + ⋅ ⋅ ⋅ + ÂyÓ

ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 139

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 ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· x-xã›ӷÈ:

ª1 = 18 cm2 . 6 cm = 108 cm3

H ÛÙ·È΋ ÚÔ‹ ª2 Ù˘ ‰È·ÙÔÌ‹˜ 2 ˆ˜ 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 – h 2

M ™ = · ⋅ 2h 1 ⋅ h 1 – · ⋅ 2h 2 ⋅ h 2 = 2· ⋅ h2

h 1 – h 2

M ™ = E 1h 1 – E 2 ⋅ h 2

M ™ =· h

2

2= E ⋅

h

2

140 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡

∂Ô̤ӈ˜ Ë ÛÙ·ÙÈ΋ ÚÔ‹ ˆ˜ 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 ˆ˜ ÚÔ˜ ¿ÍÔÓ· x-xã›ӷÈ:

ª1 = 24 cm2 . 2,5 cm = 60 cm3

∏ ÛÙ·ÙÈ΋ ÚÔ‹ ª2 Ù˘ ‰È·ÙÔÌ‹˜ 2 ˆ˜ 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

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 =

bh3

12h2

=bh3

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¡

(7.4.1.3)

∂›Ó·È ‡ÎÔÏÔ Ó· Û˘Ó¿ÁÔ˘ÌÂ, fiÙÈ ‰È·ÙËÚÒÓÙ·˜ ÙÔ h = ÛÙ·ıÂÚfi Î·È ‰ÈÏ·-ÛÈ¿˙ÔÓÙ·˜ ÙÔ b, ı· ¤¯Ô˘ÌÂ:

‰ËÏ·‰‹ ÙÒÚ· Ë ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wx ‰ÈÏ·ÛÈ¿˙ÂÙ·È.∆È Û˘Ì¤Ú·ÛÌ· ‚Á¿˙ÂÙ ·Ó ‰È·ÙËÚ‹ÛÔ˘Ì ÙÔ b = ÛÙ·ıÂÚfi Î·È ‰ÈÏ·ÛÈ¿-

ÛÔ˘Ì ÙÔ h;

¶ÂÚÈÛÛfiÙÂÚ· ı· ԇ̠ÁÈ· ‰È·ÙÔ̤˜ Ì ‰È·ÊÔÚÂÙÈο Û¯‹Ì·Ù· fiÛÔÓ ·-ÊÔÚ¿ ÙËÓ ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ ˆ˜ ÚÔ˜ ¤Ó· ·fi ÙÔ˘˜ ‰‡Ô ¿ÍÔÓ˜ Û˘ÌÌÂÙÚ›·˜,·Ú·Î¿Ùˆ.

❑ ¶∞ƒ∞¢∂π°ª∞

∞ÌÊȤÚÂÈÛÙË ‰ÔÎfi˜, Ì‹ÎÔ˘˜ l ʤÚÂÈ Î·Ù·ÎfiÚ˘ÊÔ Û˘ÁÎÂÓÙڈ̤ÓÔ ÊÔÚÙ›Ô PÛÙÔ Ì¤ÛÔÓ Ù˘ ‰ÔÎÔ‡. √È ‰È·ÛÙ¿ÛÂȘ Ù˘ ÔÚıÔÁˆÓÈ΋˜ Ù˘ ‰È·ÙÔÌ‹˜ ›ӷÈbxh fiÔ˘ h = 3b Î·È ÙÔÔıÂÙÂ›Ù·È fiˆ˜ ÛÙ· Û¯‹Ì·Ù·:

¶ÔÈ· ·fi ÙȘ ‰‡Ô ·Ú·¿Óˆ ÂÚÈÙÒÛÂȘ ÙÔÔı¤ÙËÛ˘ Ù˘ ‰È·ÙÔÌ‹˜,ÎÚ›ÓÂÙ fiÙÈ Â›Ó·È Ë ‰˘ÛÌÂÓ¤ÛÙÂÚË Î·È ÁÈ·Ù›;

h

h

b

b

Wx =2 ⋅ bh2

6=

bh2

3

Wy =

hb3

12b2

=hb3

6b=

hb2

6‰ËÏ. Wy =

hb2

6

ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 143

§‡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

™fiÚ2 = I x + I y ‰ËÏ.

Ú 2 = x 2 + y 2 ‰ËÏ. fiÚ2 = fix

2+ fiy

2

Wb =h ⋅b2

6=

3 ⋅ b ⋅ b2

6=

3b3

6

W· =b ⋅h2

6=

b ⋅ 9b2

6=

9b3

6

144 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡

™¯‹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 Û 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˘ Wp ÌÈ·˜‰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

I p =hb12

= h 2 + b 2

I p =bh 3

12+

hb 3

12= hb

12h 2 + b 2

G1

G2

G

y

y

f1

xxh

b

ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 145

Afi ›Ó·Î˜ ·Ú¤¯ÔÓÙ·È ÔÈ ÚÔ¤˜ ·‰Ú·Ó›·˜ π ‰È·ÊfiÚˆÓ ‰È·ÙÔÌÒÓ, ÔÈÚÔ¤˜ ·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 Ì ÙËÓ ÙÂ-ÙÚ·ÁˆÓÈ΋ Ú›˙· Ù˘ ÚÔ‹˜ ·‰Ú·Ó›·˜ Ù˘ ‰È·ÙÔÌ‹˜ ‰È· Ù˘ ÂÈÊ·Ó›·˜ FÙ˘ ‰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›·˜ Ì ÙËÓ·Ú¯È΋ ‰È·ÙÔÌ‹.

ÕÚ· ÌÈ· ÂÈÊ¿ÓÂÈ· ¤¯ÂÈ ‰‡Ô ·ÎÙ›Ó˜ ·‰Ú·Ó›·˜ fiˆ˜ ÔÚ›˙ÔÓÙ·È ·fi ÙÔÓÙ‡Ô 7.5.1.

°È· ÔÚıÔÁˆÓÈ΋ ‰È·ÙÔÌ‹ ‰È·ÛÙ¿ÛÂˆÓ b = 4 cm Î·È h = 6 cm ı· ¤¯Ô˘ÌÂ:

i x =

bh 3

12bh

=h 2

12= 0,289h ‰ËÏ·‰‹ i x = 1,734 cm

ix =IxF

Î·È iy =IyF

Wp = 66

6 2 + 4 2 cm 3 = 52 cm 3

Wp =b h 2 + b 2

b

Wp =

bh12

⋅ h 2 + b 2

h2

=b h 2 + b 2

6

146 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡

°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˘˜ GÙ˘ ‰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 · = ™fi B° 2

B° 2 = G° 2 + BG 2 – 2 BG G° Û˘ÓBG°

Â

Â

Â'

fi

y'

y

°

AGB

i y =

hb 3

12bh

=b 2

12= 0,289b ‰ËÏ·‰‹ i y = 1,156 cm

ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 147

(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›·˜ Î·È 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)I x 1+ I x 2

=bh 3

12

I x 1= I G 1

+ h3

2⋅ bh

2

I x 2= I G 2

+ 23

h2

⋅ bh2

I Â =3 3 ⋅ 10

3cm 4 = 90 cm 4

IÂ =hb3

12+ h ⋅ b ⋅ b

4

2=

hb3

12+ 3 ⋅ h ⋅ b3

12=

4b3h12

=b3h

3

IÂ = IG + F ⋅ Â2

I · = I G + F ⋅ Â2

ÁÈ·Ù› ™fiGA = 0

™fi B° 2 = ™fi G° 2 + ™fi BG 2 – 2™fi –G°Û˘ÓÊ ⋅ G° Û˘ÓÊ

Ê = BG° Î·È GA = G° Û˘Ó 180° – Ê = – G° Û˘ÓÊ

I G = ™fi G° 2

148 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡

™¯‹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

§‡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 cm 3

Wx =4 3

6cm 3 = 10,666 cm 3

Wx =bh 2

6

I y = 21,333 cm 4

I x =4 4

12cm 4 = 21,333 cm 4. ÕÚ· I x = 21,333 cm 4

I x =bh 3

12fiÔ˘ b = h

xã

G

x

y

yã

4 cm

4 cm

A B

°¢

150 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡

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‹˜ ˆ˜ 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=

85,33316

⋅ cm 2

cm = 2,31 cm

I ¢° = 85,333 cm 4

I ¢° = 21,333 cm 4 + 16 ⋅ 2 2 cm 4 = 21,333 cm 4 + 64 cm 4 = 85,333 cm 4

I ¢° = I x + F ⋅ Â2

Wp = 21,333 cm 3

Wp =I p· , Wp =

42,6662

cm 3 = 21,333 cm 3.

ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 151

§‡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

I y = 180 cm 4

I y =10 ⋅ 6 3

12cm 4 = 180 cm 4.

I x = 500 cm 4

I x =6 ⋅ 10 3

12cm 4 =

10 3

2cm 4 = 500 cm 4.

Ix =bh3

12

xã

G

x

y

yã

6 cm

cm

A

10

B

°¢

152 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡

∞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 =I p· , Wp = 680

5cm 3 = 136 cm 3

Wy =hb 2

6= 60 cm 3

‰ËÏ·‰‹ Wx = 100 cm 3

ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 153

3. ∫À∫§π∫∏ ¢π∞∆√ª∏

❑ ¶∞ƒ∞¢∂π°ª∞

∆˘ ·Ú·Î¿Ùˆ ΢ÎÏÈ΋˜ ‰È·ÙÔÌ‹˜ η̇Ϙ R = 3 cm Ó· ‚ÚÂıÔ‡Ó:

·) √È ÚÔ¤˜ ·‰Ú·Ó›·˜ Ix Î·È πy ˆ˜ ÚÔ˜ ÙÔ˘˜ ·ÚÈÔ˘˜ ¿ÍÔÓ˜ ·‰Ú·Ó›·˜Ù˘

‚) ∏ ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wx ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· ∞µ

Á) ∏ ÔÏÈ΋ ÚÔ‹ ·‰Ú·Ó›·˜ Ip ˆ˜ ÚÔ˜ G

‰) ∏ ÔÏÈ΋ ÚÔ‹ ·ÓÙ›ÛÙ·Û˘ Wp Ù˘ ‰È·ÙÔÌ‹˜ ·˘Ù‹˜

Â) ∏ ·ÎÙ›Ó· ·‰Ú·Ó›·˜ Ù˘ i¢° ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· ¢°

§‡ÛË

·) ∏ ÚÔ‹ ·‰Ú·Ó›·˜ Ù˘ ΢ÎÏÈ΋˜ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· x-xã›-Ó·È*

fiÔ˘ D = 6 cm

ÕÚ·: I x = 63,585 cm 4

I x = I y =3,14 ⋅ 6

4

64cm 4 = 63,585 cm 4

I x = I y =D

4

64

xã

G

x

y

yã

3 cm

A B

°¢

154 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡

* ™ÙÔ Ù¤ÏÔ˜ Ù˘ ¿ÛÎËÛ˘ ı· ‰Ôı› Û˘ÓÔÙÈ΋ ·fi‰ÂÈÍË ÙÔ˘ Ù‡Ô˘

‚) ∏ 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=

317,92528,26

cm

I ¢° = 317,925 cm 4

Wp = 42,39 cm 3

Wp =127,17

3cm 3 = 42,39 cm 3

Wp =I p· fiÔ˘ · = R =

D

2

Wx = 21,195 cm 3

Wx = ⋅ D

3

32=

3,14 ⋅ 63

32cm 3 = 21,195 cm 3

ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 155

∏ 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

I x = I y =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

§‡ÛË

∂‰Ò ·Ú·ÙËÚÔ‡ÌÂ, fiÙÈ Ë ‰È·ÙÔÌ‹ Ì·˜ ‰ÂÓ Â›Ó·È ·Ï‹ –‰ÂÓ ¤¯ÂÈ ‰ËÏ·‰‹ ·-Ïfi ÁˆÌÂÙÚÈÎfi Û¯‹Ì· fiˆ˜ ÚÔËÁÔ‡ÌÂÓ· ÂÍÂÙ¿Û·Ì ÙÔ ÙÂÙÚ¿ÁˆÓÔ, ÙÔÔÚıÔÁÒÓÈÔ, ÙÔÓ Î‡ÎÏÔ– ·ÏÏ¿ ·ÔÙÂÏÂ›Ù·È ·fi ÙÌ‹Ì·Ù· ·ÏÒÓ ÂÈÊ·ÓÂÈÒÓÔÚıÔÁˆÓ›ˆÓ, fiˆ˜ Â›Ó·È Ù· ∞µ°¢ Î·È ∂∑∏∂. ∏ Û˘ÓÔÏÈ΋ ÚÔ‹ ·‰Ú·Ó›·˜Ù˘ ‰È·ÙÔÌ‹˜ ·˘Ù‹˜ ÙÔ˘ ·ÏÔ‡ Ù·˘ ˆ˜ ÚÔ˜ ¿ÍÔÓ· ·‰Ú·Ó›·˜, ı· Â›Ó·È ›ÛËÌ ÙÔ ¿ıÚÔÈÛÌ· ÙˆÓ ÚÔÒÓ ·‰Ú·Ó›·˜ ÙˆÓ ‰‡Ô ·˘ÙÒÓ ÙÌËÌ¿ÙˆÓ (1) Î·È (2)Ù˘ ‰È·ÙÔÌ‹˜, ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ·. ∏ Û˘ÓÙÂÙ·Á̤ÓË yG ÙÔ˘ ΤÓÙÚÔ˘ ‚¿-ÚÔ˘˜ Ù˘ ‰È·ÙÔÌ‹˜ ı· ›ӷÈ:

ÕÚ·,

∞fi ÙÔ Û¯‹Ì· ÚÔ·ÙÔ˘Ó: GG1 = 5 cm Î·È GG2 = 7 cm. Œ¯Ô˘Ì ÏÔÈfiÓ(οÓÔ˘Ì ¯Ú‹ÛË ÙÔ˘ £. Steiner):

y G = 7 cm

yG =20 ⋅ 4 ⋅ 2 + 20 ⋅ 4 ⋅ 12 cm3

20 ⋅ 4 + 20 ⋅ 4 cm2= 7 cm

xã x

y

yã

4cm

3cmÂ1= 7 cm

G1

G2

G

Â2 = 17 cm

20 cm

8cm 4cm 8cm

A B

°¢ E Z

£

1

H2

ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 157

ÕÚ·, Ë ÚÔ‹ ·‰Ú·Ó›·˜ Ù˘ Û‡ÓıÂÙ˘ ·˘Ù‹˜ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ÙÔÓ Î‡-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 cm 3

= 4026,666 cm 3

I 2 = 2666,666 + 1360 cm 3 == 106,666 + 560 cm 3 = 666,666 cm 3

I 2 =4 ⋅ 20 3

12cm 3+ 4 ⋅ 20 ⋅ 17 cm 3

I 1 =20 ⋅ 4 3

12cm 3 + 4 ⋅ 20 ⋅ 7 cm 3 =

I 2 =b h 3

12+ E 2Â 2

2I 1 =bh 3

12+ E 1 Â 1

2

158 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡

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˜ ˆ˜ ÚÔ˜ ¿ÍÔÓ· x-xã

£ÂˆÚÒÓÙ·˜ ÙË ‰È·ÙÔÌ‹ ·˘Ù‹ ˆ˜ ‰È·ÊÔÚ¿ Ù˘ (2) ·fi ÙËÓ (1) ‚Ú›Ù fiÙÈ:

I1 = 4096 cm4 Î·È I2 = 1152 cm4

¯' ¯

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 ⋅ 203

12= 13333,33 cm4

ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 159

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 ⋅ 83

12–

12 ⋅ 63

12cm4

Ix = 1866,7 cm4

Ix =8 ⋅ 163

12–

6 ⋅ 123

12cm4

Â1 = 2,71 cm

Â1 =32 ⋅ 1 + 12 ⋅ 5 + 12 ⋅ 5

32 + 12 + 12cm

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¡

°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·-Ì fiÙÈ:

Ix = Iy =D

4

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 cm3Wx =

8 ⋅ 162

6–

6 ⋅ 122

6cm3 = 197,4 cm3

Iy = 466,7 cm4

ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 161

∂‰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 =4

D2

– d2

ix = iy =

64D

4 – d4

4D

2 – d2

= 14

D2 + d2 D

2 – d2

D2 – d2

Wx = Wy = 376,8 cm3

Wx = Wy =

32D

4 – d4 = 376,8 cm3

Ix = Iy = 188,4 cm4

Ix = Iy =3,1464

84

– 44 cm4 = 188,4 cm4

Ix = Iy =

64D

4 – d4

162 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡

¶∂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 ·fiÙÔ˘˜ ›Ó·Î˜ ÌÔÚԇ̠ӷ ‚Úԇ̠ÙȘ ÚÔ¤˜ ·‰Ú·Ó›·˜.

2. °È· Û‡ÓıÂÙ˜ ‰È·ÙÔ̤˜, ¯Ú‹ÛÈÌÔ Â›Ó·È ÙÔ £. Steiner, Ô˘ Ì·˜ ·Ú¤¯ÂÈÙË ÚÔ‹ ·‰Ú·Ó›·˜ Ù˘ ‰È·ÙÔÌ‹˜ ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· Ô˘ ·¤¯ÂÈ ·fiÙÔÓ 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˘Ṳ̂ÓÔÛËÌÂ›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· =I··

ƒ√¶∂™ ∞¢ƒ∞¡∂π∞™ ∂¶πº∞¡∂πø¡ 163

ÃÚ‹ÛÈ̘ Â›Ó·È ÔÈ ÂÊ·ÚÌÔÁ¤˜ ·˘ÙÒÓ ÙˆÓ ÂÓÓÔÈÒÓ Û ·Ï¤˜ Î·È Û‡Ó-ıÂÙ˜ ‰È·ÙÔ̤˜ ÛÙ· ÎÂÊ¿Ï·È· Ù˘ ο̄˘ Î·È Ù˘ ÛÙÚ¤„˘ Ô˘ ı·ÁÓˆÚ›ÛÔ˘Ì ·ÚÁfiÙÂÚ·. ™Â ‰È·ÙÔ̤˜ Ì ‰È·ÊÔÚÂÙÈο Û¯‹Ì·Ù· ÙÔ Ì¤-ÁÂıÔ˜ ÙˆÓ ÚÔÒÓ ·‰Ú·Ó›·˜ Î·È ·ÓÙ›ÛÙ·Û˘ ˆ˜ ÚÔ˜ ¤Ó· ·fi ÙÔ˘˜‰‡Ô ¿ÍÔÓ˜ Û˘ÌÌÂÙÚ›·˜ Â›Ó·È ÌÂÁ·Ï‡ÙÂÚÔ. °È’ ·˘Ùfi, ηٿ ÙËÓ ÙÔÔ-ı¤ÙËÛË ÙˆÓ ‰ÔÎÒÓ – fiˆ˜ ı· ‰Ô‡Ì ÛÙÔ ÂfiÌÂÓÔ ÎÂÊ¿Ï·ÈÔ– Ú¤ÂÈÓ· ÂÍ·ÛÊ·Ï›˙Ô˘Ì ·fi ÙÔ Û¯‹Ì· Ù˘ ‰È·ÙÔÌ‹˜, Ù· ÌÂÁ·Ï‡ÙÂÚ· ‰˘-Ó·Ù¿ ÏÂÔÓÂÎÙ‹Ì·Ù·.

∞™∫∏™∂π™

✒ 1. ∆ÚÈÁˆÓÈ΋˜ ÈÛÔÛÎÂÏÔ‡˜ ‰È·ÙÔÌ‹˜, Ì ÏÂ˘Ú¿ b Î·È ‡„Ô˜ h Ó· ‚Ú›-Ù ÙË ÚÔ‹ ·‰Ú·Ó›·˜ Ù˘ ˆ˜ ÚÔ˜ ¿ÍÔÓ· Ô˘ ÂÚÓ¿ÂÈ ·fi ÙËÓ ÎÔÚ˘Ê‹ ∞.∂Ê·ÚÌÔÁ‹: b = 4 cm Î·È h = 6 cm (Û¯. 1).

(§¿‚ÂÙ ˘’ fi„Ë fiÙÈ Î·È ÙÔ £. Steiner)

∞¿ÓÙËÛË:

✒ 2. ¢›‰ÂÙ·È ÔÚıÔÁˆÓÈ΋ ‰È·ÙÔÌ‹ ‰È·ÛÙ¿ÛÂˆÓ b = 4 cm, h = 6 cm. ¡·‚Ú›Ù ÙËÓ ·ÎÙ›Ó· ·‰Ú·Ó›·˜ Ù˘ ‰È·ÙÔÌ‹˜ ·˘Ù‹˜ ˆ˜ ÚÔ˜ ÙË ‚¿ÛË Ù˘.

(§¿‚ÂÙ ˘’ fi„Ë fiÙÈ: , Î·È ÙÔ £. Steiner)Ix =bh3

12

Ix – x′ =bh3

4216 cm4

xã xA

B °

6 cm

4 cm

IG =bh3

36

164 ∆∂áπ∫∏ M∏Ã∞¡π∫∏ - ∞¡∆√Ã∏ ∆ø¡ À§π∫ø¡

∞¿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 ηÈÔÈ· Â›Ó·È Ë ÈÔ ÛÙ·ıÂÚ‹ ˆ˜ ÚÔ˜ ¿ÍÔÓ· y-yã.

A¿ÓÙËÛË: ∏ ÚÒÙË ‰È·ÙÔÌ‹ Â›Ó·È Ë ÈÔÛÙ·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›ÙÂÙËÓ 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

∞¿ÓÙËÛË: Ix = 1459,58 cm4, Iy = 4958,33 cm4

✒ 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 cm3

✒ 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¡

∞¿ÓÙËÛË: πx = 9583,34 cm4, πy = 5520,83 cm4

✒ 8. ¢›‰ÂÙ·È ·ÌÊÈ·ÚıÚˆÙfi˜ ÛÙ‡ÏÔ˜, Û‡ÓıÂÙ˘ ‰È·ÙÔÌ‹˜.

¡· ‚ÚÂı›:

·) Ë ı¤ÛË ÙÔ˘ ∫µ Ù˘ ‰È·ÙÔÌ‹˜

‚) Ë ÚÔ‹ ·‰Ú·Ó›·˜ ˆ˜ 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