Ch13 universal gravitation

‫ﻋﺸﺭ‬ ‫ﺍﻟﺜﺎﻟﺙ‬ ‫ﺍﻟﺠﺯﺀ‬:‫ﺍﻟﻌﺎﻡ‬ ‫ﺍﻟﺠﺫﺏ‬ ‫ﻗﺎﻨﻭﻥ‬
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‫ﺘﻭﺜﻴــﻕ‬
‫اﻟﺒﯿﺎن‬
Physics for Scientists and Engineers
By
Raymond A. Serway & John W. Jewett
6th Edition
‫اﻟﻮﺣﺪة‬‫اﻷوﻟ‬‫ﻰ‬‫اﻟﻤﯿﻜﺎﻧﯿﻜﺎ‬
‫اﻟﺠﺰء‬‫اﻟﺜﺎﻟﺚ‬‫ﻋﺸﺮ‬‫اﻟﺠﺬب‬ ‫ﻗﺎﻧﻮن‬‫اﻟﻌﺎم‬
‫أ‬/.‫دﺧﺎن‬ ‫إﺑﺮاھﯿﻢ‬ ‫ﺗﻤﺎم‬
‫د‬/.‫ﺳﻜﯿﻚ‬ ‫ﻓﻼح‬ ‫ﺣﺎزم‬
‫ﻓﯿﺰﯾﺎء‬
‫ﻛﺘﺎب‬
25-3-2009
‫ﺍﻷﻭﻟﻰ‬ ‫ﺍﻟﻭﺤﺩﺓ‬:‫ﺍﻟﻤﻴﻜﺎﻨﻴﻜﺎ‬
‫أ‬.‫دﺧﺎن‬ ‫إﺑراھﯾم‬ ‫ﺗﻣﺎم‬
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‫م‬‫اﻟﺒﻨﺪ‬
1‫اﻟﻤﺼﺪر‬Physics for Scientists and Engineers
Raymond A. Serway & John W. Jewett
2‫اﻟﻤﻮﺿﻮع‬
3‫اﻟﻤﺘﺮﺟﻢ‬
4‫اﻟﻌﻠﻤﯿﺔ‬ ‫اﻟﻤﺮاﺟﻌﺔ‬
5‫اﻟﻠﻐﻮﯾﺔ‬ ‫اﻟﻤﺮاﺟﻌﺔ‬
6‫اﻟﺘﺼﻨﯿﻒ‬
7‫اﻟﻔﺌﺔ‬
11‫اﻟﺘﺎرﯾﺦ‬

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Universal Gravitation
‫اﻟﻌﺎم‬ ‫اﻟﺟذب‬ ‫ﻗﺎﻧون‬
‫دﻗﯿ‬ ‫ﻗﯿﺎﺳﺎت‬ ‫ﻷﺧﺬ‬ ‫ﻓﻀﺎﺋﯿﺔ‬ ‫ﻣﺮﻛﺒﺔ‬ ‫ﺑﺈرﺳﺎل‬ ‫ﻟﮭﻢ‬ ‫ﺳﻤﺢ‬‫ﻧﻈﺎﻣﻨﺎ‬ ‫ﻣﻦ‬ ‫أﺧﺮى‬ ‫ﻷﺟﺰاء‬ ‫ﻘﺔ‬
‫ﺑﻤﺮﻛﺒﺔ‬ ‫أﺧﺬت‬ ‫اﻟﻤﺸﺘﺮي‬ ‫ﻗﻤﺮ‬ ‫ﻋﻠﻰ‬ ‫ﺑﺮﻛﺎن‬ ‫ﺻﻮرة‬ ‫ھﺬه‬‫ﺟ‬‫ﺎﻟﯿﻠ‬‫ﻮ‬‫ﻋﺎم‬ ‫اﻟﻤﺸﺘﺮي‬ ‫ﺣﻮل‬ ‫ﺗﺪور‬ ‫ﻛﺎﻧﺖ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻔﻀﺎﺋﯿﺔ‬
‫ﻧﻔﺬت‬‫اﻟﺴﻄﺢ‬ ‫ﺗﺤﺖ‬ ‫ﻣﻦ‬.)‫أرﯾﺰوﻧﺎ‬ ‫ﺟﺎﻣﻌﺔ‬-‫ﻧﺎﺳﺎ‬((Univ. of
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Universal Gravitation Law
‫اﻟﻌﺎم‬ ‫اﻟﺟذب‬ ‫ﻗﺎﻧون‬
‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻟﻘﺎﻧﻮن‬ ‫اﻟﻌﻠﻤﺎء‬ ‫ﻓﮭﻢ‬ ‫إن‬‫اﻟﻌﺎم‬‫دﻗﯿ‬ ‫ﻗﯿﺎﺳﺎت‬ ‫ﻷﺧﺬ‬ ‫ﻓﻀﺎﺋﯿﺔ‬ ‫ﻣﺮﻛﺒﺔ‬ ‫ﺑﺈرﺳﺎل‬ ‫ﻟﮭﻢ‬ ‫ﺳﻤﺢ‬
‫اﻟﺸﻤﺴﻲ‬.‫ﺑﻤﺮﻛﺒﺔ‬ ‫أﺧﺬت‬ ‫اﻟﻤﺸﺘﺮي‬ ‫ﻗﻤﺮ‬ ‫ﻋﻠﻰ‬ ‫ﺑﺮﻛﺎن‬ ‫ﺻﻮرة‬ ‫ھﺬه‬
1995‫اﻟﺤﻤﺮاء‬ ‫اﻟﻤﺎدة‬ ‫ھﺬه‬ ‫طﺒﻌﺎ‬ ،‫ﻧﻔﺬت‬
Arizona/JPL/NASA)

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‫ﺍﻟﻭﺤﺩﺓ‬‫ﺍﻷﻭﻟ‬‫ﻰ‬:‫ﺍﻟﻤﻴﻜﺎﻨﻴﻜﺎ‬
‫ﻋﺸﺭ‬ ‫ﺍﻟﺜﺎﻟﺙ‬ ‫ﺍﻟﺠﺯﺀ‬:‫ﻗﺎﻨﻭﻥ‬‫ﺍﻟﻌﺎﻡ‬ ‫ﺍﻟﺠﺫﺏ‬
‫ﻧﯿﻮﺗﻦ‬ ‫ﻗﺎﻧﻮن‬‫اﻟﻌﺎم‬ ‫ﻟﻠﺠﺬب‬
‫ﺛﺎﺑﺖ‬ ‫ﻗﯿﺎس‬‫اﻟﻌﺎم‬ ‫اﻟﺠﺬب‬
‫اﻟﺠﺎذﺑﯿﺔ‬ ‫وﻗﻮة‬ ‫اﻟﺤﺮ‬ ‫اﻟﺴﻘﻮط‬ ‫ﺗﺴﺎرع‬
‫اﻟﻜﻮاﻛﺐ‬ ‫وﺣﺮﻛﺔ‬ ‫ﻛﺒﻠﺮ‬ ‫ﻗﻮاﻧﯿﻦ‬
‫اﻷرﺿﯿﺔ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻣﺠﺎل‬
‫وﺿﻊ‬ ‫طﺎﻗﺔ‬‫اﻟﺠﺎذﺑﯿﺔ‬‫اﻷرﺿﯿﺔ‬
‫ﻓﻲ‬ ‫اﻟﻄﺎﻗﺔ‬ ‫اﻋﺘﺒﺎرات‬‫اﻟﺼﻨﺎﻋﻲ‬ ‫اﻟﻘﻤﺮ‬ ‫وﺣﺮﻛﺔ‬ ‫اﻟﻜﻮاﻛﺐ‬
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‫ﺍﻟﻭﺤﺩﺓ‬
‫ﻋﺸﺭ‬ ‫ﺍﻟﺜﺎﻟﺙ‬ ‫ﺍﻟﺠﺯﺀ‬
‫ﻧﯿﻮﺗﻦ‬ ‫ﻗﺎﻧﻮن‬ 1.13
‫ﺛﺎﺑﺖ‬ ‫ﻗﯿﺎس‬ 2.13
‫اﻟﺠﺎذﺑﯿﺔ‬ ‫وﻗﻮة‬ ‫اﻟﺤﺮ‬ ‫اﻟﺴﻘﻮط‬ ‫ﺗﺴﺎرع‬ 3.13
‫اﻟﻜﻮاﻛﺐ‬ ‫وﺣﺮﻛﺔ‬ ‫ﻛﺒﻠﺮ‬ ‫ﻗﻮاﻧﯿﻦ‬ 4.13
‫اﻷرﺿﯿﺔ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻣﺠﺎل‬ 5.13
‫وﺿﻊ‬ ‫طﺎﻗﺔ‬ 6.13
‫ﻓﻲ‬ ‫اﻟﻄﺎﻗﺔ‬ ‫اﻋﺘﺒﺎرات‬ 7.13

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‫ﻟﻠﻘﻭﻯ‬ ‫ﻭﺍﻀﺢ‬ ‫ﻓﻬﻡ‬ ‫ﻴﺘﻭﻓﺭ‬ ‫ﻟﻡ‬ ‫ﻟﻜﻨﻪ‬ ،‫ﻭﺍﻟﻜﻭﺍﻜﺏ‬ ‫ﺍﻟﻘﻤﺭ‬ ‫ﺤﺭﻜﺔ‬ ‫ﺤﻭل‬ ‫ﺍﻟﻤﻌﻠﻭﻤﺎﺕ‬ ‫ﻤﻥ‬ ‫ﻜﺒﻴﺭﺓ‬ ‫ﻜﻤﻴﺔ‬ ‫ﺠﻤﻊ‬ ‫ﺘﻡ‬ ،
‫ﺠﺎﺀ‬ ‫ﺍﻟﺴﻨﺔ‬ ‫ﺘﻠﻙ‬ ‫ﻓﻲ‬ ،‫ﺍﻟﺤﺭﻜﺎﺕ‬ ‫ﺒﻬﺫﻩ‬ ‫ﺍﻟﻤﺘﻌﻠﻘﺔ‬‫ﺍﻟﻌﺎﻟﻡ‬‫ﻑ‬‫ﺭ‬‫ﻭﻋ‬ ،‫ﺍﻟﺴﻤﺎﺀ‬ ‫ﺃﺴﺭﺍﺭ‬ ‫ﻓﺘﺢ‬ ‫ﺍﻟﺫﻱ‬ ‫ﺒﺎﻟﻤﻔﺘﺎﺡ‬ ‫ﻭﺯﻭﺩﻨﺎ‬ ‫ﻨﻴﻭﺘﻥ‬ ‫ﺍﺴﺤﻕ‬
‫ﺨﻁ‬ ‫ﻭﻓﻕ‬ ‫ﺍﻟﻘﻤﺭ‬ ‫ﺴﻴﺴﻴﺭ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻫﺫﻩ‬ ‫ﺒﺩﻭﻥ‬ ‫ﻷﻨﻪ‬ ‫ﺍﻟﻘﻤﺭ‬ ‫ﺴﻁﺢ‬ ‫ﻋﻠﻰ‬ ‫ﺘﺅﺜﺭ‬ ‫ﻗﻭﺓ‬ ‫ﻤﻥ‬ ‫ﺩ‬‫ﺒ‬ ‫ﻻ‬ ‫ﺃﻨﻪ‬ ‫ﻴﺭﻯ‬ ‫ﻜﺎﻥ‬ ‫ﺤﻴﺙ‬ ،
‫ﺘﻘﺭﻴﺒﺎ‬ ‫ﺍﻟﺩﺍﺌﺭﻱ‬ ‫ﻤﺩﺍﺭﻩ‬ ‫ﻓﻲ‬ ‫ﻴﻜﻭﻥ‬ ‫ﺃﻥ‬ ‫ﻤﻥ‬ ‫ﺒﺩﻻ‬ ‫ﻤﺴﺘﻘﻴﻡ‬.
‫ﺇﻻ‬ ،‫ﻟﻠﻘﻤﺭ‬ ‫ﺍﻷﺭﺽ‬ ‫ﺠﺎﺫﺒﻴﺔ‬ ‫ﺒﺄﻨﻬﺎ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻫﺫﻩ‬ ‫ﻨﻴﻭﺘﻥ‬ ‫ﻓﺴﺭ‬‫ﻭﺠﺎﺫﺒﻴﺔ‬ ‫ﻟﻠﻘﻤﺭ‬ ‫ﺍﻷﺭﺽ‬ ‫ﺠﺎﺫﺒﻴﺔ‬ ‫ﻋﻥ‬ ‫ﺍﻟﻤﺴﺌﻭﻟﺔ‬ ‫ﺍﻟﻘﻭﻯ‬ ‫ﺃﻥ‬ ‫ﺃﺩﺭﻙ‬ ‫ﺃﻨﻪ‬
‫ﺍﻷﺠﺴﺎﻡ‬ ‫ﺒﻴﻥ‬ ‫ﻭﺍﻟﻌﺎﻟﻤﻲ‬ ‫ﺍﻟﻌﺎﻡ‬ ‫ﺍﻟﺠﺫﺏ‬ ‫ﻤﻥ‬ ‫ﺨﺎﺼﺔ‬ ‫ﺤﺎﻟﺔ‬ ‫ﻫﻲ‬ ‫ﺇﻨﻤﺎ‬ ‫ﻓﻘﻁ‬ ‫ﺍﻟﻨﻅﻡ‬ ‫ﺒﺘﻠﻙ‬ ‫ﺨﺎﺼﺎ‬ ‫ﺸﻴﺌﺎ‬ ‫ﻟﻴﺴﺕ‬ ،‫ﻟﻠﻜﻭﺍﻜﺏ‬ ‫ﺍﻟﺸﻤﺱ‬.
‫ﺴﻘﻭﻁ‬ ‫ﺇﻟﻰ‬ ‫ﺃﺩﺕ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻨﻔﺴﻬﺎ‬ ‫ﻫﻲ‬ ‫ﺍﻷﺭﺽ‬ ‫ﺤﻭل‬ ‫ﺍﻟﻘﻤﺭ‬ ‫ﺩﻭﺭﺍﻥ‬ ‫ﺘﺴﺒﺏ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﺃﻥ‬ ‫ﻨﻴﻭﺘﻥ‬ ‫ﺭﺃﻯ‬
‫ﺃﻥ‬ ‫ﻴﺠﺏ‬ ‫ﻤﺩﺍﺭﺍﺘﻬﺎ‬ ‫ﻓﻲ‬ ‫ﺍﻟﻜﻭﺍﻜﺏ‬ ‫ﺘﺒﻘﻲ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﺃﻥ‬ ‫ﻨﺴﺘﻨﺘﺞ‬‫ﻭ‬ ‫ﺒﻴﻨﻬﺎ‬ ‫ﺍﻟﻤﺴﺎﻓﺔ‬ ‫ﻤﺭﺒﻊ‬ ‫ﻤﻊ‬ ‫ﺘﺘﻨﺎﺴﺏ‬‫ﺒﻴﻥ‬
‫ﻭﺠﺩ‬ ،‫ﺍﻷﺭﺽ‬ ‫ﺴﻁﺢ‬ ‫ﺠﺎﺫﺒﻴﺔ‬ ‫ﻗﻭﺓ‬ ‫ﻤﻊ‬ ‫ﻤﺩﺍﺭﻩ‬ ‫ﻓﻲ‬ ‫ﺍﻟﻘﻤﺭ‬ ‫ﻟﺒﻘﺎﺀ‬ ‫ﺍﻟﻼﺯﻤﺔ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻤﻘﺎﺭﻨﺔ‬ ‫ﻋﻨﺩ‬ ‫ﻭﺒﺫﻟﻙ‬ ،‫ﺤﻭﻟﻬﺎ‬ ‫ﺘﺩﻭﺭ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻤﺭﻜﺯ‬
‫ﺤﻭل‬ ‫ﺒﻤﻌﻠﻭﻤﺎﺕ‬ ‫ﺴﺘﺯﻭﺩﻨﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻜﻭﺍﻜﺏ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻭﺼﻑ‬ ‫ﻋﻠﻰ‬ ‫ﻭﻨﺅﻜﺩ‬ ،‫ﺍﻟﻌﺎﻟﻤﻲ‬ ‫ﺍﻟﺠﺎﺫﺒﻴﺔ‬ ‫ﻗﺎﻨﻭﻥ‬ ‫ﺍﻟﻔﺼل‬ ‫ﻫﺫﺍ‬
‫ﻗﺒل‬ ‫ﻤﻥ‬ ‫ﻤﺘﺎﺒﻌﺘﻬﺎ‬ ‫ﺘﻤﺕ‬ ‫ﻜﺒﻠﺭ‬ ‫ﻴﻭﻫﺎﻨﺱ‬ ‫ﻭﻀﻌﻬﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻜﻭﺍﻜﺏ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻗﻭﺍﻨﻴﻥ‬ ‫ﺒﺄﻥ‬ ‫ﻨﺒﻴﻥ‬ ‫ﺜﻡ‬
‫ﺍﻟﺯ‬ ‫ﺍﻻﻨﺩﻓﺎﻉ‬ ‫ﺤﻔﻅ‬‫ﺍﻭ‬،‫ﻱ‬‫ﺍﻟﻌﺎﻡ‬ ‫ﺍﻟﺘﻌﺒﻴﺭ‬ ‫ﺒﺎﺴﺘﻨﺘﺎﺝ‬ ‫ﻭﺴﻨﺨﺘﻡ‬‫ﺍﻟﻭﻀﻊ‬ ‫ﻟﻁﺎﻗﺔ‬،‫ﻭ‬‫ﺤﺭﻜﺔ‬‫ﺍﻷﻗﻤﺎﺭ‬
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‫ﻋﺎﻡ‬ ‫ﻗﺒل‬1687‫ﻟﻠﻘﻭﻯ‬ ‫ﻭﺍﻀﺢ‬ ‫ﻓﻬﻡ‬ ‫ﻴﺘﻭﻓﺭ‬ ‫ﻟﻡ‬ ‫ﻟﻜﻨﻪ‬ ،‫ﻭﺍﻟﻜﻭﺍﻜﺏ‬ ‫ﺍﻟﻘﻤﺭ‬ ‫ﺤﺭﻜﺔ‬ ‫ﺤﻭل‬ ‫ﺍﻟﻤﻌﻠﻭﻤﺎﺕ‬ ‫ﻤﻥ‬ ‫ﻜﺒﻴﺭﺓ‬ ‫ﻜﻤﻴﺔ‬ ‫ﺠﻤﻊ‬ ‫ﺘﻡ‬ ،
‫ﺠﺎﺀ‬ ‫ﺍﻟﺴﻨﺔ‬ ‫ﺘﻠﻙ‬ ‫ﻓﻲ‬ ،‫ﺍﻟﺤﺭﻜﺎﺕ‬ ‫ﺒﻬﺫﻩ‬ ‫ﺍﻟﻤﺘﻌﻠﻘﺔ‬
‫ﺒﻘﺎﻨﻭﻨﻪ‬‫ﺍﻷﻭل‬‫ﺨﻁ‬ ‫ﻭﻓﻕ‬ ‫ﺍﻟﻘﻤﺭ‬ ‫ﺴﻴﺴﻴﺭ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻫﺫﻩ‬ ‫ﺒﺩﻭﻥ‬ ‫ﻷﻨﻪ‬ ‫ﺍﻟﻘﻤﺭ‬ ‫ﺴﻁﺢ‬ ‫ﻋﻠﻰ‬ ‫ﺘﺅﺜﺭ‬ ‫ﻗﻭﺓ‬ ‫ﻤﻥ‬ ‫ﺩ‬‫ﺒ‬ ‫ﻻ‬ ‫ﺃﻨﻪ‬ ‫ﻴﺭﻯ‬ ‫ﻜﺎﻥ‬ ‫ﺤﻴﺙ‬ ،
‫ﺘﻘﺭﻴﺒﺎ‬ ‫ﺍﻟﺩﺍﺌﺭﻱ‬ ‫ﻤﺩﺍﺭﻩ‬ ‫ﻓﻲ‬ ‫ﻴﻜﻭﻥ‬ ‫ﺃﻥ‬ ‫ﻤﻥ‬ ‫ﺒﺩﻻ‬ ‫ﻤﺴﺘﻘﻴﻡ‬
‫ﺇﻻ‬ ،‫ﻟﻠﻘﻤﺭ‬ ‫ﺍﻷﺭﺽ‬ ‫ﺠﺎﺫﺒﻴﺔ‬ ‫ﺒﺄﻨﻬﺎ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻫﺫﻩ‬ ‫ﻨﻴﻭﺘﻥ‬ ‫ﻓﺴﺭ‬
‫ﺍﻷﺠﺴﺎﻡ‬ ‫ﺒﻴﻥ‬ ‫ﻭﺍﻟﻌﺎﻟﻤﻲ‬ ‫ﺍﻟﻌﺎﻡ‬ ‫ﺍﻟﺠﺫﺏ‬ ‫ﻤﻥ‬ ‫ﺨﺎﺼﺔ‬ ‫ﺤﺎﻟﺔ‬ ‫ﻫﻲ‬ ‫ﺇﻨﻤﺎ‬ ‫ﻓﻘﻁ‬ ‫ﺍﻟﻨﻅﻡ‬ ‫ﺒﺘﻠﻙ‬ ‫ﺨﺎﺼﺎ‬ ‫ﺸﻴﺌﺎ‬ ‫ﻟﻴﺴﺕ‬ ،‫ﻟﻠﻜﻭﺍﻜﺏ‬ ‫ﺍﻟﺸﻤﺱ‬
‫ﺃﺨﺭﻯ‬ ‫ﺒﻌﺒﺎﺭﺓ‬:‫ﺴﻘﻭﻁ‬ ‫ﺇﻟﻰ‬ ‫ﺃﺩﺕ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻨﻔﺴﻬﺎ‬ ‫ﻫﻲ‬ ‫ﺍﻷﺭﺽ‬ ‫ﺤﻭل‬ ‫ﺍﻟﻘﻤﺭ‬ ‫ﺩﻭﺭﺍﻥ‬ ‫ﺘﺴﺒﺏ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﺃﻥ‬ ‫ﻨﻴﻭﺘﻥ‬ ‫ﺭﺃﻯ‬
‫ﺍ‬‫ﺍﻟﺸﺠﺭﺓ‬ ‫ﻤﻥ‬ ‫ﻟﺘﻔﺎﺤﺔ‬.
‫ﺘﻌﺒﻴﺭﻩ‬ ‫ﺤﺩ‬ ‫ﻋﻠﻰ‬:"‫ﺃﻥ‬ ‫ﻴﺠﺏ‬ ‫ﻤﺩﺍﺭﺍﺘﻬﺎ‬ ‫ﻓﻲ‬ ‫ﺍﻟﻜﻭﺍﻜﺏ‬ ‫ﺘﺒﻘﻲ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﺃﻥ‬ ‫ﻨﺴﺘﻨﺘﺞ‬
‫ﻭﺠﺩ‬ ،‫ﺍﻷﺭﺽ‬ ‫ﺴﻁﺢ‬ ‫ﺠﺎﺫﺒﻴﺔ‬ ‫ﻗﻭﺓ‬ ‫ﻤﻊ‬ ‫ﻤﺩﺍﺭﻩ‬ ‫ﻓﻲ‬ ‫ﺍﻟﻘﻤﺭ‬ ‫ﻟﺒﻘﺎﺀ‬ ‫ﺍﻟﻼﺯﻤﺔ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻤﻘﺎﺭﻨﺔ‬ ‫ﻋﻨﺩ‬ ‫ﻭﺒﺫﻟﻙ‬ ،‫ﺤﻭﻟﻬﺎ‬ ‫ﺘﺩﻭﺭ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻤﺭﻜﺯ‬
‫ﺘﻘﺭﻴﺒﺎ‬ ‫ﺫﺍﺘﻪ‬ ‫ﺍﻟﺠﻭﺍﺏ‬."
‫ﻓﻲ‬ ‫ﺴﻨﺩﺭﺱ‬‫ﺤﻭل‬ ‫ﺒﻤﻌﻠﻭﻤﺎﺕ‬ ‫ﺴﺘﺯﻭﺩﻨﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻜﻭﺍﻜﺏ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻭﺼﻑ‬ ‫ﻋﻠﻰ‬ ‫ﻭﻨﺅﻜﺩ‬ ،‫ﺍﻟﻌﺎﻟﻤﻲ‬ ‫ﺍﻟﺠﺎﺫﺒﻴﺔ‬ ‫ﻗﺎﻨﻭﻥ‬ ‫ﺍﻟﻔﺼل‬ ‫ﻫﺫﺍ‬
‫ﺍﻟﻘﺎﻨﻭﻥ‬ ‫ﻫﺫﺍ‬ ‫ﺼﺤﺔ‬ ‫ﺍﺨﺘﺒﺎﺭ‬ ‫ﺃﻫﻤﻴﺔ‬.‫ﻗﺒل‬ ‫ﻤﻥ‬ ‫ﻤﺘﺎﺒﻌﺘﻬﺎ‬ ‫ﺘﻤﺕ‬ ‫ﻜﺒﻠﺭ‬ ‫ﻴﻭﻫﺎﻨﺱ‬ ‫ﻭﻀﻌﻬﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻜﻭﺍﻜﺏ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻗﻭﺍﻨﻴﻥ‬ ‫ﺒﺄﻥ‬ ‫ﻨﺒﻴﻥ‬ ‫ﺜﻡ‬
‫ﻭ‬ ‫ﺍﻟﻌﺎﻟﻤﻲ‬ ‫ﺍﻟﺠﺎﺫﺒﻴﺔ‬ ‫ﻗﺎﻨﻭﻥ‬‫ﻤﻔﻬﻭﻡ‬‫ﺍﻟﺯ‬ ‫ﺍﻻﻨﺩﻓﺎﻉ‬ ‫ﺤﻔﻅ‬
‫ﺍﻟﺼﻨﺎﻋﻴﺔ‬.

6
‫ﻴﻔﺘﺭﺽ‬ ‫ﺍﻷﺴﻁﻭﺭﺓ‬ ‫ﻫﺫﻩ‬ ‫ﺇﻥ‬ ،‫ﺍﻟﺸﺠﺭﺓ‬ ‫ﺘﺤﺕ‬ ‫ﻗﻴﻠﻭﻟﺔ‬ ‫ﻓﻲ‬ ‫ﻜﺎﻥ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﻨﻴﻭﺘﻥ‬ ‫ﺭﺃﺱ‬ ‫ﻋﻠﻰ‬ ‫ﺍﻟﺘﻔﺎﺤﺔ‬ ‫ﺴﻘﻭﻁ‬ ‫ﺒﺄﺴﻁﻭﺭﺓ‬ ‫ﺴﻤﻌﺕ‬ ‫ﺭﺒﻤﺎ‬
‫ﺒﻌﻀﻬﺎ‬ ‫ﺇﻟﻰ‬ ‫ﺘﻨﺠﺫﺏ‬ ‫ﺍﻟﻜﻭﻥ‬ ‫ﻓﻲ‬ ‫ﺍﻷﺠﺴﺎﻡ‬ ‫ﻜل‬ ‫ﺒﺄﻥ‬ ‫ﻟﻠﺘﺨﻴل‬ ‫ﺩﻓﻌﺘﻪ‬ ‫ﺒﺄﻨﻬﺎ‬‫ﺍﻷﺭﺽ‬ ‫ﺇﻟﻰ‬ ‫ﺍﻟﺘﻔﺎﺤﺔ‬ ‫ﺠﺫﺏ‬ ‫ﻁﺭﻴﻘﺔ‬ ‫ﺒﻨﻔﺱ‬ ،‫ﺍﻟﺒﻌﺽ‬.‫ﺤﻠل‬
‫ﻴﺤﻜﻡ‬ ‫ﺍﻟﺫﻱ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻗﺎﻨﻭﻥ‬ ‫ﺒﺄﻥ‬ ،‫ﺍﻟﺠﺭﻱﺀ‬ ‫ﺒﻘﻭﻟﻪ‬ ‫ﺨﺭﺝ‬ ‫ﺍﻟﺘﺤﻠﻴل‬ ‫ﻫﺫﺍ‬ ‫ﻭﻤﻥ‬ ،‫ﺍﻷﺭﺽ‬ ‫ﺤﻭل‬ ‫ﺍﻟﻘﻤﺭ‬ ‫ﺒﺤﺭﻜﺔ‬ ‫ﺍﻟﻤﺘﻌﻠﻘﺔ‬ ‫ﺍﻟﺒﻴﺎﻨﺎﺕ‬ ‫ﻨﻴﻭﺘﻥ‬
‫ﺍﻷﺭﺽ‬ ‫ﺇﻟﻰ‬ ‫ﺴﻘﻭﻁﻬﺎ‬ ‫ﻋﻨﺩ‬ ‫ﺍﻟﺘﻔﺎﺤﺔ‬ ‫ﺠﺫﺏ‬ ‫ﺍﻟﺫﻱ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻗﺎﻨﻭﻥ‬ ‫ﻨﻔﺴﻪ‬.‫ﻫﺫﻩ‬ ‫ﻜﺎﻨﺕ‬‫ﺃﻭل‬‫ﻓﻴﻬﺎ‬ ‫ﺘﺘﻭﺤﺩ‬ ‫ﻤﺭﺓ‬
‫ﻨﻴﻭﺘﻥ‬ ‫ﻟﺘﺤﻠﻴﻼﺕ‬ ‫ﺭﻴﺎﻀﻴﺔ‬ ‫ﺘﻔﺎﺼﻴل‬ ‫ﺍﻟﻔﺼل‬ ‫ﻫﺫﺍ‬ ‫ﻓﻲ‬ ‫ﻨﺭﻯ‬ ‫ﺴﻭﻑ‬ ،‫ﻭﺍﻷﺭﻀﻴﺔ‬ ‫ﻴﺔ‬.
‫ﻤﺒﺎﺩﺉ‬ ‫ﺍﻟﺭﻴﺎﻀﻲ‬ ‫ﺒﺤﺜﻪ‬ ‫ﻓﻲ‬ ‫ﺍﻟﺠﺎﺫﺒﻴﺔ‬ ‫ﻗﺎﻨﻭﻥ‬ ‫ﻋﻥ‬ ‫ﻋﻤﻠﻪ‬ ‫ﻨﻴﻭﺘﻥ‬ ‫ﻨﺸﺭ‬‫ﻗﺎﻨﻭﻥ‬ ،‫ﺍﻟﻁﺒﻴﻌﻴﺔ‬ ‫ﺍﻟﻔﻠﺴﻔﺔ‬‫ﺍﻟﻌﺎﻡ‬ ‫ﺍﻟﺠﺫﺏ‬
‫ﻣﺮﺑﻊ‬ ‫ﻣﻊ‬ ‫وﻋﻜﺴﺎ‬ ‫ﻛﺘﻠﮭﺎ‬ ‫ﻣﻊ‬ ‫طﺮدا‬ ‫ﺗﺘﻨﺎﺳﺐ‬ ‫ﻣﺒﺎﺷﺮة‬ ‫ﻗﻮة‬ ‫ﺑﺘﺄﺛﯿﺮ‬ ‫اﻷﺧﺮى‬ ‫اﻟﺠﺴﯿﻤﺎت‬ ‫ﻣﻊ‬ ‫ﺗﺘﺠﺎذب‬ ‫اﻟﻜﻮن‬ ‫ﻓﻲ‬
‫ﻣﺴﺎﻓﺔ‬ ‫ﺑﯿﻨﮭﻤﺎ‬ ‫وﺗﻔﺼﻞ‬r‫ﺑﺎﻟﻌﻼﻗﺔ‬ ‫ﯾﻌﻄﻰ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫ﻣﻘﺪار‬ ،:
‫ﻓﻲ‬ ‫اﻟﺘﺠﺮﯾﺒﯿﺔ‬ ‫ﻗﯿﻤﺘﮫ‬‫ﻧﻈﺎم‬‫ھﻲ‬ ‫اﻟﺪوﻟﯿﺔ‬ ‫اﻟﻮﺣﺪات‬:
www.trgma.com
1.13‫ﻗﺎﻧﻮن‬‫اﻟﻌﺎم‬ ‫اﻟﺠﺬب‬‫ﻟﻨﯿﻮﺗﻦ‬
‫ﻴﻔﺘﺭﺽ‬ ‫ﺍﻷﺴﻁﻭﺭﺓ‬ ‫ﻫﺫﻩ‬ ‫ﺇﻥ‬ ،‫ﺍﻟﺸﺠﺭﺓ‬ ‫ﺘﺤﺕ‬ ‫ﻗﻴﻠﻭﻟﺔ‬ ‫ﻓﻲ‬ ‫ﻜﺎﻥ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﻨﻴﻭﺘﻥ‬ ‫ﺭﺃﺱ‬ ‫ﻋﻠﻰ‬ ‫ﺍﻟﺘﻔﺎﺤﺔ‬ ‫ﺴﻘﻭﻁ‬ ‫ﺒﺄﺴﻁﻭﺭﺓ‬ ‫ﺴﻤﻌﺕ‬ ‫ﺭﺒﻤﺎ‬
‫ﺒﻌﻀﻬﺎ‬ ‫ﺇﻟﻰ‬ ‫ﺘﻨﺠﺫﺏ‬ ‫ﺍﻟﻜﻭﻥ‬ ‫ﻓﻲ‬ ‫ﺍﻷﺠﺴﺎﻡ‬ ‫ﻜل‬ ‫ﺒﺄﻥ‬ ‫ﻟﻠﺘﺨﻴل‬ ‫ﺩﻓﻌﺘﻪ‬ ‫ﺒﺄﻨﻬﺎ‬
‫ﻴﺤﻜﻡ‬ ‫ﺍﻟﺫﻱ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻗﺎﻨﻭﻥ‬ ‫ﺒﺄﻥ‬ ،‫ﺍﻟﺠﺭﻱﺀ‬ ‫ﺒﻘﻭﻟﻪ‬ ‫ﺨﺭﺝ‬ ‫ﺍﻟﺘﺤﻠﻴل‬ ‫ﻫﺫﺍ‬ ‫ﻭﻤﻥ‬ ،‫ﺍﻷﺭﺽ‬ ‫ﺤﻭل‬ ‫ﺍﻟﻘﻤﺭ‬ ‫ﺒﺤﺭﻜﺔ‬ ‫ﺍﻟﻤﺘﻌﻠﻘﺔ‬ ‫ﺍﻟﺒﻴﺎﻨﺎﺕ‬ ‫ﻨﻴﻭﺘﻥ‬
‫ﺍﻟﻜﻭﺍﻜﺏ‬ ‫ﺤﺭﻜﺔ‬‫ﻫﻭ‬‫ﺍﻷﺭﺽ‬ ‫ﺇﻟﻰ‬ ‫ﺴﻘﻭﻁﻬﺎ‬ ‫ﻋﻨﺩ‬ ‫ﺍﻟﺘﻔﺎﺤﺔ‬ ‫ﺠﺫﺏ‬ ‫ﺍﻟﺫﻱ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻗﺎﻨﻭﻥ‬ ‫ﻨﻔﺴﻪ‬
‫ﺍﻟﺤ‬‫ﺍﻟﺴﻤ‬ ‫ﺭﻜﺎﺕ‬‫ﺎ‬‫ﻭ‬‫ﻨﻴﻭﺘﻥ‬ ‫ﻟﺘﺤﻠﻴﻼﺕ‬ ‫ﺭﻴﺎﻀﻴﺔ‬ ‫ﺘﻔﺎﺼﻴل‬ ‫ﺍﻟﻔﺼل‬ ‫ﻫﺫﺍ‬ ‫ﻓﻲ‬ ‫ﻨﺭﻯ‬ ‫ﺴﻭﻑ‬ ،‫ﻭﺍﻷﺭﻀﻴﺔ‬ ‫ﻴﺔ‬
‫ﻋﺎﻡ‬ ‫ﻓﻲ‬1987‫ﻤﺒﺎﺩﺉ‬ ‫ﺍﻟﺭﻴﺎﻀﻲ‬ ‫ﺒﺤﺜﻪ‬ ‫ﻓﻲ‬ ‫ﺍﻟﺠﺎﺫﺒﻴﺔ‬ ‫ﻗﺎﻨﻭﻥ‬ ‫ﻋﻥ‬ ‫ﻋﻤﻠﻪ‬ ‫ﻨﻴﻭﺘﻥ‬ ‫ﻨﺸﺭ‬
‫ﺃﻥ‬ ‫ﺫﻜﺭ‬ ‫ﻟﻨﻴﻭﺘﻥ‬:
‫ﻛﻞ‬‫اﻷﺟﺴﺎم‬‫ﻣﺮﺑﻊ‬ ‫ﻣﻊ‬ ‫وﻋﻜﺴﺎ‬ ‫ﻛﺘﻠﮭﺎ‬ ‫ﻣﻊ‬ ‫طﺮدا‬ ‫ﺗﺘﻨﺎﺳﺐ‬ ‫ﻣﺒﺎﺷﺮة‬ ‫ﻗﻮة‬ ‫ﺑﺘﺄﺛﯿﺮ‬ ‫اﻷﺧﺮى‬ ‫اﻟﺠﺴﯿﻤﺎت‬ ‫ﻣﻊ‬ ‫ﺗﺘﺠﺎذب‬ ‫اﻟﻜﻮن‬ ‫ﻓﻲ‬
‫ﺑﯿﻨﮭﺎ‬ ‫اﻟﻤﺴﺎﻓﺔ‬.
‫أﻧ‬ ‫أي‬‫ﮫ‬‫ﻟﺪﯾﻨﺎ‬ ‫ﻛﺎن‬ ‫إذا‬‫ﻛﺘﻠﺘﯿﻦ‬m1،m2‫ﻣﺴﺎﻓﺔ‬ ‫ﺑﯿﻨﮭﻤﺎ‬ ‫وﺗﻔﺼﻞ‬
‫ﺣﯿﺚ‬G‫ﺛﺎﺑﺖ‬ ‫وﯾﺪﻋﻰ‬ ،‫ﺛﺎﺑﺖ‬‫و‬ ‫اﻟﻌﺎم‬ ‫اﻟﺠﺬب‬‫ﻓﻲ‬ ‫اﻟﺘﺠﺮﯾﺒﯿﺔ‬ ‫ﻗﯿﻤﺘﮫ‬
‫رﻗﻢ‬ ‫اﻟﺸﻜﻞ‬1.13‫ﺑﯿﻦ‬ ‫اﻟﺘﺠﺎذب‬ ‫ﻗﻮة‬
‫اﻟﻮاﺣﺪة‬ ‫ﺷﻌﺎع‬ ،‫ﺟﺴﻤﯿﻦ‬rˆ12‫ﻣﻦ‬ ‫ﻣﻮﺟﮫ‬
‫اﻟﺠﺴﻢ‬1‫اﻟﺠﺴﻢ‬ ‫إﻟﻰ‬2،
‫أن‬ ‫ﻻﺣﻆ‬:-F21=F12
‫رﻗﻢ‬ ‫اﻟﺸﻜﻞ‬
‫اﻟﻮاﺣﺪة‬ ‫ﺷﻌﺎع‬ ،‫ﺟﺴﻤﯿﻦ‬

7
‫ﻣﺮﺑﻊ‬ ‫ﻣﻊ‬ ‫ﯾﺘﻨﺎﺳﺐ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫ﻣﻘﺪار‬ ‫ﻷن‬ ،‫اﻟﻌﻜﺴﻲ‬ ‫اﻟﺘﺮﺑﯿﻊ‬ ‫ﺑﻘﺎﻧﻮن‬ ‫ﻏﺎﻟﺒﺎ‬ ‫ﯾﺴﻤﻰ‬
،‫ﺷﻌﺎع‬ ‫ﺷﻜﻞ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻘﻮة‬ ‫ھﺬه‬ ‫ﻧﻤﺜﻞ‬ ‫أن‬ ‫ﯾﻤﻜﻨﻨﺎ‬‫اﻟﻮﺣﺪة‬ ‫ﺷﻌﺎع‬ ‫ﺑﻮاﺳﻄﺔ‬ ‫وذﻟﻚ‬rˆ12)‫اﻟﺸﻜﻞ‬1.13(‫ﺣﯿﺚ‬
2‫ﯾ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻘﻮة‬ ‫وﺑﺎﻟﺘﺎﻟﻲ‬ ،‫ﺘﺄ‬‫ﺑﮭﺎ‬ ‫ﺛﺮ‬‫اﻟﺠﺴﻢ‬1‫ﻣﻦ‬‫اﻟﺠﺴﻢ‬2‫ھﻲ‬:
‫ﯾﺠﺬب‬‫اﻟﺠﺴﻢ‬1‫ﻗﻮة‬ ‫وﺑﺎﻟﺘﺎﻟﻲ‬ ،‫اﻟﺠﺴﻢ‬2‫اﻟﺠﺴﻢ‬ ‫ﺑﺎﺗﺠﺎه‬ ‫ﺗﻜﻮن‬ ‫أن‬ ‫ﯾﺠﺐ‬1‫وذﻟﻚ‬ ،
‫اﻟﺠﺴﻢ‬2‫اﻟﺠﺴﻢ‬ ‫ﻋﻠﻰ‬1.
F‫و‬‫اﻟﻤﻌﺎﻛﺲ‬ ‫اﻻﺗﺠﺎه‬ ‫ﻓﻲ‬ ‫ﻟﻜﻦ‬.‫ﺗﺸﻜﻞ‬ ‫اﻟﻘﻮى‬ ‫ھﺬه‬ ‫أن‬ ‫ﯾﻌﻨﻲ‬ ‫وھﺬا‬‫اﻟﻔﻌﻞ‬ ‫ﻣﻦ‬ ‫زوﺟﺎ‬
‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬ ‫اﻟﺬﻛﺮ‬ ‫ﺗﺴﺘﺤﻖ‬ ‫ﻋﺪﯾﺪة‬ ‫ﻣﯿﺰات‬)3.13.(‫ﻗﻮة‬ ‫ھﻲ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬‫اﻟﻤﺠﺎل‬
‫ﺑﯿﻨﮭﻤﺎ‬ ‫اﻟﻔﺎﺻﻞ‬ ‫اﻟﻮﺳﻂ‬ ‫ﻋﻦ‬ ‫اﻟﻨﻈﺮ‬ ‫ﺑﻐﺾ‬ ،.‫ﺑﯿﻦ‬ ‫اﻟﻤﺴﺎﻓﺔ‬ ‫ﻣﺮﺑﻊ‬ ‫ﻣﻊ‬ ‫ﺗﺘﻨﺎﺳﺐ‬ ‫اﻟﻘﻮة‬ ‫ﻷن‬ ‫وذﻟﻚ‬
‫اﻟﻤﺴﺎﻓﺔ‬ ‫زﯾﺎدة‬ ‫ﻣﻊ‬ ‫ﻣﻠﺤﻮظ‬ ‫ﺑﺸﻜﻞ‬.
‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬ ‫ﻧﺮاھﺎ‬ ‫أن‬ ‫ﻧﺴﺘﻄﯿﻊ‬ ‫أﺧﺮى‬ ‫ھﺎﻣﺔ‬ ‫ﻧﻘﻄﺔ‬ ‫ھﻨﺎك‬)3.13(‫و‬‫ھﻮ‬‫ا‬ ‫ﻗﻮة‬ ‫أن‬‫ﻛﺮوﯾﺔ‬ ‫ﻛﺘﻠﺔ‬ ‫ﻋﻦ‬ ‫اﻟﻨﺎﺗﺠﺔ‬ ‫ﻟﺠﺎذﺑﯿﺔ‬
‫ﻟ‬ ‫ﻛﻤﺎ‬ ‫ﻧﻔﺴﮭﺎ‬ ‫ھﻲ‬ ‫اﻟﺘﻮزﯾﻊ‬ ‫ﺧﺎرج‬ ‫ﺟﺴﻢ‬ ‫ﻋﻠﻰ‬ ‫وﺗﺆﺛﺮ‬ ‫اﻟﺤﺠﻢ‬ ‫وﻣﺤﺪودة‬ ‫ﻣﺘﺠﺎﻧﺴﺔ‬‫ﻮ‬‫أن‬‫اﻟ‬ ‫ﻛﻞ‬‫ﻜﺘﻠﺔ‬‫ﻣﺮﻛﺰة‬‫اﻟﻤﺮﻛﺰ‬ ‫ﻓﻲ‬.‫ﺳﺒﯿﻞ‬ ‫ﻋﻠﻰ‬
‫ﻛﺘﻠﺘﮫ‬ ‫ﺟﺴﻢ‬ ‫ﻋﻠﻰ‬ ‫اﻷرض‬ ‫ﺑﮭﺎ‬ ‫ﺗﺆﺛﺮ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻘﻮة‬ ‫ﻣﻘﺪار‬ ‫إن‬ ،m‫ھﻲ‬ ‫اﻷرض‬ ‫ﺳﻄﺢ‬ ‫ﻗﺮب‬:
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‫اﻟﻘﻮة‬ ‫ﻗﺎﻧﻮن‬ ‫ﺷﻜﻞ‬ ‫إن‬‫ﻓﻲ‬‫اﻟﻤﻌﺎدﻟﺔ‬1.13‫ﻣﺮﺑﻊ‬ ‫ﻣﻊ‬ ‫ﯾﺘﻨﺎﺳﺐ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫ﻣﻘﺪار‬ ‫ﻷن‬ ،‫اﻟﻌﻜﺴﻲ‬ ‫اﻟﺘﺮﺑﯿﻊ‬ ‫ﺑﻘﺎﻧﻮن‬ ‫ﻏﺎﻟﺒﺎ‬ ‫ﯾﺴﻤﻰ‬
‫اﻷﺟﺴﺎم‬ ‫ﺑﯿﻦ‬ ‫اﻟﻤﺴﺎﻓﺔ‬.،‫ﺷﻌﺎع‬ ‫ﺷﻜﻞ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻘﻮة‬ ‫ھﺬه‬ ‫ﻧﻤﺜﻞ‬ ‫أن‬ ‫ﯾﻤﻜﻨﻨﺎ‬
‫اﻟﺠﺴﻢ‬ ‫ﻣﻦ‬ ‫ﻣﻮﺟﮫ‬ ‫اﻟﻮاﺣﺪة‬ ‫ﺷﻌﺎع‬1‫إﻟﻰ‬‫اﻟﺠﺴﻢ‬2
‫اﻟﺠﺴﻢ‬ ‫أن‬ ‫ﻋﻠﻰ‬ ‫اﻟﺴﺎﻟﺒﺔ‬ ‫اﻹﺷﺎرة‬ ‫ﺗﺪل‬ ‫ﺣﯿﺚ‬2‫ﯾﺠﺬب‬
‫وﻓﻘ‬‫ﺎ‬‫ﻟ‬‫و‬ ،‫اﻟﺜﺎﻟﺚ‬ ‫ﻧﯿﻮﺗﻦ‬ ‫ﻘﺎﻧﻮن‬‫ھﺬ‬‫ه‬‫اﻟﻘﻮة‬‫ﯾﻘﺎﺑﻠﮭﺎ‬‫اﻟﺠﺴﻢ‬
‫اﻟﻘﻮة‬ ‫ﺗﻌﯿﯿﻦ‬ ‫ﻋﻨﺪ‬F21‫ﻣﺴ‬ ‫ﻧﺠﺪھﺎ‬‫ﺎ‬‫و‬‫ﻟﻠﻘﻮة‬ ‫ﯾﺔ‬F12
‫و‬‫اﻟﻔﻌﻞ‬ ‫رد‬.‫وﻟﻠﻘﻮة‬F21-=F12‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬ ‫اﻟﺬﻛﺮ‬ ‫ﺗﺴﺘﺤﻖ‬ ‫ﻋﺪﯾﺪة‬ ‫ﻣﯿﺰات‬
‫ﺑﯿﻦ‬ ‫داﺋﻤﺎ‬ ‫اﻟﻤﻮﺟﻮد‬‫ﻛﺘﻠﺘﯿﻦ‬ ‫أي‬‫ﺑﯿﻨﮭﻤﺎ‬ ‫اﻟﻔﺎﺻﻞ‬ ‫اﻟﻮﺳﻂ‬ ‫ﻋﻦ‬ ‫اﻟﻨﻈﺮ‬ ‫ﺑﻐﺾ‬ ،
‫اﻷﺟﺴﺎم‬‫وﺗﺘﻨﺎﻗﺺ‬‫اﻟﻤﺴﺎﻓﺔ‬ ‫زﯾﺎدة‬ ‫ﻣﻊ‬ ‫ﻣﻠﺤﻮظ‬ ‫ﺑﺸﻜﻞ‬
‫إن‬ ‫ﻛﻤﺎ‬‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬ ‫ﻧﺮاھﺎ‬ ‫أن‬ ‫ﻧﺴﺘﻄﯿﻊ‬ ‫أﺧﺮى‬ ‫ھﺎﻣﺔ‬ ‫ﻧﻘﻄﺔ‬ ‫ھﻨﺎك‬
‫ﻟ‬ ‫ﻛﻤﺎ‬ ‫ﻧﻔﺴﮭﺎ‬ ‫ھﻲ‬ ‫اﻟﺘﻮزﯾﻊ‬ ‫ﺧﺎرج‬ ‫ﺟﺴﻢ‬ ‫ﻋﻠﻰ‬ ‫وﺗﺆﺛﺮ‬ ‫اﻟﺤﺠﻢ‬ ‫وﻣﺤﺪودة‬ ‫ﻣﺘﺠﺎﻧﺴﺔ‬
‫اﻟﻤﺜﺎل‬‫ﻛﺘﻠﺘﮫ‬ ‫ﺟﺴﻢ‬ ‫ﻋﻠﻰ‬ ‫اﻷرض‬ ‫ﺑﮭﺎ‬ ‫ﺗﺆﺛﺮ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻘﻮة‬ ‫ﻣﻘﺪار‬ ‫إن‬ ،

8
،‫ﻗﻄﺮھﺎ‬ ‫ﻧﺼﻒ‬‫و‬‫ﻣﻮﺟﮭﺔ‬ ‫اﻟﻘﻮة‬ ‫ھﺬه‬‫ﻧﺤﻮ‬‫اﻷرض‬ ‫ﻣﺮﻛﺰ‬.
‫ﻗﺎﻧﻮن‬ ‫ﺻﯿﺎﻏﺘﮫ‬ ‫ﻓﻲ‬ ‫ﻧﯿﻮﺗﻦ‬ ‫اﺳﺘﺨﺪم‬‫اﻟﻌﺎم‬ ‫اﻟﺠﺬب‬‫ﻣﺮﺑﻊ‬ ‫ﻣﻊ‬ ‫ﻋﻜﺴﯿﺎ‬ ‫ﺗﺘﻨﺎﺳﺐ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫أن‬ ‫ﯾﻔﺘﺮض‬ ‫اﻟﺬي‬ ‫اﻟﻤﻨﻄﻖ‬
‫اﻟﻤﺘﻔﺎﻋﻠﯿﻦ‬ ‫اﻟﺠﺴﻤﯿﻦ‬ ‫ﺑﯿﻦ‬ ‫ﻟﻤﺴﺎﻓﺔ‬‫ﻗﺎرن‬ ‫ﺣﯿﺚ‬ ،‫ﻋﺠﻠﺔ‬‫ﻣﺪاره‬ ‫ﻓﻲ‬ ‫اﻟﻘﻤﺮ‬‫ﺑﻌﺠﻠﺔ‬‫ﺳﻄﺢ‬ ‫ﻣﻦ‬ ‫ﺑﺎﻟﻘﺮب‬ ‫ﯾﺴﻘﻂ‬ ‫ﺟﺴﻢ‬
‫اﻷﺳﻄﻮرﯾﺔ‬ ‫اﻟﺘﻔﺎﺣﺔ‬ ‫ﻣﺜﻞ‬ ،)‫ﺷﻜﻞ‬13.2(.
‫اﻟﺘﺮﺑﯿﻊ‬ ‫ﻗﺎﻧﻮن‬ ‫ﻧﯿﻮﺗﻦ‬ ‫اﺳﺘﺨﺪم‬ ،‫اﻷرض‬ ‫ﺟﺎذﺑﯿﺔ‬ ‫ﺟﺬب‬ ‫ﯾﻌﻨﻲ‬ ‫اﻟﺴﺒﺐ‬ ‫ﻧﻔﺲ‬ ‫ﻟﮭﻤﺎ‬ ‫ﻛﺎن‬
‫ﻧﺤﻮ‬‫اﻷرض‬)‫ﻣﺮﻛﺰي‬ ‫ﺗﺴﺎرع‬(‫ﻣﻊ‬ ‫ﻣﺘﻨﺎﺳﺒﺎ‬ ‫ﯾﻜﻮن‬ ‫أن‬ ‫ﯾﺠﺐ‬ ‫أي‬ ،1/rM
2
،‫ﺣﯿﺚ‬
‫اﻷرض‬ ‫ﻣﺮﻛﺰ‬ ‫ﺑﯿﻦ‬ ‫اﻟﻤﺴﺎﻓﺔ‬ ‫ھﻲ‬‫و‬‫اﻟﻘﻤﺮ‬.
‫اﻟﺘﻔﺎﺣﺔ‬ ‫ﺗﺴﺎرع‬ ‫ﯾﻜﻮن‬ ‫أن‬ ‫ﯾﺠﺐ‬ ‫ذﻟﻚ‬ ‫ﻋﻠﻰ‬ ‫ة‬‫ﻧﺤﻮ‬‫ﻣﻊ‬ ‫ﻣﺘﻨﺎﺳﺒﺎ‬ ‫اﻷرض‬1/Ra
2
‫ﺣﯿﺚ‬ ،Ra‫ﺑﯿﻦ‬ ‫اﻟﻤﺴﺎﻓﺔ‬ ‫ھﻲ‬
‫أي‬ ‫اﻷرض‬ ‫ﺳﻄﺢ‬ ‫ﻋﻠﻰ‬ ‫ﺗﻘﻊ‬ ‫اﻟﺘﻔﺎﺣﺔ‬ ‫أن‬ ‫ﺑﻤﺎ‬RaRE‫ﻗﻄﺮ‬ ‫ﻧﺼﻒ‬‫اﺳﺘﺨﺪم‬ ‫اﻷرض‬
RE=6.370x10،‫ﻧﺴﺒﺔ‬ ‫ﺑﺄن‬ ‫ﻧﯿﻮﺗﻦ‬ ‫ﺗﻨﺒﺄ‬ ‫ﻟﻘﺪ‬‫ﻋﺠﻠﺔ‬‫اﻟﻘﻤﺮ‬aM‫إﻟﻰ‬‫ﻋﺠﻠﺔ‬‫اﻟﺘﻔﺎﺣﺔ‬g
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‫ﺣﯿﺚ‬ME،‫اﻷرض‬ ‫ﻛﺘﻠﺔ‬ ‫ھﻲ‬‫و‬RE‫ھﻮ‬،‫ﻗﻄﺮھﺎ‬ ‫ﻧﺼﻒ‬
‫ﻗﺎﻧﻮن‬ ‫ﺻﯿﺎﻏﺘﮫ‬ ‫ﻓﻲ‬ ‫ﻧﯿﻮﺗﻦ‬ ‫اﺳﺘﺨﺪم‬
‫ا‬‫اﻟﻤﺘﻔﺎﻋﻠﯿﻦ‬ ‫اﻟﺠﺴﻤﯿﻦ‬ ‫ﺑﯿﻦ‬ ‫ﻟﻤﺴﺎﻓﺔ‬
‫اﻷرض‬‫اﻷﺳﻄﻮرﯾﺔ‬ ‫اﻟﺘﻔﺎﺣﺔ‬ ‫ﻣﺜﻞ‬ ،
‫ﻛﻼ‬ ‫أن‬ ‫ﻓﺮض‬ ‫ﻋﻠﻰ‬‫اﻟﻌﺠﻠﺘﯿﻦ‬‫اﻟﺘﺮﺑﯿﻊ‬ ‫ﻗﺎﻧﻮن‬ ‫ﻧﯿﻮﺗﻦ‬ ‫اﺳﺘﺨﺪم‬ ،‫اﻷرض‬ ‫ﺟﺎذﺑﯿﺔ‬ ‫ﺟﺬب‬ ‫ﯾﻌﻨﻲ‬ ‫اﻟﺴﺒﺐ‬ ‫ﻧﻔﺲ‬ ‫ﻟﮭﻤﺎ‬ ‫ﻛﺎن‬
‫اﻟﻘﻤﺮ‬ ‫ﻟﺘﺴﺎرع‬ ‫ﻛﺴﺒﺐ‬ ‫اﻟﻌﻜﺴﻲ‬‫ﻧﺤﻮ‬
rM‫اﻷرض‬ ‫ﻣﺮﻛﺰ‬ ‫ﺑﯿﻦ‬ ‫اﻟﻤﺴﺎﻓﺔ‬ ‫ھﻲ‬
‫ﻋﻼ‬‫و‬‫اﻟﺘﻔﺎﺣﺔ‬ ‫ﺗﺴﺎرع‬ ‫ﯾﻜﻮن‬ ‫أن‬ ‫ﯾﺠﺐ‬ ‫ذﻟﻚ‬ ‫ﻋﻠﻰ‬ ‫ة‬
‫و‬ ‫اﻷرض‬ ‫ﻣﺮﻛﺰ‬‫و‬ ،‫اﻟﺘﻔﺎﺣﺔ‬‫أي‬ ‫اﻷرض‬ ‫ﺳﻄﺢ‬ ‫ﻋﻠﻰ‬ ‫ﺗﻘﻊ‬ ‫اﻟﺘﻔﺎﺣﺔ‬ ‫أن‬ ‫ﺑﻤﺎ‬
‫اﻟﻘﯿﻢ‬rM=3.84x108
m‫و‬=6.370x106
m
‫ﺑﺎﻟﻌﻼﻗﺔ‬ ‫ﯾﻌﻄﻰ‬:
‫ﻓ‬ ‫وﻟﺬﻟﻚ‬‫ﺎﻟﻌﺠﻠﺔ‬‫اﻟﻤﺮﻛﺰﯾ‬‫ﺔ‬‫ھ‬ ‫ﻟﻠﻘﻤﺮ‬‫ﻲ‬
‫اﻟﺸﻜﻞ‬2.13‫ﺣﻮل‬ ‫اﻟﻘﻤﺮ‬ ‫دوران‬ ‫إن‬‫اﻷرض‬
‫ﻣﺮﻛﺰي‬ ‫ﺗﺴﺎرع‬ ‫ﯾﻜﺴﺒﮫ‬aM‫اﻷرض‬ ‫ﻧﺤﻮ‬ ‫ﺟﮭﺘﮫ‬.
‫ﺗﻜﺘﺴﺐ‬ ‫ھﻨﺎ‬ ‫ﻛﺎﻟﺘﻔﺎﺣﺔ‬ ‫اﻷرض‬ ‫ﺳﻄﺢ‬ ‫ﻗﺮب‬ ‫ﺟﺴﻢ‬
‫ﺗﺴﺎرﻋﺎ‬g.
‫اﻟﺸﻜﻞ‬
‫ﻣﺮﻛﺰي‬ ‫ﺗﺴﺎرع‬ ‫ﯾﻜﺴﺒﮫ‬
‫ﺗﻜﺘﺴﺐ‬ ‫ھﻨﺎ‬ ‫ﻛﺎﻟﺘﻔﺎﺣﺔ‬ ‫اﻷرض‬ ‫ﺳﻄﺢ‬ ‫ﻗﺮب‬ ‫ﺟﺴﻢ‬
‫ﺗﺴﺎرﻋﺎ‬

9
‫ﻗﯿﻤﺔ‬ ‫وﻣﻌﺮﻓﺔ‬ ‫اﻷرض‬ ‫وﺑﯿﻦ‬ ‫ﺑﯿﻨﮫ‬ ‫اﻟﻤﺴﺎﻓﺔ‬ ‫ﻣﻌﺮﻓﺔ‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬ ‫ﻟﻠﻘﻤﺮ‬‫زﻣﻨﮫ‬‫اﻟﺪوري‬،
.‫و‬‫ﻓﻲ‬‫اﻟﻔﺘﺮة‬ ‫ھﺬه‬ ‫ﺧﻼل‬‫اﻟﺰﻣﻨﯿﺔ‬T‫ﻣﺴﺎﻓﺔ‬ ‫اﻟﻘﻤﺮ‬ ‫ﯾﻘﻄﻊ‬‫ﻣﻘﺪارھﺎ‬2πrM‫وھﻲ‬ ،
‫اﻟﻤﺪارﯾﺔ‬ ‫ﺳﺮﻋﺘﮫ‬ ‫ﻓﺈن‬ ‫ﻟﺬﻟﻚ‬orbital speed‫ھﻲ‬2πrM /T‫و‬‫اﻟﻤﺮﻛﺰﯾﺔ‬ ‫اﻟﻌﺠﻠﺔ‬‫ھ‬‫ﻲ‬
‫ﻋﻠﻰ‬ ‫وﯾﺆﻛﺪ‬ ‫اﻟﻌﺎم‬ ‫اﻟﺠﺬب‬ ‫ﻗﺎﻧﻮن‬ ‫ﺻﺤﺔ‬‫ﻓﻲ‬ ‫اﻟﻌﻜﺴﻲ‬ ‫اﻟﺘﺮﺑﯿﻊ‬ ‫طﺒﯿﻌﺔ‬‫اﻟ‬‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻘﻮة‬‫اﻷﺟﺴﺎم‬ ‫ﺑﯿﻦ‬.
‫ا‬ ‫ﺗﺠﺎه‬ ‫ﺑﺸﺪة‬ ‫ﻗﻠﻘﺎ‬ ‫ﻛﺎن‬ ‫أﻧﮫ‬ ‫إﻻ‬ ،‫ﻟﻨﯿﻮﺗﻦ‬ ‫ﺟﺪا‬ ‫ﻣﺸﺠﻌﺔ‬ ‫ﺗﻜﻮن‬ ‫أن‬ ‫ﯾﺠﺐ‬ ‫اﻟﻨﺘﺎﺋﺞ‬ ‫ھﺬه‬ ‫أن‬ ‫ﻣﻦ‬ ‫اﻟﺮﻏﻢ‬ ‫ﻋﻠﻰ‬‫ﻓﻲ‬ ‫ﺑﮫ‬ ‫أدﻟﻰ‬ ‫اﻟﺬي‬ ‫ﻻﻓﺘﺮاض‬
‫اﻷرض‬ ‫ﺳﻄﺢ‬ ‫ﻋﻠﻰ‬ ‫ﺟﺴﻢ‬ ‫ﺗﺴﺎرع‬ ‫ﻟﺘﻘﺪﯾﺮ‬‫اﻷرض‬ ‫ﻣﻊ‬ ‫ﻧﯿﻮﺗﻦ‬ ‫ﺗﻌﺎﻣﻞ‬ ،‫ﻟ‬ ‫ﻛﻤﺎ‬‫ﻮ‬‫اﻟﻤﺮﻛﺰ‬ ‫ﻓﻲ‬ ‫ﺗﺠﻤﻌﺖ‬ ‫ﻛﺘﻠﺘﮭﺎ‬ ‫ﻛﺎﻣﻞ‬ ‫أن‬.‫إﻧﮫ‬
‫ﻛﺠﺴﯿ‬ ‫اﻟﻤﺴﺘﻄﺎع‬ ‫ﺑﻘﺪر‬ ‫ﺗﺆﺛﺮ‬ ‫اﻷرض‬ ‫أن‬ ‫اﻓﺘﺮض‬‫ﺑﮫ‬ ‫ﺗﻌﻠﻖ‬ ‫ﺧﺎرﺟﻲ‬ ‫ﺟﺴﻢ‬ ‫ﻋﻠﻰ‬ ‫ﯾﺴﯿﻄﺮ‬ ‫ﻢ‬.
،‫واﻟﺘﻜﺎﻣﻞ‬ ‫اﻟﺘﻔﺎﺿﻞ‬ ‫ﺣﺴﺎب‬ ‫ﺗﻄﻮﯾﺮ‬ ‫ﻓﻲ‬ ‫اﻟﺮاﺋﺪ‬ ‫ﻋﻤﻠﮫ‬ ‫أﺳﺎس‬ ‫وﻋﻠﻰ‬‫ھﺬا‬ ‫أن‬ ‫ﻧﯿﻮﺗﻦ‬ ‫أﺛﺒﺖ‬
‫ﻟﻘﺎﻧﻮن‬ ‫طﺒﯿﻌﯿﺔ‬ ‫وﻧﺘﯿﺠﺔ‬ ‫ﺻﺤﯿﺤﺎ‬ ‫ﻛﺎن‬ ‫اﻻﻓﺘﺮاض‬‫اﻟﺠﺬ‬‫اﻟﻌﺎم‬ ‫ب‬.
‫ﺟﺴﻢ‬ ‫ﻓﻲ‬ ‫اﻟﻤﺆﺛﺮة‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫أن‬ ‫ﻋﻠﻰ‬ ‫دﻟﯿﻞ‬ ‫ﻟﺪﯾﻨﺎ‬‫طﺮد‬ ‫ﺗﺘﻨﺎﺳﺐ‬‫ﯾ‬،‫اﻷﺟﺴﺎم‬ ‫ﻟﺴﻘﻮط‬ ‫ﻣﻼﺣﻈﺘﻨﺎ‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬ ‫وذﻟﻚ‬ ‫اﻟﻜﺘﻠﺔ‬ ‫ﻣﻊ‬ ‫ﺎ‬)‫ﺗﻤﺖ‬
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‫ﺑﺤﺴﺎب‬ ‫أﯾﻀﺎ‬ ‫ﻧﯿﻮﺗﻦ‬ ‫ﻗﺎم‬‫اﻟﻌﺠﻠﺔ‬‫اﻟﻤﺮﻛﺰﯾ‬‫ﺔ‬‫ﻗﯿﻤﺔ‬ ‫وﻣﻌﺮﻓﺔ‬ ‫اﻷرض‬ ‫وﺑﯿﻦ‬ ‫ﺑﯿﻨﮫ‬ ‫اﻟﻤﺴﺎﻓﺔ‬ ‫ﻣﻌﺮﻓﺔ‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬ ‫ﻟﻠﻘﻤﺮ‬
T=27.32‫ﯾﻮم‬‫ﺗﺴﺎوي‬ ‫أو‬2.36x106
‫ﺛﺎﻧﯿﺔ‬.
‫ﻣﺪاره‬ ‫ﻣﺤﯿﻂ‬ ‫ﺗﻌﺎدل‬.‫و‬‫اﻟﻤﺪارﯾﺔ‬ ‫ﺳﺮﻋﺘﮫ‬ ‫ﻓﺈن‬ ‫ﻟﺬﻟﻚ‬
‫و‬‫ﻋﻠﻰ‬ ‫ﻗﻮﯾﺎ‬ ‫دﻟﯿﻼ‬ ‫ﯾﻌﻄﯿﻨﺎ‬ ‫ھﺬا‬‫ﻋﻠﻰ‬ ‫وﯾﺆﻛﺪ‬ ‫اﻟﻌﺎم‬ ‫اﻟﺠﺬب‬ ‫ﻗﺎﻧﻮن‬ ‫ﺻﺤﺔ‬
‫ا‬ ‫ﺗﺠﺎه‬ ‫ﺑﺸﺪة‬ ‫ﻗﻠﻘﺎ‬ ‫ﻛﺎن‬ ‫أﻧﮫ‬ ‫إﻻ‬ ،‫ﻟﻨﯿﻮﺗﻦ‬ ‫ﺟﺪا‬ ‫ﻣﺸﺠﻌﺔ‬ ‫ﺗﻜﻮن‬ ‫أن‬ ‫ﯾﺠﺐ‬ ‫اﻟﻨﺘﺎﺋﺞ‬ ‫ھﺬه‬ ‫أن‬ ‫ﻣﻦ‬ ‫اﻟﺮﻏﻢ‬ ‫ﻋﻠﻰ‬
‫اﻟﺘﺤﻠﯿﻞ‬.‫اﻷرض‬ ‫ﺳﻄﺢ‬ ‫ﻋﻠﻰ‬ ‫ﺟﺴﻢ‬ ‫ﺗﺴﺎرع‬ ‫ﻟﺘﻘﺪﯾﺮ‬
‫ﻛﺠﺴﯿ‬ ‫اﻟﻤﺴﺘﻄﺎع‬ ‫ﺑﻘﺪر‬ ‫ﺗﺆﺛﺮ‬ ‫اﻷرض‬ ‫أن‬ ‫اﻓﺘﺮض‬
‫ﻋﺎم‬ ‫وﻓﻲ‬ ‫ﺳﻨﻮات‬ ‫ﻋﺪة‬ ‫ﺑﻌﺪ‬1687،‫واﻟﺘﻜﺎﻣﻞ‬ ‫اﻟﺘﻔﺎﺿﻞ‬ ‫ﺣﺴﺎب‬ ‫ﺗﻄﻮﯾﺮ‬ ‫ﻓﻲ‬ ‫اﻟﺮاﺋﺪ‬ ‫ﻋﻤﻠﮫ‬ ‫أﺳﺎس‬ ‫وﻋﻠﻰ‬
‫ﻟﻘﺎﻧﻮن‬ ‫طﺒﯿﻌﯿﺔ‬ ‫وﻧﺘﯿﺠﺔ‬ ‫ﺻﺤﯿﺤﺎ‬ ‫ﻛﺎن‬ ‫اﻻﻓﺘﺮاض‬
‫ﺟﺴﻢ‬ ‫ﻓﻲ‬ ‫اﻟﻤﺆﺛﺮة‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫أن‬ ‫ﻋﻠﻰ‬ ‫دﻟﯿﻞ‬ ‫ﻟﺪﯾﻨﺎ‬
‫اﻟﻔﺼﻞ‬ ‫ﻓﻲ‬ ‫ﻣﻨﺎﻗﺸﺘﮭﺎ‬2.(

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‫وﺑﺈھﻤﺎل‬ ،‫ﻣﻘﺎوﻣﺔ‬‫اﻟﮭﻮاء‬‫اﻟ‬ ‫ﺑﻨﻔﺲ‬ ‫ﺗﺴﻘﻂ‬‫ﻌﺠﻠﺔ‬g‫اﻷرض‬ ‫ﺳﻄﺢ‬ ‫ﻗﺮب‬.‫ﻟﻘﺎﻧﻮن‬ ‫وﻓﻘﺎ‬ ‫وذﻟﻚ‬
g = Fg /m،‫ﺣﯿﺚ‬m،‫اﻟﺴﺎﻗﻂ‬ ‫اﻟﺠﺴﻢ‬ ‫ﻛﺘﻠﺔ‬ ‫ھﻲ‬‫ﻟﻜﻞ‬ ‫ﻧﻔﺴﮭﺎ‬ ‫ھﻲ‬ ‫اﻟﻨﺴﺒﺔ‬ ‫ھﺬه‬ ‫إن‬
‫اﻟﻜﺘﻠﺔ‬ ‫ﻣﻊ‬ ‫ﻣﺒﺎﺷﺮة‬ ‫ﻣﺘﻨﺎﺳﺒﺔ‬ ‫ﺗﻜﻮن‬ ‫أن‬ ‫ﯾﺠﺐ‬m‫اﻟﻨﺴﺒﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻜﺘﻠﺔ‬ ‫ﻧﻠﻐﻰ‬ ‫ﻟﺬﻟﻚ‬ ،.
‫ﺗﻄﺒﯿﻘﮭﺎ‬ ‫ﯾﻤﻜﻨﻨﺎ‬ ‫اﻟﺤﺎﻟﺔ‬ ‫ھﺬه‬ ‫ﻧﻔﺲ‬ ،‫ﻛﻮﻛﺒﯿﻦ‬ ‫ﻣﺜﻞ‬ ،‫اﻟﻜﺘﻠﺔ‬ ‫ﻣﻊ‬ ‫ﺟﺴﻤﯿﻦ‬ ‫أي‬ ‫ﺑﯿﻦ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻟﻘﻮة‬ ‫ﻋﻤﻮﻣﯿﺔ‬ ‫اﻷﻛﺜﺮ‬
.‫ﻛﺎﻧﺖ‬ ‫ﻣﮭﻤﺎ‬ ‫اﻟﻜﺘﻞ‬ ‫ﻣﻦ‬ ‫أي‬ ‫اﺧﺘﯿﺎر‬ ‫ﯾﻤﻜﻨﻨﺎ‬ ‫اﻟﺪﻟﯿﻞ‬ ‫ھﺬا‬ ‫ﻋﻠﻰ‬ ‫وﺑﻨﺎءا‬،‫و‬‫ﻗﻮة‬ ‫ﺑﺎﻟﺘﺎﻟﻲ‬
‫اﻟﻜﺘﻠﺘ‬ ‫ﻛﻼ‬ ‫ﻣﻊ‬ ‫طﺮدا‬ ‫ﻣﺘﻨﺎﺳﺒﺔ‬ ‫ﺗﻜﻮن‬ ‫أن‬ ‫ﯾﺠﺐ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ﻓﻲ‬ ‫ذﻟﻚ‬ ‫رؤﯾﺔ‬ ‫ﯾﻤﻜﻨﻨﺎ‬ ‫ﻛﻤﺎ‬ ،‫ﯿﻦ‬3.13.
‫ﺑﺴﺒﺐ‬ ‫اﻷرض‬ ‫ﻋﻠﻰ‬ ‫ﯾﺴﻘﻂ‬ ‫أن‬ ‫دون‬ ‫رض‬)a(،‫اﻷرض‬ ‫ﺟﺎذﺑﯿﺔ‬ ‫ﺗﺄﺛﯿﺮ‬ ‫ﺧﺎرج‬ ‫اﻧﮫ‬)b(‫ﻣﻊ‬ ‫ﻣﺘﻮازن‬
،‫ﻣﻌﺪوﻣﺔ‬ ‫اﻟﻘﻤﺮ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻘﻮة‬ ‫ﻣﺤﺼﻠﺔ‬)d(،‫ذﻟﻚ‬ ‫ﻣﻦ‬ ‫ﻻﺷﻲء‬)e(‫ﻣﺎ‬ ‫ﻛﻞ‬‫ﺳﺒﻖ‬.
‫ﻗﻄﺮه‬ ‫ﻧﺼﻒ‬ ‫داﺋﺮي‬ ‫ﻣﺪار‬ ‫ﻓﻲ‬ ‫ل‬r‫ﻗﻄﺮه‬ ‫ﻧﺼﻒ‬ ‫داﺋﺮي‬ ‫ﻣﺪار‬ ‫ﻓﻲ‬ ‫اﻟﺜﺎﻧﻲ‬ ‫واﻟﻘﻤﺮ‬ ،
‫ھﻲ‬ ‫اﻟﺜﺎﻧﻲ‬ ‫اﻟﻘﻤﺮ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻜﻮﻛﺐ‬ ‫ﺑﮭﺎ‬ ‫ﯾﺆﺛﺮ‬ ‫اﻟﺘﻲ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫ﻣﻘﺪار‬ ‫إن‬)a(‫اﻟﻘﻤﺮ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻤﺆﺛﺮة‬ ‫اﻟﻘﻮة‬ ‫ﻣﻦ‬ ‫ﻣﺮات‬ ‫أرﺑﻊ‬ ‫أﻛﺒﺮ‬
‫اﻟﻘﻤﺮ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻤﺆﺛﺮة‬ ‫اﻟﻘﻮة‬ ‫ﻣﻦ‬ ‫ﻣﺮﺗﯿﻦ‬ ‫أﻛﺒﺮ‬‫اﻷول‬،)c(‫ﺗﺴﺎوي‬‫اﻟﻘﻤﺮ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻤﺆﺛﺮة‬ ‫اﻟﻘﻮة‬‫اﻷول‬،)d(‫اﻟﻘﻮة‬ ‫ﻧﺼﻒ‬
‫اﻟﻘﻤﺮ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻤﺆﺛﺮة‬ ‫اﻟﻘﻮة‬ ‫رﺑﻊ‬‫اﻷول‬
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‫ﻋﻦ‬ ‫اﻟﻨﻈﺮ‬ ‫ﺑﻐﺾ‬ ‫اﻷﺟﺴﺎم‬ ‫ﺟﻤﯿﻊ‬‫ﻛﺘﻠﺘﮭﺎ‬‫وﺑﺈھﻤﺎل‬ ،
‫و‬ ،‫اﻟﺜﺎﻧﻲ‬ ‫ﻧﯿﻮﺗﻦ‬‫ﺗ‬‫ﻌﻄﻰ‬‫اﻟﻌﺠﻠﺔ‬ ‫ھﺬه‬‫ﺑﺎﻟﻌﻼﻗﺔ‬m
‫ﺣﯿﺚ‬ ‫اﻟﺴﺎﻗﻄﺔ‬ ‫اﻷﺟﺴﺎم‬Fg‫اﻟﻜﺘﻠﺔ‬ ‫ﻣﻊ‬ ‫ﻣﺒﺎﺷﺮة‬ ‫ﻣﺘﻨﺎﺳﺒﺔ‬ ‫ﺗﻜﻮن‬ ‫أن‬ ‫ﯾﺠﺐ‬
‫اﻟﺤﺎﻟﺔ‬ ‫اﻋﺘﺒﺮﻧﺎ‬ ‫ﻣﺎ‬ ‫إذا‬‫ﺗﻄﺒﯿﻘﮭﺎ‬ ‫ﯾﻤﻜﻨﻨﺎ‬ ‫اﻟﺤﺎﻟﺔ‬ ‫ھﺬه‬ ‫ﻧﻔﺲ‬ ،‫ﻛﻮﻛﺒﯿﻦ‬ ‫ﻣﺜﻞ‬ ،‫اﻟﻜﺘﻠﺔ‬ ‫ﻣﻊ‬ ‫ﺟﺴﻤﯿﻦ‬ ‫أي‬ ‫ﺑﯿﻦ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻟﻘﻮة‬ ‫ﻋﻤﻮﻣﯿﺔ‬ ‫اﻷﻛﺜﺮ‬
‫اﻟﻜﺘﻞ‬ ‫أﺣﺪ‬ ‫ﻣﻊ‬ ‫ﺗﺘﻨﺎﺳﺐ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫أن‬ ‫ﻟﻨﺜﺒﺖ‬.
‫اﻟﻜﺘﻠﺘ‬ ‫ﻛﻼ‬ ‫ﻣﻊ‬ ‫طﺮدا‬ ‫ﻣﺘﻨﺎﺳﺒﺔ‬ ‫ﺗﻜﻮن‬ ‫أن‬ ‫ﯾﺠﺐ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬
‫ﻟﻠﺘﻔﻜﻴﺭ‬ ‫ﺴﺅﺍل‬1.13
‫اﻷ‬ ‫ﺣﻮل‬ ‫ﻣﺪاره‬ ‫ﻓﻲ‬ ‫اﻟﻘﻤﺮ‬ ‫ﯾﺒﻘﻰ‬‫ﺑﺴﺒﺐ‬ ‫اﻷرض‬ ‫ﻋﻠﻰ‬ ‫ﯾﺴﻘﻂ‬ ‫أن‬ ‫دون‬ ‫رض‬
،‫اﻷﺧﺮى‬ ‫واﻟﻜﻮاﻛﺐ‬ ‫اﻟﺸﻤﺲ‬ ‫ﺟﺬب‬ ‫ﻗﻮى‬)c(،‫ﻣﻌﺪوﻣﺔ‬ ‫اﻟﻘﻤﺮ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻘﻮة‬ ‫ﻣﺤﺼﻠﺔ‬
‫ﻣ‬ ‫ﻗﻤﺮان‬ ‫ﻟﺪﯾﮫ‬ ‫ﻛﻮﻛﺐ‬‫ﺘﺴﺎوﯾ‬‫اﻟﻜﺘﻠﺔ‬ ‫ﺎ‬‫اﻷ‬ ‫اﻟﻘﻤﺮ‬ ،‫و‬‫ﻗﻄﺮه‬ ‫ﻧﺼﻒ‬ ‫داﺋﺮي‬ ‫ﻣﺪار‬ ‫ﻓﻲ‬ ‫ل‬
2r.‫ھﻲ‬ ‫اﻟﺜﺎﻧﻲ‬ ‫اﻟﻘﻤﺮ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻜﻮﻛﺐ‬ ‫ﺑﮭﺎ‬ ‫ﯾﺆﺛﺮ‬ ‫اﻟﺘﻲ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫ﻣﻘﺪار‬ ‫إن‬
‫اﻷول‬،)b(‫اﻟﻘﻤﺮ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻤﺆﺛﺮة‬ ‫اﻟﻘﻮة‬ ‫ﻣﻦ‬ ‫ﻣﺮﺗﯿﻦ‬ ‫أﻛﺒﺮ‬
‫اﻷ‬ ‫اﻟﻘﻤﺮ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻤﺆﺛﺮة‬‫و‬‫ل‬،)e(‫اﻟﻘﻤﺮ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻤﺆﺛﺮة‬ ‫اﻟﻘﻮة‬ ‫رﺑﻊ‬

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‫ﻁﺎﻭﻟﺔ‬‫ﻗﺎﺌﻡ‬ ‫ﻤﺜﻠﺙ‬ ‫ﺯﻭﺍﻴﺎ‬ ‫ﻓﻲ‬‫ﻜﻤﺎ‬
‫ﺍﻟﻜﺭﺘﻴﻥ‬ ‫ﻋﻥ‬ ‫ﺍﻟﻨﺎﺘﺠﺔ‬m2‫ﻭ‬m3.
‫ﺍﻟﻜﺭﺓ‬ ‫ﻋﻠﻰ‬ ‫ﺍﻷﺨﺭﻴﻴﻥ‬ ‫ﺍﻟﻜﺭﺘﻴﻥ‬ ‫ﻤﻥ‬ ‫ﻜل‬ ‫ﺒﻬﺎ‬ ‫ﺘﺅﺜﺭ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻤﻨﻔﺼل‬ ‫ﺒﺸﻜل‬ ‫ﻨﺤﺴﺏ‬ ‫ﻻ‬m1،‫ﺜﻡ‬
‫ﺍﻟﺒﻴﺎﻨﻲ‬ ‫ﺍﻟﺭﺴﻡ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬ ‫ﻨﺭﻯ‬ ‫ﺃﻥ‬ ‫ﻴﻤﻜﻨﻨﺎ‬
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﻜﻤﺎ‬3.13،‫ﻤﺭﻜﺯ‬ ‫ﻨﻀﻊ‬
‫ﺒـ‬ ‫ﺘﻌﻁﻰ‬ ‫ﻤﺒﺎﺸﺭﺓ‬ ‫ﺍﻷﻋﻠﻰ‬ ‫ﺒﺎﺘﺠﺎﻩ‬:‫اﻟﺸﻜﻞ‬3.13‫اﻟﻜﺘﻠﺔ‬ ‫ﻋﻦ‬ ‫اﻟﻨﺎﺗﺠﺔ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬
m1‫اﻟﻤﺤﺼﻠﺔ‬ ‫ﺷﻌﺎع‬ ‫ﻋﻨﮭﺎ‬ ‫ﯾﻨﻮب‬F21+F31.
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‫ﻤﺜﺎل‬1.13‫ﺍﻟﺒﻠﻴﺎﺭﺩﻭ‬ ‫ﻜﺭﺍﺕ‬
‫ﺒﻠﻴﺎﺭﺩ‬ ‫ﻜﺭﺍﺕ‬ ‫ﺜﻼﺙ‬‫ﻭ‬‫ﻤﻨﻬﺎ‬ ‫ﻜل‬ ‫ﻜﺘﻠﺔ‬0.300‫ﻜﻴﻠ‬‫ﻭ‬‫ﺠ‬‫ﻋﻠﻰ‬ ‫ﻤﻭﻀﻭﻋﺔ‬ ‫ﺭﺍﻡ‬‫ﻁﺎﻭﻟﺔ‬
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬3.13.‫ﺍﻟﻜﺭﺓ‬ ‫ﻋﻠﻰ‬ ‫ﺍﻟﺠﺎﺫﺒﻴﺔ‬ ‫ﻗﻭﺓ‬ ‫ﺃﺤﺴﺏ‬m1‫ﺍﻟﻜﺭﺘﻴﻥ‬ ‫ﻋﻥ‬ ‫ﺍﻟﻨﺎﺘﺠﺔ‬
‫ﺍﻟﺤل‬:‫ﺃﻭ‬‫ﺍﻟﻜﺭﺓ‬ ‫ﻋﻠﻰ‬ ‫ﺍﻷﺨﺭﻴﻴﻥ‬ ‫ﺍﻟﻜﺭﺘﻴﻥ‬ ‫ﻤﻥ‬ ‫ﻜل‬ ‫ﺒﻬﺎ‬ ‫ﺘﺅﺜﺭ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻤﻨﻔﺼل‬ ‫ﺒﺸﻜل‬ ‫ﻨﺤﺴﺏ‬ ‫ﻻ‬
‫ﺍﻟﻤﺤﺼﻠﺔ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻋﻠﻰ‬ ‫ﻟﻨﺤﺼل‬ ‫ﺍﻟﻤﺘﺠﻬﺔ‬ ‫ﺍﻟﻜﻤﻴﺎﺕ‬ ‫ﻤﺠﻤﻭﻉ‬ ‫ﻨﻭﺠﺩ‬،‫ﺍﻟﺒﻴﺎﻨﻲ‬ ‫ﺍﻟﺭﺴﻡ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬ ‫ﻨﺭﻯ‬ ‫ﺃﻥ‬ ‫ﻴﻤﻜﻨﻨﺎ‬
‫ﻭ‬ ‫ﺍﻷﻋﻠﻰ‬ ‫ﺇﻟﻰ‬ ‫ﺘﺸﻴﺭ‬ ‫ﻭﻫﻲ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻫﺫﻩ‬‫ﻨﺤﻭ‬‫ﺍﻟﻴﻤﻴﻥ‬.‫ﻨﺨﺘﺎﺭ‬‫ﺍﻟﻤﺤﺎﻭﺭ‬‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﻜﻤﺎ‬
‫ﺍﻟﻤﺤﺎﻭﺭ‬‫ﺍﻟﻜﺭﺓ‬ ‫ﻓﻲ‬m1
‫ﺍﻟﻜﺘﻠﺔ‬ ‫ﺒﻬﺎ‬ ‫ﺘﺅﺜﺭ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻘﻭﺓ‬m2‫ﺍﻟﻜﺭﺓ‬ ‫ﻋﻠﻰ‬m1‫ﻭ‬‫ﺒـ‬ ‫ﺘﻌﻁﻰ‬ ‫ﻤﺒﺎﺸﺭﺓ‬ ‫ﺍﻷﻋﻠﻰ‬ ‫ﺒﺎﺘﺠﺎﻩ‬

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‫ﺠﺩﺍ‬ ‫ﻀﺌﻴﻠﺔ‬ ‫ﻗﻴﻤﺔ‬ ‫ﻟﻬﺎ‬.
‫ﺒـ‬ ‫ﺘﻌﻁﻰ‬ ‫ﻤﺒﺎﺸﺭﺓ‬ ‫ﺍﻟﻴﻤﻴﻥ‬ ‫ﻭﺒﺎﺘﺠﺎﻩ‬:
‫ﺍﻟﺠﺎﺫﺒﻴﺔ‬ ‫ﻗﻭﺓ‬ ‫ﻤﺤﺼﻠﺔ‬ ‫ﺍﺘﺠﺎﻩ‬ ‫ﻴﻜﻭﻥ‬Ө=29.3°‫ﺍﻟﻤﺤﻭﺭ‬ ‫ﻤﻥ‬ ‫ﺍﻟﺴﺎﻋﺔ‬ ‫ﻋﻘﺎﺭﺏ‬ ‫ﻟﺤﺭﻜﺔ‬ ‫ﻤﻌﺎﻜﺱ‬x.
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‫ﺍﻟﻌﺎﺩﻴﺔ‬ ‫ﺍﻷﺠﺴﺎﻡ‬ ‫ﺒﻴﻥ‬ ‫ﺍﻟﺠﺎﺫﺒﻴﺔ‬ ‫ﻗﻭﺓ‬ ‫ﺃﻥ‬ ‫ﻨﺭﻯ‬ ‫ﺍﻟﻨﺘﻴﺠﺔ‬ ‫ﻫﺫﻩ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬‫ﺠﺩﺍ‬ ‫ﻀﺌﻴﻠﺔ‬ ‫ﻗﻴﻤﺔ‬ ‫ﻟﻬﺎ‬
‫ﺍﻟﻜﺘﻠﺔ‬ ‫ﺒﻬﺎ‬ ‫ﺘﺅﺜﺭ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﺍﻵﻥ‬m3‫ﺍﻟﻜﺘﻠﺔ‬ ‫ﻋﻠﻰ‬m1‫ﺒـ‬ ‫ﺘﻌﻁﻰ‬ ‫ﻤﺒﺎﺸﺭﺓ‬ ‫ﺍﻟﻴﻤﻴﻥ‬ ‫ﻭﺒﺎﺘﺠﺎﻩ‬
‫ﻋﻠﻰ‬ ‫ﺍﻟﻤﺅﺜﺭﺓ‬ ‫ﺍﻟﺠﺎﺫﺒﻴﺔ‬ ‫ﻗﻭﺓ‬ ‫ﻤﺤﺼﻠﺔ‬ ‫ﺇﺫﺍ‬m1‫ﺒﺎﻟﻌﻼﻗﺔ‬ ‫ﺘﻌﻁﻰ‬:
‫ﺍﻟﻘﻭﺓ‬ ‫ﻫﺫﻩ‬ ‫ﻤﻘﺩﺍﺭ‬ ‫ﻭﺒﺎﻟﺘﺎﻟﻲ‬‫ﻫﻭ‬:
‫ﺍﻟﺯ‬ ‫ﻅل‬ ‫ﻤﻥ‬‫ﺍ‬‫ﻭ‬‫ﻴﺔ‬tanӨ = 3.75/6.67 = 0.562‫ﺍﻟﺠﺎﺫﺒﻴﺔ‬ ‫ﻗﻭﺓ‬ ‫ﻤﺤﺼﻠﺔ‬ ‫ﺍﺘﺠﺎﻩ‬ ‫ﻴﻜﻭﻥ‬

13
Measuring the Gravitational Constant
‫ﻗﺎم‬ ‫ھﺎﻣﺔ‬ ‫ﺗﺠﺮﺑﺔ‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬‫ﻛﺎﻓﯿﻨﺪش‬ ‫ھﻨﺮي‬ ‫اﻟﻌﺎﻟﻢ‬ ‫ﺑﮭﺎ‬)1810-1737(‫ﻋﺎم‬ ‫ﻓﻲ‬1798.‫ﺟﮭﺎز‬
‫ﻣﻨﮭﻤﺎ‬ ‫ﻛﻞ‬ ‫ﻛﺘﻠﺔ‬ ‫ﺻﻐﯿﺮﺗﯿﻦ‬m‫ﻣﻌ‬ ‫أﻓﻘﻲ‬ ‫ﻗﻀﯿﺐ‬ ‫طﺮﻓﻲ‬ ‫إﻟﻰ‬ ‫ﺛﺒﺘﺘﺎ‬ ،‫رﻓﯿﻊ‬ ‫ﺑﺨﯿﻂ‬ ‫ﻠﻖ‬‫أو‬‫ﻣﻌﺪﻧﻲ‬ ‫ﺑﺴﻠﻚ‬
‫ﻣﻨﮭﻤﺎ‬ ‫ﻛﻞ‬ ‫ﻛﺘﻠﺔ‬ ‫ﻛﺒﯿﺮﺗﺎن‬ ‫ﻛﺮﺗﺎن‬ ‫ﺗﻮﺿﻊ‬ ‫ﻋﻨﺪﻣﺎ‬M‫واﻷﺻﻐﺮ‬ ‫اﻷﻛﺒﺮ‬ ‫اﻟﻜﺮات‬ ‫ﺑﯿﻦ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫ﻓﺈن‬ ،‫اﻷﺻﻐﺮ‬ ‫اﻟﻜﺮات‬ ‫ﺑﺠﺎﻧﺐ‬
‫ﺟﺪﯾﺪ‬ ‫ﺗﻮازن‬ ‫وﺿﻊ‬ ‫إﻟﻰ‬ ‫اﻟﺘﻌﻠﯿﻖ‬ ‫ﺳﻠﻚ‬.
‫اﻟﻤﺮ‬ ‫ﻋﻦ‬ ‫اﻟﻀﻮﺋﻲ‬ ‫اﻟﺸﻌﺎع‬ ‫اﻧﻌﻜﺎس‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬ ‫اﻟﺪوران‬‫اﻟﺘﻌﻠﯿﻖ‬ ‫ﺳﻠﻚ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻤﺮﺑﻮطﺔ‬ ‫آة‬)‫اﻟﺸﻌﺎع‬ ‫ﯾﻨﺤﺮف‬ ‫ﺣﯿﺚ‬
‫اﻟﻀﻮﺋﻲ‬ ‫اﻟﺸﻌﺎع‬ ‫اﻧﺤﺮاف‬ ‫إن‬‫ھﻮ‬‫ودراﺳﺘﮭﺎ‬ ‫اﻟﺤﺮﻛﺔ‬ ‫ﻟﺘﻀﺨﯿﻢ‬ ‫ﻓﻌﺎل‬ ‫أﺳﻠﻮب‬.‫اﻟﺘﺠﺮﺑﺔ‬ ‫ﺗﻜﺮار‬ ‫ﺗﻢ‬ ‫ﻟﻘﺪ‬
‫ﺣﺎﻻت‬ ‫ﻋﺪة‬ ‫وﻓﻲ‬.‫طﺮد‬ ‫ﺗﺘﻨﺎﺳﺐ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫أن‬ ‫أظﮭﺮت‬ ‫واﻟﻨﺘﯿﺠﺔ‬‫ﯾ‬‫ﻣﻊ‬ ‫ﺎ‬‫اﻟﻜﺘﻠﺘﯿﻦ‬ ‫ﺿﺮب‬ ‫ﺣﺎﺻﻞ‬
‫ﺑﯿﻨﮭﻤﺎ‬.
‫ﯾﻤﺜﻞ‬ ‫اﻟﻤﻨﻘﻂ‬ ‫اﻟﺨﻂ‬
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2.13‫ﺛﺎﺑﺖ‬ ‫ﻗﯿﺎس‬‫اﻟﻌﺎم‬ ‫اﻟﺠﺬب‬Measuring the Gravitational Constant
‫ﺗﻢ‬ ‫ﻟﻘﺪ‬‫ﺛﺎﺑﺖ‬ ‫ﻗﯿﺎس‬‫اﻟﻌﺎم‬ ‫اﻟﺠﺬب‬‫ﻗﺎم‬ ‫ھﺎﻣﺔ‬ ‫ﺗﺠﺮﺑﺔ‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬
‫ﻛﺮﺗﯿﻦ‬ ‫ﻣﻦ‬ ‫ﯾﺘﺄﻟﻒ‬ ‫ﻛﺎﻓﯿﻨﺪش‬‫ﻣﻨﮭﻤﺎ‬ ‫ﻛﻞ‬ ‫ﻛﺘﻠﺔ‬ ‫ﺻﻐﯿﺮﺗﯿﻦ‬
‫رﻗﯿﻖ‬‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫ﻛﻤﺎ‬ ،4.13.
‫ﻣﻨﮭﻤﺎ‬ ‫ﻛﻞ‬ ‫ﻛﺘﻠﺔ‬ ‫ﻛﺒﯿﺮﺗﺎن‬ ‫ﻛﺮﺗﺎن‬ ‫ﺗﻮﺿﻊ‬ ‫ﻋﻨﺪﻣﺎ‬
‫و‬ ‫دوران‬ ‫وﺑﺎﻟﺘﺎﻟﻲ‬ ‫اﻟﻘﻀﯿﺐ‬ ‫دوران‬ ‫ﺗﺴﺒﺐ‬‫ﺑﺮم‬‫ﺟﺪﯾﺪ‬ ‫ﺗﻮازن‬ ‫وﺿﻊ‬ ‫إﻟﻰ‬ ‫اﻟﺘﻌﻠﯿﻖ‬ ‫ﺳﻠﻚ‬
‫ﺗﻘﺎس‬‫زاوﯾﺔ‬‫اﻟﻤﺮ‬ ‫ﻋﻦ‬ ‫اﻟﻀﻮﺋﻲ‬ ‫اﻟﺸﻌﺎع‬ ‫اﻧﻌﻜﺎس‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬ ‫اﻟﺪوران‬
‫اﻟﺴﻠﻚ‬ ‫ﺑﺪوران‬ ‫اﻟﻀﻮﺋﻲ‬.(‫اﻟﻀﻮﺋﻲ‬ ‫اﻟﺸﻌﺎع‬ ‫اﻧﺤﺮاف‬ ‫إن‬
‫وذﻟﻚ‬ ‫ﺑﻌﻨﺎﯾﺔ‬‫ﺑ‬‫ﻣﺨﺘﻠﻔﺔ‬ ‫ﻛﺘﻞ‬ ‫ﺄﺧﺬ‬‫ﺣﺎﻻت‬ ‫ﻋﺪة‬ ‫وﻓﻲ‬
mM‫و‬‫ﻋ‬ ‫ﺗﺘﻨﺎﺳﺐ‬‫ﻜﺴﯿﺎ‬‫اﻟﻤﺴﺎﻓﺔ‬ ‫ﻣﺮﺑﻊ‬ ‫ﻣﻊ‬r‫ﺑﯿﻨﮭﻤﺎ‬
‫ﺷﻜﻞ‬4.13‫ﻟﻘﯿﺎس‬ ‫ﻛﺎﻓﻨﺪش‬ ‫ﺟﮭﺎز‬G.‫ﯾﻤﺜﻞ‬ ‫اﻟﻤﻨﻘﻂ‬ ‫اﻟﺨﻂ‬
‫ﻟﻠﻘﻀﯿﺐ‬ ‫اﻷﺻﻠﻲ‬ ‫اﻟﻤﻮﻗﻊ‬

14
‫اﻟﺠﺎذﺑﯿﺔ‬ ‫وﻗﻮة‬ ‫اﻟﺤﺮ‬ ‫اﻟﺴﻘﻮط‬ ‫ﺗﺴﺎرع‬Free-Fall Acceleration and the
‫اﻷرﺿﯿﺔ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻋﺠﻠﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻜﺘﻠﺔ‬ ‫ﺿﺮب‬ ‫ﺣﺎﺻﻞ‬mg‫ﻛﺘﻠﺘﮫ‬ ‫ﻟﺠﺴﻢ‬ ‫ﻛﻮزن‬m،‫إﻟﻰ‬ ‫أﺷﺮﻧﺎ‬
‫ﻟـ‬ ‫ﻋﻤﻘﺎ‬ ‫أﻛﺜﺮ‬ ‫ﻓﮭﻢ‬ ‫ﻋﻠﻰ‬ ‫اﻟﺤﺼﻮل‬ ‫ﯾﻤﻜﻨﻨﺎ‬ ‫اﻵن‬g.‫ﺳﻘﻮط‬ ‫ﻋﻠﻰ‬ ‫ﺗﻌﻤﻞ‬ ‫اﻟﺘﻲ‬ ‫اﻟﻘﻮة‬ ‫ﻣﻘﺪار‬ ‫أن‬ ‫ﺑﻤﺎ‬
‫ﺑﺎﻟﻌﻼﻗﺔ‬ ‫ﺗﻌﻄﻰ‬ ‫اﻷرض‬ ‫ﺳﻄﺢ‬ ‫ﻣﻦ‬ ‫ﺑﺎﻟﻘﺮب‬ ‫ﺣﺮا‬ ‫ﺳﻘﻮطﺎ‬4.13‫ﯾﻤﻜﻨﻨﺎ‬ ،‫ﻣﺴﺎواة‬mg‫ﻋﻠﻰ‬ ‫ﻟﻨﺤﺼﻞ‬ ‫اﻟﻘﻮة‬ ‫ھﺬه‬ ‫ﻣﻊ‬:
h‫اﻷرض‬ ‫ﺳﻄﺢ‬ ‫ﻋﻦ‬‫أو‬‫ارﺗﻔﺎع‬ ‫ﻋﻠﻰ‬r‫اﻷرض‬ ‫ﻣﺮﻛﺰ‬ ‫ﻋﻦ‬‫ﺣﯿﺚ‬ ،RE + h=r.
‫ھﻲ‬ ‫اﻟﺠﺴﻢ‬ ‫ﻋﻠﻰ‬ ‫ﺗﺆﺛﺮ‬ ‫اﻟﺘﻲ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬:
‫أﯾﻀﺎ‬ ‫ﺗﻌﻄﻰ‬ ‫اﻟﻤﻮﺿﻊ‬ ‫ھﺬا‬ ‫ﻓﻲ‬ ‫اﻟﺠﺴﻢ‬ ‫ﻋﻠﻰ‬ ‫ﺗﺆﺛﺮ‬ ‫اﻟﺘﻲ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬‫ﺑﺎﻟﻌﻼﻗﺔ‬mg=Fg‫ﺣﯿﺚ‬g‫ﺗﺴﺎرع‬ ‫ﻗﯿﻤﺔ‬ ‫ھﻲ‬
‫ﻟـ‬ ‫اﻟﻌﻼﻗﺔ‬ ‫ھﺬه‬ ‫ﺑﺘﻌﻮﯾﺾ‬Fg‫أن‬ ‫ﻧﺠﺪ‬ ‫اﻟﺴﺎﺑﻘﺔ‬ ‫ﺑﺎﻟﻌﻼﻗﺔ‬g‫ھﻲ‬:
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3.13‫اﻟﺠﺎذﺑﯿﺔ‬ ‫وﻗﻮة‬ ‫اﻟﺤﺮ‬ ‫اﻟﺴﻘﻮط‬ ‫ﺗﺴﺎرع‬
Gravitational Force
‫ﺣﺪدﻧﺎ‬ ‫ﻋﻨﺪﻣﺎ‬ ،‫اﻟﺨﺎﻣﺲ‬ ‫اﻟﻔﺼﻞ‬ ‫ﻓﻲ‬‫اﻷرﺿﯿﺔ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻋﺠﻠﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻜﺘﻠﺔ‬ ‫ﺿﺮب‬ ‫ﺣﺎﺻﻞ‬
g‫ﻛﻤﻘﺪار‬‫ﻟﻌﺠﻠﺔ‬‫اﻟﺤﺮ‬ ‫اﻟﺴﻘﻮط‬.‫ﻟـ‬ ‫ﻋﻤﻘﺎ‬ ‫أﻛﺜﺮ‬ ‫ﻓﮭﻢ‬ ‫ﻋﻠﻰ‬ ‫اﻟﺤﺼﻮل‬ ‫ﯾﻤﻜﻨﻨﺎ‬ ‫اﻵن‬
‫ﻛﺘﻠﺘﮫ‬ ‫ﺟﺴﻢ‬m‫ﺑﺎﻟﻌﻼﻗﺔ‬ ‫ﺗﻌﻄﻰ‬ ‫اﻷرض‬ ‫ﺳﻄﺢ‬ ‫ﻣﻦ‬ ‫ﺑﺎﻟﻘﺮب‬ ‫ﺣﺮا‬ ‫ﺳﻘﻮطﺎ‬
‫ﻛﺘﻠﺘﮫ‬ ‫ﺟﺴﻢ‬ ‫ﺑﻔﺮض‬ ‫اﻵن‬m‫ارﺗﻔﺎع‬ ‫ﻋﻠﻰ‬ ‫ﯾﻘﻊ‬h
‫ﻗﻮة‬ ‫ﻣﻘﺪار‬ ‫ﻓﺈن‬‫ھﻲ‬ ‫اﻟﺠﺴﻢ‬ ‫ﻋﻠﻰ‬ ‫ﺗﺆﺛﺮ‬ ‫اﻟﺘﻲ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬
‫ﻣﻘﺪار‬ ‫إن‬‫أﯾﻀﺎ‬ ‫ﺗﻌﻄﻰ‬ ‫اﻟﻤﻮﺿﻊ‬ ‫ھﺬا‬ ‫ﻓﻲ‬ ‫اﻟﺠﺴﻢ‬ ‫ﻋﻠﻰ‬ ‫ﺗﺆﺛﺮ‬ ‫اﻟﺘﻲ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬
‫اﻻرﺗﻔﺎع‬ ‫ﻣﻦ‬ ‫اﻟﺤﺮ‬ ‫اﻟﺴﻘﻮط‬h.‫ﻟـ‬ ‫اﻟﻌﻼﻗﺔ‬ ‫ھﺬه‬ ‫ﺑﺘﻌﻮﯾﺾ‬

15
‫ﻋﻨﺪﻣﺎ‬ ‫أﻧﮫ‬ ‫أي‬ ،r→∞‫اﻟﺼ‬ ‫ﻣﻦ‬ ‫ﯾﻘﺘﺮب‬ ‫اﻟﻮزن‬ ‫ﻓﺈن‬ ،‫ﻔﺮ‬‫اﻟﺠﺴﻢ‬ ‫وزن‬ ‫ﻷن‬ ‫وذﻟﻚ‬ ،‫ھﻮ‬
‫اﻟﻔﻀﺎء‬ ‫ﻣﺮﻛﺒﺔ‬ ‫ﺗﻮﺿﺢ‬ ‫اﻟﺼﻮرة‬ ‫ھﺬه‬Endeavor‫ﻣﺪارھﻤﺎ‬ ‫ﻓﻲ‬ ‫ھﻤﺎ‬ ‫ﺑﯿﻨﻤﺎ‬ ‫ﺣﺮ‬ ‫ﺳﻘﻮط‬ ‫ﺣﺎﻟﺔ‬ ‫ﻓﻲ‬ ‫وﻛﻼھﻤﺎ‬ ‫ھﺎﺑﻞ‬ ‫وﺗﻠﺴﻜﻮب‬
‫ﺟﺪا‬ ‫ﻣﺮﺗﻔﻌﺔ‬ ‫ﺟﺒﻞ‬ ‫ﻗﻤﺔ‬ ‫ﻋﻠﻰ‬ ‫ﺳﻮﺑﺮﻣﺎن‬ ‫ﯾﻘﻒ‬‫و‬‫أﻓﻘﯿﺎ‬ ‫اﻟﺒﺴﺒﻮل‬ ‫ﻛﺮة‬ ‫ﺑﺮﻣﻲ‬ ‫ﯾﻘﻮم‬‫و‬‫اﻟﻜﺮة‬ ‫ﺗﺘﺨﺬ‬ ‫ﺑﺤﯿﺚ‬ ‫ﻛﺒﯿﺮة‬ ‫ﺑﺴﺮﻋﺔ‬
‫ﻓﻲ‬ ‫اﻟﻜﺮة‬ ‫ﺗﻜﻮن‬ ‫ﻋﻨﺪﻣﺎ‬ ،‫اﻷرض‬ ‫ﺣﻮل‬ ‫ﻣﺪارا‬‫ﺗﺴﺎرﻋﮭﺎ‬ ‫ﻓﺈن‬ ‫اﻟﻤﺪار‬) :a(‫ﺑﮭﺎ‬ ‫رﻣﯿﺖ‬ ‫اﻟﺘﻲ‬ ‫اﻟﺴﺮﻋﺔ‬ ‫ﻋﻠﻰ‬ ‫ﯾﺘﻮﻗﻒ‬
،‫اﻷرض‬ ‫ﻋﻠﻰ‬ ‫ﺗﺴﻘﻂ‬ ‫ﻟﻢ‬ ‫اﻟﻜﺮة‬ ‫ﻷن‬ ‫ﺻﻔﺮ‬)c(‫ﻣﻦ‬ ‫ﻗﻠﯿﻼ‬ ‫أﻗﻞ‬9.80 m/s2
،)d(‫ﯾﺴﺎوي‬9.80
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‫وﺑﺎﻟﺘﺎﻟﻲ‬‫ﻗﯿﻤﺔ‬ ‫ﺗﺘﻨﺎﻗﺺ‬ ،g‫اﻻرﺗﻔﺎع‬ ‫ﺑﺰﯾﺎدة‬‫ﻋﻨﺪﻣﺎ‬ ‫أﻧﮫ‬ ‫أي‬ ،
mg.
‫اﻟﻔﻀﺎء‬ ‫ﻣﺮﻛﺒﺔ‬ ‫ﺗﻮﺿﺢ‬ ‫اﻟﺼﻮرة‬ ‫ھﺬه‬
‫اﻷرض‬ ‫ﺣﻮل‬.
‫ﺟﺪا‬ ‫ﻣﺮﺗﻔﻌﺔ‬ ‫ﺟﺒﻞ‬ ‫ﻗﻤﺔ‬ ‫ﻋﻠﻰ‬ ‫ﺳﻮﺑﺮﻣﺎن‬ ‫ﯾﻘﻒ‬
‫ﻓﻲ‬ ‫اﻟﻜﺮة‬ ‫ﺗﻜﻮن‬ ‫ﻋﻨﺪﻣﺎ‬ ،‫اﻷرض‬ ‫ﺣﻮل‬ ‫ﻣﺪارا‬
،‫اﻟﻜﺮة‬)b(‫ھﻮ‬،‫اﻷرض‬ ‫ﻋﻠﻰ‬ ‫ﺗﺴﻘﻂ‬ ‫ﻟﻢ‬ ‫اﻟﻜﺮة‬ ‫ﻷن‬ ‫ﺻﻔﺮ‬
m/s2
.

16
‫ﺇﻨﺸﺎﺌﻬﺎ‬ ‫ﻤﻥ‬ ‫ﺍﻻﻨﺘﻬﺎﺀ‬ ‫ﻋﻨﺩ‬‫ﻭﺯﻨﻬﺎ‬ ‫ﺴﻴﻜﻭﻥ‬ ،‫ﺒﺎﻟﻜﺎﻤل‬)‫ﺍﻷﺭﺽ‬ ‫ﺴﻁﺢ‬ ‫ﻋﻠﻰ‬ ‫ﻤﻘﺎﺴﺎ‬(‫ﻫﻭ‬
‫ﺃﻴﺔ‬ ‫ﻋﻠﻰ‬ ‫ﻓﺈﻨﻨﺎ‬ ،‫ﺍﻷﺭﺽ‬ ‫ﺴﻁﺢ‬ ‫ﻓﻭﻕ‬ ‫ﺍﻟﻔﻀﺎﺌﻴﺔ‬ ‫ﺍﻟﻤﺤﻁﺔ‬ ‫ﺃﻥ‬ ‫ﻭﺒﻤﺎ‬ ،‫ﺍﻟﻤﺤﻁﺔ‬ ‫ﻤﻜﺎﻥ‬ ‫ﻋﻥ‬ ‫ﺍﻟﻨﻅﺭ‬ ‫ﺒﻐﺽ‬ ‫ﺜﺎﺒﺘﺔ‬ ‫ﺍﻟﻜﺘﻠﺔ‬ ‫ﻫﺫﻩ‬
h350 km‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻓﻲ‬6.13‫ﻨﺠﺩ‬
‫اﻟﺠﺪول‬1.13
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‫ﻤﺜﺎل‬2.13‫ﺘﻐﻴﺭ‬g‫ﺍﻻﺭﺘﻔﺎﻉ‬ ‫ﻤﻊ‬h
‫ﺘﻌﻤل‬ ‫ﺍﻟﺩﻭﻟﻴﺔ‬ ‫ﺍﻟﻔﻀﺎﺀ‬ ‫ﻤﺤﻁﺔ‬‫ﺍﺭﺘﻔﺎﻉ‬ ‫ﻋﻠﻰ‬350 km.‫ﺇﻨﺸﺎﺌﻬﺎ‬ ‫ﻤﻥ‬ ‫ﺍﻻﻨﺘﻬﺎﺀ‬ ‫ﻋﻨﺩ‬
4.22×106
N.‫ﻤﺎ‬‫ﻫﻭ‬‫ﺍﻟﻤﺩﺍﺭ؟‬ ‫ﻓﻲ‬ ‫ﺘﻜﻭﻥ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﻭﺯﻨﻬﺎ‬
‫ﺍﻟﺤل‬:‫ﻧﻮﺟﺪ‬‫أوﻻ‬‫اﻷرض‬ ‫ﺳﻄﺢ‬ ‫ﻋﻠﻰ‬ ‫وزﻧﮭﺎ‬ ‫ﻣﻦ‬ ‫اﻟﻔﻀﺎﺋﯿﺔ‬ ‫اﻟﻤﺮﻛﺒﺔ‬ ‫ﻛﺘﻠﺔ‬:
‫ﺃﻴﺔ‬ ‫ﻋﻠﻰ‬ ‫ﻓﺈﻨﻨﺎ‬ ،‫ﺍﻷﺭﺽ‬ ‫ﺴﻁﺢ‬ ‫ﻓﻭﻕ‬ ‫ﺍﻟﻔﻀﺎﺌﻴﺔ‬ ‫ﺍﻟﻤﺤﻁﺔ‬ ‫ﺃﻥ‬ ‫ﻭﺒﻤﺎ‬ ،‫ﺍﻟﻤﺤﻁﺔ‬ ‫ﻤﻜﺎﻥ‬ ‫ﻋﻥ‬ ‫ﺍﻟﻨﻅﺭ‬ ‫ﺒﻐﺽ‬ ‫ﺜﺎﺒﺘﺔ‬ ‫ﺍﻟﻜﺘﻠﺔ‬ ‫ﻫﺫﻩ‬
‫ﺍﻷﺭﺽ‬ ‫ﻋﻠﻰ‬ ‫ﻭﺯﻨﻬﺎ‬ ‫ﻤﻥ‬ ‫ﺃﻗل‬ ‫ﺍﻟﻤﺩﺍﺭ‬ ‫ﻓﻲ‬ ‫ﻭﺯﻨﻬﺎ‬ ‫ﺃﻥ‬ ‫ﻨﺘﻭﻗﻊ‬ ‫ﺤﺎل‬.‫ﺒﺘﻌﻭﻴﺽ‬h
‫ﺃﻥ‬

17
‫ﺍﺭﺘﻔﺎﻉ‬ ‫ﻋﻠﻰ‬ ‫ﻤﺩﺍﺭﻫﺎ‬ ‫ﻓﻲ‬ ‫ﺍﻟﻤﺤﻁﺔ‬ ‫ﻭﺯﻥ‬ ‫ﺃﻥ‬ ‫ﻨﺘﻭﻗﻊ‬ ‫ﺒﺎﻟﺘﺎﻟﻲ‬ ‫ﺍﻷﺭﺽ‬ ‫ﺴﻁﺢ‬ ‫ﻋﻠﻰ‬350 km‫ﻫﻭ‬90%‫ﻭﺯﻨﻬﺎ‬ ‫ﻗﻴﻤﺔ‬ ‫ﻤﻥ‬
،‫ﺍﻷﺭﺽ‬ ‫ﺴﻁﺢ‬ ‫ﻋﻠﻰ‬‫ﺃﻭ‬‫ﻟﻸﺭﺽ‬ ‫ﺍﻟﻭﺴﻁﻴﺔ‬ ‫ﺍﻟﻜﺜﺎﻓﺔ‬ ‫ﺠﺩ‬.
‫ﻗﯿﻤﺔ‬ ‫ﻣﻦ‬ ‫اﻷرض‬ ‫ﻛﺘﻠﺔ‬ ‫إﻟﻰ‬ ‫ﻧﺼﻞ‬ ‫أن‬ ‫ﻧﺴﺘﻄﯿﻊ‬ ،g:
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‫ﺍﻟﻘﻴﻤﺔ‬ ‫ﻫﺫﻩ‬‫ﺘﺴﺎﻭﻱ‬90%‫ﻗﻴﻤﺔ‬ ‫ﻤﻥ‬g‫ﺍﺭﺘﻔﺎﻉ‬ ‫ﻋﻠﻰ‬ ‫ﻤﺩﺍﺭﻫﺎ‬ ‫ﻓﻲ‬ ‫ﺍﻟﻤﺤﻁﺔ‬ ‫ﻭﺯﻥ‬ ‫ﺃﻥ‬ ‫ﻨﺘﻭﻗﻊ‬ ‫ﺒﺎﻟﺘﺎﻟﻲ‬ ‫ﺍﻷﺭﺽ‬ ‫ﺴﻁﺢ‬ ‫ﻋﻠﻰ‬
‫ﻋﻠﻰ‬‫ﺍﻷﺭﺽ‬ ‫ﺴﻁﺢ‬.
‫ﻗﻴﻤﺔ‬ ‫ﻨﺴﺘﺨﺩﻡ‬g‫ﺍﻟﻤﺩﺍﺭ‬ ‫ﻓﻲ‬ ‫ﺍﻟﻤﺤﻁﺔ‬ ‫ﻭﺯﻥ‬ ‫ﻓﻴﻜﻭﻥ‬ ،‫ﺍﻟﻤﺤﻁﺔ‬ ‫ﻤﻭﻗﻊ‬ ‫ﻓﻲ‬‫ﻫﻭ‬:
‫ﻤﺜﺎل‬3.13‫ﺍﻷﺭﺽ‬ ‫ﻜﺜﺎﻓﺔ‬
‫ﻟﻸﺭﺽ‬ ‫ﺍﻟﻤﻌﺭﻭﻑ‬ ‫ﺍﻟﻘﻁﺭ‬ ‫ﻨﺼﻑ‬ ‫ﺒﺎﺴﺘﺨﺩﺍﻡ‬‫ﻭ‬‫ﻗﻴﻤﺔ‬g= 9.80 m/s2
،‫ﺍﻷﺭﺽ‬ ‫ﺴﻁﺢ‬ ‫ﻋﻠﻰ‬
‫ﺍﻟﺤل‬:‫اﻟﻌﻼﻗﺔ‬ ‫ﻣﻦ‬ ‫ﻧﻌﻠﻢ‬ ‫ﻧﺤﻦ‬1.1‫ﺑﺄن‬‫اﻟﻮﺳﻂ‬ ‫ﻛﺜﺎﻓﺔ‬‫ھﻲ‬:
‫ﺣﯿﺚ‬ME‫اﻷرض‬ ‫ﻛﺘﻠﺔ‬ ‫ھﻲ‬،‫و‬VE‫ھﻮ‬‫ﺣﺠﻤﮭﺎ‬.‫اﻟﻌﻼﻗﺔ‬ ‫ﻣﻦ‬5.13‫ﻗﯿﻤﺔ‬ ‫ﻣﻦ‬ ‫اﻷرض‬ ‫ﻛﺘﻠﺔ‬ ‫إﻟﻰ‬ ‫ﻧﺼﻞ‬ ‫أن‬ ‫ﻧﺴﺘﻄﯿﻊ‬ ،

18
‫ھﻲ‬ ‫اﻷرض‬ ‫ﺳﻄﺢ‬ ‫ﻋﻠﻰ‬ ‫ﺮاﻧﯿﺖ‬2.75×103
kg/m3
‫داﺧﻞ‬ ‫اﻟﻤﺤﺴﻮﺑﺔ‬ ‫اﻟﻤﺎدة‬ ‫ﻛﺜﺎﻓﺔ‬ ‫ﻣﻊ‬ ‫ﺗﺘﻮاﻓﻖ‬ ‫ﻛﯿﻒ‬ ،
‫ﻣﻦ‬ ‫ﺑﻜﺜﯿﺮ‬ ‫أﻋﻠﻰ‬ ‫ﻛﺜﺎﻓﺔ‬ ‫ﻟﮭﺎ‬ ‫ﻟﻸرض‬ ‫اﻷﺳﺎﺳﯿﺔ‬ ‫اﻟﺪاﺧﻠﯿﺔ‬ ‫اﻟﺒﻨﯿﺔ‬ ‫أن‬ ‫ﻧﺴﺘﻨﺘﺞ‬ ،‫ﻛﺎﻣﻠﺔ‬ ‫اﻷرض‬ ‫ﻟﻜﺜﺎﻓﺔ‬ ‫ﻛﻤﺘﻮﺳﻂ‬ ‫ﻣﺤﺴﻮﺑﺔ‬ ‫وھﻲ‬ ‫وﺳﻄﯿﺔ‬ ‫اﻟﻘﯿﻤﺔ‬ ‫ھﺬه‬ ‫ﻷن‬
‫ﺳﻄﺢ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻌﻤﻞ‬ ‫ﯾﻤﻜﻨﮭﺎ‬ ‫واﻟﺘﻲ‬‫طﺎوﻟﺔ‬.‫اﻟﺒﺴﯿﻄﺔ‬ ‫اﻟﺤﺮ‬ ‫اﻟﺴﻘﻮط‬ ‫ﻣﻘﺎﯾﯿﺲ‬ ‫ﻣﻊ‬ ‫اﻧﺪﻣﺠﺖ‬‫ﻟﺘﺰودﻧﺎ‬
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‫ﻋﻠﻰ‬ ‫ﻓﻨﺤﺼﻞ‬ ‫اﻟﻜﺜﺎﻓﺔ‬ ‫ﺑﻌﻼﻗﺔ‬ ‫اﻟﻘﯿﻤﺔ‬ ‫ھﺬه‬ ‫ﻧﻌﻮض‬:
‫ﻟﻮ؟‬ ‫ﻣﺎذا‬‫ﻟ‬ ‫ﻣﺎذا‬‫ﻮ‬‫ﻟﻠ‬ ‫اﻟﻨﻤﻮذﺟﯿﺔ‬ ‫اﻟﻜﺜﺎﻓﺔ‬ ‫ﺑﺄن‬ ‫ﻟﻚ‬ ‫ﻗﯿﻞ‬‫ﺠ‬‫ھﻲ‬ ‫اﻷرض‬ ‫ﺳﻄﺢ‬ ‫ﻋﻠﻰ‬ ‫ﺮاﻧﯿﺖ‬
‫اﻷرض؟‬
‫اﻟﺠﻮاب‬:‫ﻣﻦ‬ ‫ﺑﻜﺜﯿﺮ‬ ‫أﻋﻠﻰ‬ ‫ﻛﺜﺎﻓﺔ‬ ‫ﻟﮭﺎ‬ ‫ﻟﻸرض‬ ‫اﻷﺳﺎﺳﯿﺔ‬ ‫اﻟﺪاﺧﻠﯿﺔ‬ ‫اﻟﺒﻨﯿﺔ‬ ‫أن‬ ‫ﻧﺴﺘﻨﺘﺞ‬ ،‫ﻛﺎﻣﻠﺔ‬ ‫اﻷرض‬ ‫ﻟﻜﺜﺎﻓﺔ‬ ‫ﻛﻤﺘﻮﺳﻂ‬ ‫ﻣﺤﺴﻮﺑﺔ‬ ‫وھﻲ‬ ‫وﺳﻄﯿﺔ‬ ‫اﻟﻘﯿﻤﺔ‬ ‫ھﺬه‬ ‫ﻷن‬
‫اﻟﻘﯿﻤﺔ‬ ‫ﻣﺘﻮﺳﻂ‬.‫ﺣﺪدت‬ ‫اﻟﺘﻲ‬ ‫ﻛﺎﻓﯿﻨﺪش‬ ‫ﺗﺠﺮﺑﺔ‬ ‫أن‬ ‫اﻟﻤﺪھﺶ‬ ‫وﻣﻦ‬G‫ﺳﻄﺢ‬ ‫ﻋﻠﻰ‬ ‫اﻟﻌﻤﻞ‬ ‫ﯾﻤﻜﻨﮭﺎ‬ ‫واﻟﺘﻲ‬
‫اﻷرض‬ ‫ﺑﺎطﻦ‬ ‫ﻋﻦ‬ ‫ﺑﻤﻌﻠﻮﻣﺎت‬.

19
Kepler’s Laws and the Motion of Planets
‫اﻟﺴﻤ‬ ‫واﻷﺟﺴﺎم‬ ‫واﻟﻨﺠﻮم‬ ‫اﻟﻜﻮاﻛﺐ‬ ‫ﺣﺮﻛﺎت‬ ‫اﻟﺴﻨﯿﻦ‬ ‫آﻻف‬ ‫ﻣﻨﺬ‬ ‫اﻟﻨﺎس‬ ‫ﻻﺣﻆ‬ ‫ﻟﻘﺪ‬‫ﺄو‬‫اﻷﺧﺮى‬ ‫ﯾﺔ‬.‫و‬‫اﻟﺘﺎرﯾﺦ‬ ‫ﻣﻦ‬ ‫ﻣﺒﻜﺮ‬ ‫وﻗﺖ‬ ‫ﻓﻲ‬،
‫ھﻮ‬‫ﻣﺎ‬‫اﻟﻔﻠﻚ‬ ‫ﻋﺎﻟﻢ‬ ‫رﺳﻤﯿﺎ‬ ‫واﻋﺘﻤﺪه‬ ‫درﺳﮫ‬ ‫ﺣﯿﺚ‬ ‫اﻷرض‬ ‫ﻣﺮﻛﺰﯾﺔ‬ ‫ﻧﻤﻮذج‬ ‫ﯾﺴﻤﻰ‬
Claudius Ptolemy(c. 100–c. 170)‫ﻣﺪة‬ ‫ﻣﻌﺘﻤﺪا‬ ‫ﺑﻘﻲ‬ ‫وﻗﺪ‬ ‫اﻟﻤﯿﻼدي‬ ‫اﻟﺜﺎﻧﻲ‬ ‫اﻟﻘﺮن‬ ‫ﻓﻲ‬
‫اﻟﺒﻮﻟﻨﺪي‬ ‫اﻟﻔﻠﻜﻲ‬ ‫اﻟﻌﺎﻟﻢ‬ ‫ﺟﺎء‬‫ﻛﻮﺑﺮﻧﯿﻜﻮس‬ ‫ﻧﯿﻜﻮﻻس‬Nicolaus Copernicus 1473–1543
‫داﺋﺮﯾﺔ‬ ‫ﻣﺪارات‬ ‫ﻓﻲ‬ ‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫ﺗﺪور‬ ‫اﻷﺧﺮى‬ ‫واﻟﻜﻮاﻛﺐ‬ ‫اﻷرض‬ ‫أن‬ ‫واﻗﺘﺮح‬)‫اﻟﺸﻤﺲ‬ ‫ﻣﺮﻛﺰﯾﺔ‬ ‫ﻧﻤﻮذج‬(.
Tycho Brahe(1546–1601)‫وﺿﻊ‬ ‫وھﻜﺬا‬ ‫اﻟﺴﻤﺎء‬ ‫ﺑﻨﺎء‬ ‫ﻛﯿﻔﯿﺔ‬ ‫ﯾﺤﺪد‬ ‫أن‬ ‫وأراد‬
،‫واﻟﻜﻮاﻛﺐ‬ ‫اﻟﻨﺠﻮم‬ ‫ﻣﻦ‬ ‫ﻛﻞ‬ ‫ﻣﻮاﻗﻊ‬ ‫ﻟﺘﺤﺪﯾﺪ‬ ‫ﺑﺮﻧﺎﻣﺠﺎ‬‫اﻟﻜﻮاﻛﺐ‬ ‫ﻣﻦ‬ ‫اﻟﻌﺪﯾﺪ‬ ‫ورﺻﺪ‬ ‫ﻣﺸﺎھﺪة‬ ‫ﺗﻢ‬ ‫أﻧﮫ‬ ‫ﻣﻼﺣﻈﺔ‬ ‫ﻟﻼھﺘﻤﺎم‬ ‫اﻟﻤﺜﯿﺮ‬ ‫وﻣﻦ‬
‫ﻛﺒﯿﺮة‬ ‫ﺳﺪﺳﯿﺔ‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬ ‫ﻓﻘﻂ‬ ‫وذﻟﻚ‬ ‫اﻟﻤﺠﺮدة‬ ‫ﺑﺎﻟﻌﯿﻦ‬ ‫اﻟﻤﺮﺋﯿﺔ‬ ‫اﻟﻨﺠﻮم‬ ‫ﻣﻦ‬)‫آﻟﺔ‬‫اﻻرﺗﻔﺎع‬ ‫ﻟﻘﯿﺎس‬‫أو‬‫ﻟ‬‫اﻻﺗﺠﺎه‬ ‫ﺘﺤﺪﯾﺪ‬(
‫اﻟﻮﻗﺖ‬ ‫ذاك‬ ‫اﺧﺘﺮع‬ ‫ﻗﺪ‬ ‫ﯾﻜﻦ‬ ‫ﻟﻢ‬ ‫اﻟﺘﻠﺴﻜﻮب‬ ‫أن‬ ‫ﻣﻦ‬ ‫اﻟﺮﻏﻢ‬ ‫ﻋﻠﻰ‬.(
‫وﺗﺠﺎرﺑﮫ‬ ‫اﻟﻔﻠﻜﻲ‬ ‫ﻣﻌﻠﻤﮫ‬ ‫ﻣﻌﻠﻮﻣﺎت‬ ‫اﻛﺘﺴﺐ‬ ‫وھﻜﺬا‬ ،‫وﻓﺎﺗﮫ‬ ‫ﻗﺒﻞ‬ ‫ﻗﺼﯿﺮة‬ ‫ﻟﻔﺘﺮة‬ ‫ﻟﺒﺮاھﻲ‬ ‫ﻣﺴﺎﻋﺪا‬ ‫ﻋﻤﻞ‬ ‫ﻛﺒﻠﺮ‬ ‫ﺟﻮھﺎﻧﺲ‬ ‫اﻷﻟﻤﺎﻧﻲ‬ ‫اﻟﻔﻠﻜﻲ‬
‫اﻟﻜﻮاﻛﺐ‬ ‫ﻟﺤﺮﻛﺔ‬ ‫اﻟﺮﯾﺎﺿﻲ‬ ‫اﻟﻨﻤﻮذج‬.‫اﻷرض‬ ‫ﻷن‬ ‫ﺗﺮﺗﯿﺒﮭﺎ‬ ‫اﻟﺼﻌﺐ‬ ‫ﻣﻦ‬ ‫اﻟﻤﻌﻠﻮﻣﺎت‬ ‫ھﺬه‬ ‫ﻣﺜﻞ‬‫أﯾﻀﺎ‬
‫اﻟ‬ ‫ﻣﺴﺎر‬ ‫ﻋﻦ‬ ‫ﺑﺮاھﻲ‬ ‫ﻣﻌﻠﻮﻣﺎت‬ ‫وﻣﻦ‬ ،‫ﺷﺎﻗﺔ‬‫ﻛﺒﻠﺮ‬ ‫وﺟﺪ‬ ‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫ﻤﺮﯾﺦ‬‫اﻹﺟﺎﺑﺔ‬.‫اﻟﻜﺎﻣﻞ‬ ‫ﻛﺒﻠﺮ‬ ‫ﺗﺤﻠﯿﻞ‬ ‫إن‬
‫ﻛﺒﻠﺮ‬ ‫ﺑﻘﻮاﻧﯿﻦ‬ ‫ﺗﻌﺮف‬ ،‫ﻗﻮاﻧﯿﻦ‬ ‫ﺛﻼث‬ ‫ﻓﻲ‬ ‫ﯾﻠﺨﺺ‬ ‫اﻟﻜﻮﻛﺒﯿﺔ‬ ‫ﻟﻠﺤﺮﻛﺔ‬:
‫ﺑﺆرﺗﯿﮫ‬ ‫إﺣﺪى‬ ‫ﻓﻲ‬ ‫ﺗﻘﻊ‬ ‫واﻟﺸﻤﺲ‬ ‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫اھﻠﯿﻠﯿﺠﯿﺔ‬ ‫ﻣﺪارات‬ ‫ﻓﻲ‬ ‫ﺗﺘﺤﺮك‬ ‫اﻟﻜﻮاﻛﺐ‬ ‫ﻛﻞ‬.
‫ﻣ‬ ‫ﻣﺴﺎﺣﺎت‬ ‫ﯾﻤﺴﺢ‬ ‫واﻟﺸﻤﺲ‬ ‫اﻟﻜﻮﻛﺐ‬ ‫ﺑﯿﻦ‬ ‫اﻟﻮاﺻﻞ‬ ‫اﻟﺨﻂ‬‫ﺘﺴﺎوﯾ‬‫ﻣ‬ ‫أزﻣﻨﺔ‬ ‫ﻓﻲ‬ ‫ﺔ‬‫ﺘﺴﺎوﯾ‬‫ﺔ‬.
‫اﻻھﻠﯿﻠﺠﻲ‬ ‫ﻟﻠﻤﺪار‬ ‫اﻟﻜﺒﯿﺮ‬ ‫اﻟﻤﺤﻮر‬ ‫ﻧﺼﻒ‬ ‫ﻣﻜﻌﺐ‬ ‫ﻣﻊ‬ ‫ﯾﺘﻨﺎﺳﺐ‬ ‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫ﻛﻮﻛﺐ‬ ‫ﻷي‬ ‫اﻟﺪوري‬ ‫اﻟﺰﻣﻦ‬ ‫ﻣﺮﺑﻊ‬.
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13.4‫اﻟﻜﻮاﻛﺐ‬ ‫وﺣﺮﻛﺔ‬ ‫ﻛﺒﻠﺮ‬ ‫ﻗﻮاﻧﯿﻦ‬Kepler’s Laws and the Motion of Planets
‫اﻟﺴﻤ‬ ‫واﻷﺟﺴﺎم‬ ‫واﻟﻨﺠﻮم‬ ‫اﻟﻜﻮاﻛﺐ‬ ‫ﺣﺮﻛﺎت‬ ‫اﻟﺴﻨﯿﻦ‬ ‫آﻻف‬ ‫ﻣﻨﺬ‬ ‫اﻟﻨﺎس‬ ‫ﻻﺣﻆ‬ ‫ﻟﻘﺪ‬
‫و‬ ‫اﻟﻜﻮن‬ ‫ﻣﺮﻛﺰ‬ ‫اﻷرض‬ ‫أن‬ ‫اﻟﻌﻠﻤﺎء‬ ‫اﻋﺘﺒﺮ‬‫ھﻮ‬
‫ﻛﻼ‬ ‫اﻹﻏﺮﯾﻘﻲ‬‫و‬‫ﺑﻄﻠﯿﻤﻮس‬ ‫دﯾﻮس‬Claudius Ptolemy
1400‫ﺳﻨﺔ‬.‫ﻋﺎم‬ ‫ﻓﻲ‬1543‫اﻟﺒﻮﻟﻨﺪي‬ ‫اﻟﻔﻠﻜﻲ‬ ‫اﻟﻌﺎﻟﻢ‬ ‫ﺟﺎء‬
‫داﺋﺮﯾﺔ‬ ‫ﻣﺪارات‬ ‫ﻓﻲ‬ ‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫ﺗﺪور‬ ‫اﻷﺧﺮى‬ ‫واﻟﻜﻮاﻛﺐ‬ ‫اﻷرض‬ ‫أن‬ ‫واﻗﺘﺮح‬
‫اﻟﺪﻧﻤﺎرﻛﻲ‬ ‫اﻟﻔﻠﻜﻲ‬ ‫ﺟﺎء‬ ‫ﺑﻌﺪھﺎ‬‫ﺗﺎﯾﻜ‬‫ﻮ‬‫ﺑﺮاھﻲ‬Tycho Brahe
،‫واﻟﻜﻮاﻛﺐ‬ ‫اﻟﻨﺠﻮم‬ ‫ﻣﻦ‬ ‫ﻛﻞ‬ ‫ﻣﻮاﻗﻊ‬ ‫ﻟﺘﺤﺪﯾﺪ‬ ‫ﺑﺮﻧﺎﻣﺠﺎ‬
‫و‬777‫ﻛﺒﯿﺮة‬ ‫ﺳﺪﺳﯿﺔ‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬ ‫ﻓﻘﻂ‬ ‫وذﻟﻚ‬ ‫اﻟﻤﺠﺮدة‬ ‫ﺑﺎﻟﻌﯿﻦ‬ ‫اﻟﻤﺮﺋﯿﺔ‬ ‫اﻟﻨﺠﻮم‬ ‫ﻣﻦ‬
‫وﺑﻮﺻﻠﺔ‬) .‫اﻟﻮﻗﺖ‬ ‫ذاك‬ ‫اﺧﺘﺮع‬ ‫ﻗﺪ‬ ‫ﯾﻜﻦ‬ ‫ﻟﻢ‬ ‫اﻟﺘﻠﺴﻜﻮب‬ ‫أن‬ ‫ﻣﻦ‬ ‫اﻟﺮﻏﻢ‬ ‫ﻋﻠﻰ‬
‫وﺗﺠﺎرﺑﮫ‬ ‫اﻟﻔﻠﻜﻲ‬ ‫ﻣﻌﻠﻤﮫ‬ ‫ﻣﻌﻠﻮﻣﺎت‬ ‫اﻛﺘﺴﺐ‬ ‫وھﻜﺬا‬ ،‫وﻓﺎﺗﮫ‬ ‫ﻗﺒﻞ‬ ‫ﻗﺼﯿﺮة‬ ‫ﻟﻔﺘﺮة‬ ‫ﻟﺒﺮاھﻲ‬ ‫ﻣﺴﺎﻋﺪا‬ ‫ﻋﻤﻞ‬ ‫ﻛﺒﻠﺮ‬ ‫ﺟﻮھﺎﻧﺲ‬ ‫اﻷﻟﻤﺎﻧﻲ‬ ‫اﻟﻔﻠﻜﻲ‬
‫ﻟﻤﺪة‬16‫ﻋﺎﻣﺎ‬‫ﻻﺳﺘﻨﺒﺎط‬‫اﻟﻜﻮاﻛﺐ‬ ‫ﻟﺤﺮﻛﺔ‬ ‫اﻟﺮﯾﺎﺿﻲ‬ ‫اﻟﻨﻤﻮذج‬
‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫ﺗﺘﺤﺮك‬.
‫ﻋﺪة‬ ‫ﺑﻌﺪ‬‫ﻣﺤﺎوﻻت‬‫اﻟ‬ ‫ﻣﺴﺎر‬ ‫ﻋﻦ‬ ‫ﺑﺮاھﻲ‬ ‫ﻣﻌﻠﻮﻣﺎت‬ ‫وﻣﻦ‬ ،‫ﺷﺎﻗﺔ‬
‫ﻛﺒﻠﺮ‬ ‫ﺑﻘﻮاﻧﯿﻦ‬ ‫ﺗﻌﺮف‬ ،‫ﻗﻮاﻧﯿﻦ‬ ‫ﺛﻼث‬ ‫ﻓﻲ‬ ‫ﯾﻠﺨﺺ‬ ‫اﻟﻜﻮﻛﺒﯿﺔ‬ ‫ﻟﻠﺤﺮﻛﺔ‬
1-‫ﺑﺆرﺗﯿﮫ‬ ‫إﺣﺪى‬ ‫ﻓﻲ‬ ‫ﺗﻘﻊ‬ ‫واﻟﺸﻤﺲ‬ ‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫اھﻠﯿﻠﯿﺠﯿﺔ‬ ‫ﻣﺪارات‬ ‫ﻓﻲ‬ ‫ﺗﺘﺤﺮك‬ ‫اﻟﻜﻮاﻛﺐ‬ ‫ﻛﻞ‬
2-‫ﻣ‬ ‫ﻣﺴﺎﺣﺎت‬ ‫ﯾﻤﺴﺢ‬ ‫واﻟﺸﻤﺲ‬ ‫اﻟﻜﻮﻛﺐ‬ ‫ﺑﯿﻦ‬ ‫اﻟﻮاﺻﻞ‬ ‫اﻟﺨﻂ‬
3-‫اﻻھﻠﯿﻠﺠﻲ‬ ‫ﻟﻠﻤﺪار‬ ‫اﻟﻜﺒﯿﺮ‬ ‫اﻟﻤﺤﻮر‬ ‫ﻧﺼﻒ‬ ‫ﻣﻜﻌﺐ‬ ‫ﻣﻊ‬ ‫ﯾﺘﻨﺎﺳﺐ‬ ‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫ﻛﻮﻛﺐ‬ ‫ﻷي‬ ‫اﻟﺪوري‬ ‫اﻟﺰﻣﻦ‬ ‫ﻣﺮﺑﻊ‬
‫ﻛﺒﻠﺮ‬ ‫ﺟﻮھﺎﻧﺲ‬
)1630-1571(
‫ﻛﺒﻠﺮ‬ ‫اﻷﻟﻤﺎﻧﻲ‬ ‫اﻟﻔﻠﻜﻲ‬
‫ﺣﺮﻛﺔ‬ ‫ﻗﻮاﻧﯿﻦ‬ ‫ﺑﺘﻄﻮﯾﺮ‬ ‫اﺷﺘﮭﺮ‬
‫ﻋﻠﻰ‬ ‫ﻣﻌﺘﻤﺪا‬ ‫اﻟﻜﻮاﻛﺐ‬
‫ﻷﺳﺘﺎذه‬ ‫اﻟﺪﻗﯿﻘﺔ‬ ‫اﻟﻤﻼﺣﻈﺎت‬
‫ﺑﺮاھﻲ‬ ‫ﺗﺎﯾﻜﻮ‬.
‫ﺣﺮﻛﺔ‬ ‫ﻗﻮاﻧﯿﻦ‬ ‫ﺑﺘﻄﻮﯾﺮ‬ ‫اﺷﺘﮭﺮ‬

20
‫اﻟﻔﺼﻞ‬ ‫ھﺬا‬ ‫ﻓﻲ‬ ‫ذﻟﻚ‬ ‫ﻧﺎﻗﺸﻨﺎ‬ ‫ﻛﻤﺎ‬ ‫داﺋﺮﯾﺔ‬ ‫ھﻲ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫ﻣﺮاﻛﺰ‬ ‫ﺣﻮل‬ ‫اﻷﺟﺴﺎم‬ ‫ﻣﺪارات‬ ‫أن‬ ‫ﻋﻠﻰ‬ ‫اﻋﺘﺪﻧﺎ‬ ‫ﻟﻘﺪ‬.‫ﻗﺎﻧﻮن‬ ‫ﯾﺸﯿﺮ‬
‫اﻟﻌﺎﻣﺔ‬ ‫اﻟﺤﺎﻟﺔ‬ ‫ھﻲ‬ ‫اﻻھﻠﯿﻠﺠﯿﺔ‬ ‫واﻟﻤﺪارات‬ ‫ﺟﺪا‬ ‫ﺧﺎﺻﺔ‬ ‫ﺣﺎﻟﺔ‬.‫اﻟﻔﻜﺮة‬ ‫ھﺬه‬ ‫ﻛﺎﻧﺖ‬ ‫ﻟﻘﺪ‬
‫ﺗﻌﻜﺲ‬ ‫ﻟﻠﻜﻮاﻛﺐ‬ ‫ﻛﺎﻣﻠﺔ‬ ‫داﺋﺮﯾﺔ‬ ‫ﻣﺪارات‬ ‫أن‬ ‫ﯾﺮون‬ ‫ﻛﺎﻧﻮا‬ ‫ﻷﻧﮭﻢ‬ ‫وذﻟﻚ‬ ،‫اﻟﻮﻗﺖ‬ ‫ذﻟﻚ‬ ‫ﻓﻲ‬ ‫اﻟﻌﻠﻤﺎء‬ ‫ﻟﺪى‬ ‫ﺗﻘﺒﻠﮭﺎ‬ ‫اﻟﺼﻌﺐ‬
‫ﻧﺎﻗﺺ‬ ‫ﻟﻘﻄﻊ‬ ‫ھﻨﺪﺳﻲ‬ ‫ﺷﻜﻞ‬.‫ﻟﻠﻜﻮﻛﺐ‬ ‫اﻻھﻠﯿﻠﺠﻲ‬ ‫ﻟﻠﻤﺪار‬ ‫ﺑﻨﻤﻮذج‬ ‫زودﻧﺎ‬ ‫ﺣﯿﺚ‬.‫ﯾﻌﺮف‬ ‫اﻟﻨﺎﻗﺺ‬ ‫اﻟﻘﻄﻊ‬
‫ﺑﺎﻟﺒﺆرة‬ ‫ﻣﻨﮭﻤﺎ‬ ‫واﺣﺪة‬ ‫ﻛﻞ‬ ‫ﺗﺪﻋﻰ‬focus.‫اﻟﻤﻨﻘﻂ‬ ‫اﻟﻤﻨﺤﻨﻰ‬ ‫رﺳﻢ‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬ ‫ذﻟﻚ‬ ‫ﺑﻌﺪ‬
r‫ﻟــ‬F1‫و‬F2‫اﻟﺘﻮاﻟﻲ‬ ‫ﻋﻠﻰ‬ ،‫و‬‫ھﻮ‬‫ﺛﺎﺑﺖ‬.‫وﺣﺪود‬ ‫اﻟﻤﺮﻛﺰ‬ ‫ﺑﯿﻦ‬ ‫اﻷطﻮل‬ ‫اﻟﻤﺴﺎﻓﺔ‬
‫اﻟﺒﺆرﺗﯿﻦ‬ ‫ﻛﻼ‬ ‫ﻣﻦ‬ ‫وﯾﻤﺮ‬(،‫اﻟﺮﺋﯿﺴﻲ‬ ‫اﻟﻤﺤﻮر‬ ‫ﺗﺴﻤﻰ‬‫ھﻲ‬ ‫اﻟﻤﺴﺎﻓﺔ‬ ‫ھﺬه‬2a.‫اﻟ‬ ‫ﻓﻲ‬‫ﺸﻜﻞ‬5.13.‫ﯾﺮﺳﻢ‬
‫اﻟﺴﯿﻨﻲ‬ ‫اﻻﺗﺠﺎه‬ ‫طﻮل‬ ‫ﻋﻠﻰ‬ ‫اﻟﺮﺋﯿﺴﻲ‬ ‫اﻟﻤﺤﻮر‬.‫اﻟﻤﺴﺎﻓﺔ‬a‫اﻟﺮﺋﯿﺴﻲ‬ ‫اﻟﻤﺤﻮر‬ ‫ﺑﻨﺼﻒ‬ ‫ﺗﺪﻋﻰ‬.
‫وﺣﺪود‬ ‫اﻟﻤﺮﻛﺰ‬ ‫ﺑﯿﻦ‬ ‫اﻷﻗﺼﺮ‬ ‫اﻟﻤﺴﺎﻓﺔ‬ ،‫و‬ ‫اﻟﺜﺎﻧﻮي‬ ‫ﺑﺎﻟﻤﺤﻮر‬ ‫ﺗﺴﻤﻰ‬ ‫اﻟﻨﺎﻗﺺ‬ ‫اﻟﻘﻄﻊ‬‫ھﻮ‬2b.‫واﻟﻤﺴﺎﻓﺔ‬b‫ﻧﺼﻒ‬ ‫ھﻲ‬
‫ﻣﺴﺎﻓﺔ‬ ‫ﻋﻠﻰ‬ ‫ﺗﻘﻊ‬ ‫اﻟﻨﺎﻗﺺ‬ ‫اﻟﻘﻄﻊ‬ ‫ﺑﺆرة‬ ‫أﻣﺎ‬c‫اﻟﻘﻄﻊ‬ ‫ﻣﺮﻛﺰ‬ ‫ﻣﻦ‬.‫ﺣﯿﺚ‬a2
b2
+c2
.‫اﻻھﻠﯿﻠﺠﻲ‬ ‫اﻟﻤﺪار‬ ‫ﻓﻲ‬
‫اﻟﻨﺎﻗﺺ‬ ‫اﻟﻘﻄﻊ‬ ‫ﺑﺆرﺗﻲ‬ ‫إﺣﺪى‬ ‫ﻓﻲ‬ ‫اﻟﺸﻤﺲ‬ ‫ﺗﻜﻮن‬ ‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫ﻟﻠﻜﻮﻛﺐ‬.‫و‬‫اﻷﺧﺮى‬ ‫اﻟﺒﺆرة‬ ‫ﻓﻲ‬ ‫ﺷﻲء‬ ‫أي‬ ‫ﯾﻮﺟﺪ‬ ‫ﻻ‬.
‫ﺑـ‬ ‫ﯾﻌﺮف‬ ‫اﻟﻨﺎﻗﺺ‬ ‫اﻟﻘﻄﻊ‬ ‫ﻓﻲ‬ ‫اﻟﻤﺮﻛﺰي‬ ‫اﻻﻧﺤﺮاف‬ ‫إن‬c/a=e‫و‬‫اﻟﻨﺎﻗﺺ‬ ‫ﻟﻠﻘﻄﻊ‬ ‫اﻟﻌﺎم‬ ‫اﻟﺸﻜﻞ‬ ‫ﯾﺮﺳﻢ‬.
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‫ھﺬه‬ ‫ﻣﻦ‬ ‫ﻛﻞ‬ ‫ﺳﻨﻨﺎﻗﺶ‬‫أدﻧﺎه‬ ‫اﻟﻘﻮاﻧﯿﻦ‬.
‫ﻛﺒﻠﺮ‬ ‫ﻗﺎﻧﻮن‬‫اﻷول‬
‫اﻟﻔﺼﻞ‬ ‫ھﺬا‬ ‫ﻓﻲ‬ ‫ذﻟﻚ‬ ‫ﻧﺎﻗﺸﻨﺎ‬ ‫ﻛﻤﺎ‬ ‫داﺋﺮﯾﺔ‬ ‫ھﻲ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫ﻣﺮاﻛﺰ‬ ‫ﺣﻮل‬ ‫اﻷﺟﺴﺎم‬ ‫ﻣﺪارات‬ ‫أن‬ ‫ﻋﻠﻰ‬ ‫اﻋﺘﺪﻧﺎ‬ ‫ﻟﻘﺪ‬
‫ﻛﺒﻠﺮ‬‫اﻷول‬‫اﻟﺪاﺋﺮي‬ ‫اﻟﻤﺪار‬ ‫أن‬ ‫إﻟﻰ‬‫ھﻮ‬‫اﻟﻌﺎﻣﺔ‬ ‫اﻟﺤﺎﻟﺔ‬ ‫ھﻲ‬ ‫اﻻھﻠﯿﻠﺠﯿﺔ‬ ‫واﻟﻤﺪارات‬ ‫ﺟﺪا‬ ‫ﺧﺎﺻﺔ‬ ‫ﺣﺎﻟﺔ‬
‫ﻣﻦ‬‫ﺗﻌﻜﺲ‬ ‫ﻟﻠﻜﻮاﻛﺐ‬ ‫ﻛﺎﻣﻠﺔ‬ ‫داﺋﺮﯾﺔ‬ ‫ﻣﺪارات‬ ‫أن‬ ‫ﯾﺮون‬ ‫ﻛﺎﻧﻮا‬ ‫ﻷﻧﮭﻢ‬ ‫وذﻟﻚ‬ ،‫اﻟﻮﻗﺖ‬ ‫ذﻟﻚ‬ ‫ﻓﻲ‬ ‫اﻟﻌﻠﻤﺎء‬ ‫ﻟﺪى‬ ‫ﺗﻘﺒﻠﮭﺎ‬ ‫اﻟﺼﻌﺐ‬
‫اﻟﺴﻤﺎء‬ ‫ﻓﻲ‬ ‫اﻟﻜﻤﺎل‬.
‫اﻟ‬‫ﺸﻜﻞ‬5.13‫ھ‬‫ﻮ‬‫ﻧﺎﻗﺺ‬ ‫ﻟﻘﻄﻊ‬ ‫ھﻨﺪﺳﻲ‬ ‫ﺷﻜﻞ‬
‫ﻧﻘﻄﺘﯿﻦ‬ ‫ﺑﺎﺧﺘﯿﺎر‬ ‫رﯾﺎﺿﯿﺎ‬F1‫و‬F2‫ﺑﺎﻟﺒﺆرة‬ ‫ﻣﻨﮭﻤﺎ‬ ‫واﺣﺪة‬ ‫ﻛﻞ‬ ‫ﺗﺪﻋﻰ‬
‫اﻟﻤﺴﺎﻓﺎت‬ ‫ﻣﺠﻤﻮع‬ ‫ﯾﻤﺜﻞ‬ ‫واﻟﺬي‬r1‫و‬r2
‫اﻟﻨﺎﻗﺺ‬ ‫اﻟﻘﻄﻊ‬)‫اﻟﺒﺆرﺗﯿﻦ‬ ‫ﻛﻼ‬ ‫ﻣﻦ‬ ‫وﯾﻤﺮ‬
‫اﻟﺴﯿﻨﻲ‬ ‫اﻻﺗﺠﺎه‬ ‫طﻮل‬ ‫ﻋﻠﻰ‬ ‫اﻟﺮﺋﯿﺴﻲ‬ ‫اﻟﻤﺤﻮر‬
‫أﯾﻀﺎ‬ ‫وﺑﺎﻟﻤﺜﻞ‬‫وﺣﺪود‬ ‫اﻟﻤﺮﻛﺰ‬ ‫ﺑﯿﻦ‬ ‫اﻷﻗﺼﺮ‬ ‫اﻟﻤﺴﺎﻓﺔ‬ ،
‫اﻟﺜﺎﻧﻮي‬ ‫اﻟﻤﺤﻮر‬.‫ﻣﺴﺎﻓﺔ‬ ‫ﻋﻠﻰ‬ ‫ﺗﻘﻊ‬ ‫اﻟﻨﺎﻗﺺ‬ ‫اﻟﻘﻄﻊ‬ ‫ﺑﺆرة‬ ‫أﻣﺎ‬
‫اﻟﻨﺎﻗﺺ‬ ‫اﻟﻘﻄﻊ‬ ‫ﺑﺆرﺗﻲ‬ ‫إﺣﺪى‬ ‫ﻓﻲ‬ ‫اﻟﺸﻤﺲ‬ ‫ﺗﻜﻮن‬ ‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫ﻟﻠﻜﻮﻛﺐ‬
‫ﺑـ‬ ‫ﯾﻌﺮف‬ ‫اﻟﻨﺎﻗﺺ‬ ‫اﻟﻘﻄﻊ‬ ‫ﻓﻲ‬ ‫اﻟﻤﺮﻛﺰي‬ ‫اﻻﻧﺤﺮاف‬ ‫إن‬
‫ﺷﻜﻞ‬5.13‫ﻧﺎﻗﺺ‬ ‫ﻗﻄﻊ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬
‫ﺣﯿﺚ‬a‫اﻟﺮﺋﯿﺴﻲ‬ ‫اﻟﻤﺤﻮر‬ ‫ﻧﺼﻒ‬ ‫ھﻲ‬
b‫وﻛﻼ‬ ‫اﻟﺜﺎﻧﻮي‬ ‫اﻟﻤﺤﻮر‬ ‫ﻧﺼﻒ‬ ‫ھﻲ‬
‫اﻟﻤﺮﻛﺰ‬ ‫ﺟﺎﻧﺒﻲ‬ ‫ﻋﻠﻰ‬ ‫ﺗﻘﻌﺎن‬ ‫اﻟﺒﺆرﺗﯿﻦ‬
‫ﻣﺴﺎﻓﺔ‬ ‫وﻋﻠﻰ‬c‫ﻣﻨﮫ‬.
‫ﺷﻜﻞ‬
‫ﺣﯿﺚ‬
‫و‬b
‫اﻟﻤﺮﻛﺰ‬ ‫ﺟﺎﻧﺒﻲ‬ ‫ﻋﻠﻰ‬ ‫ﺗﻘﻌﺎن‬ ‫اﻟﺒﺆرﺗﯿﻦ‬

21
‫اﻟﻤﺮﻛﺰي‬ ‫ﻓﺎﻻﻧﺤﺮاف‬‫ﯾﺴﺎوي‬‫اﻟﺼﻔﺮ‬.b‫ھﻮ‬‫ﻣﻦ‬ ‫اﻷﺻﻐﺮ‬a‫اﻟﻨﺎﻗﺺ‬ ‫اﻟﻘﻄﻊ‬ ‫ﻓﺈن‬ ‫ﺑﺎﻟﺘﺎﻟﻲ‬ ،‫ھﻮ‬‫اﻷﻗﺼﺮ‬
‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫ﻛﻤﺎ‬5.13.‫ﺗﺘﻨﺎﻗﺺ‬ ‫ﻋﻨﺪﻣﺎ‬b‫ﺗﺘﺰاﯾﺪ‬ ،c‫اﻟﻤﺮﻛﺰي‬ ‫واﻻﻧﺤﺮاف‬ ،e
‫ﻋﺮﺿﺎ‬ ‫واﻷﻗﻞ‬ ‫اﻷطﻮل‬ ‫اﻟﻨﺎﻗﺼﺔ‬ ‫اﻟﻘﻄﻮع‬ ‫ﻣﻊ‬ ‫ﺗﺘﻮاﻓﻖ‬ ‫اﻟﻤﺮﻛﺰي‬ ‫ﻟﻼﻧﺤﺮاف‬ ‫اﻟﻌﻈﻤﻰ‬ ‫اﻟﻘﯿﻢ‬ ‫ﻓﺈن‬ ‫وھﻜﺬا‬.
‫اﻟﻨﺎﻗﺺ‬ ‫ﻟﻠﻘﻄﻊ‬ ‫اﻟﻤﺮﻛﺰي‬ ‫اﻻﻧﺤﺮاف‬ ‫ﻗﯿﻢ‬ ‫ﻧﻄﺎق‬ ‫أن‬ ‫ﺣﯿﺚ‬‫ھﻮ‬:0 < e < 1..
‫ﺗﻔ‬ ‫ت‬‫ﺎ‬‫و‬‫اﻟﻤﺮﻛﺰي‬ ‫اﻻﻧﺤﺮاف‬ ‫ﯾﺒﻠﻎ‬ ‫ﺣﯿﺚ‬ ،‫اﻟﺸﻤﺴﯿﺔ‬ ‫اﻟﻤﺠﻤﻮﻋﺔ‬ ‫ﻛﻮاﻛﺐ‬ ‫ﺑﯿﻦ‬ ‫ﻛﺒﯿﺮا‬ ‫ﺗﺎ‬
‫أﺧﺮى‬ ‫ﻧﺎﺣﯿﺔ‬ ‫ﻣﻦ‬ ،‫داﺋﺮي‬ ‫ﺷﺒﮫ‬ ‫ﻣﻨﮫ‬ ‫ﯾﺠﻌﻞ‬ ‫اﻟﺬي‬‫ﺑﻠﻮﺗ‬ ‫ﻟﻤﺪار‬ ‫اﻟﻤﺮﻛﺰي‬ ‫اﻻﻧﺤﺮاف‬ ‫ﯾﺒﻠﻎ‬‫ﻮ‬0.25‫و‬‫ھﻮ‬
‫اﻟﺸﻜﻞ‬6.13a‫ﺑﻠﻮﺗﻮ‬ ‫ﻟﻤﺪار‬ ‫اﻟﻤﺮﻛﺰي‬ ‫اﻻﻧﺤﺮاف‬ ‫ﻣﻊ‬ ‫ﻧﺎﻗﺺ‬ ‫ﻗﻄﻊ‬ ‫ﯾﻈﮭﺮ‬.‫أﻧﮫ‬ ‫ﻻﺣﻆ‬
‫اﻟﺪاﺋﺮة‬ ‫ﻋﻦ‬ ‫ﺗﻤﯿﯿﺰه‬ ‫اﻟﺼﻌﺐ‬ ‫ﻣﻦ‬ ‫ﯾﻜﻮن‬ ‫ﻟﻠﻤﺪار‬ ‫ﻣﺮﻛﺰي‬ ‫اﻧﺤﺮاف‬ ‫درﺟﺔ‬ ‫أﻋﻠﻰ‬ ‫ﻋﻨﺪ‬ ‫ﺣﺘﻰ‬.
‫ﻣﺮﻛﺰي‬ ‫اﻧﺤﺮاف‬ ‫درﺟﺔ‬ ‫أﻋﻠﻰ‬ ‫ﯾﻤﻠﻚ‬ ‫واﻟﺬي‬(e = 0.25)‫اﻟﺸﻤﺴﯿﺔ‬ ‫اﻟﻤﺠﻤﻮﻋﺔ‬ ‫ﻓﻲ‬ ‫اﻟﻜﻮاﻛﺐ‬ ‫ﺑﯿﻦ‬.
‫اﻟﺰرﻗﺎء‬ ‫اﻟﻨﻘﻄﺔ‬ ‫ھﻲ‬ ‫اﻷﺧﺮى‬ ‫واﻟﺒﺆرة‬ ‫اﻟﻨﺎﻗﺺ‬ ‫اﻟﻘﻄﻊ‬ ‫ﺑﺆرة‬ ‫ھﻲ‬ ‫واﻟﺘﻲ‬ ‫اﻟﻜﺒﯿﺮة‬ ‫اﻟﺼﻔﺮاء‬ ‫اﻟﻨﻘﻄﺔ‬ ‫ﻓﻲ‬ ‫ﺗﻘﻊ‬ ‫اﻟﺸﻤﺲ‬.)b(‫ﻣﺬﻧﺐ‬ ‫ﻣﺪار‬ ‫ﺗﻤﺜﻞ‬
‫ﺑﺎﻹﻋﺠﺎب‬ ‫ﺟﺪﯾﺮ‬ ‫اﻧﺠﺎز‬.‫ھﺎﻟﻲ‬ ‫ﻣﺬﻧﺐ‬ ‫ﻟﻤﺪار‬ ‫اﻟﻤﺮﻛﺰي‬ ‫اﻻﻧﺤﺮاف‬ ‫إن‬‫ھﻮ‬0.97‫ﻣﺪاره‬ ‫إن‬ ‫ﺣﯿﺚ‬ ،
‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫ﻛﻤﺎ‬ ‫اﻟﺜﺎﻧﻮي‬ ‫اﻟﻤﺤﻮر‬ ‫ﻣﻦ‬ ‫ﺑﻜﺜﯿﺮ‬ ‫أطﻮل‬ ‫رﺋﯿﺴﻲ‬ ‫ﺑﻤﺤﻮر‬ ‫ﯾﻮﺻﻒ‬6.13b،‫و‬‫ﻓﺘﺮة‬ ‫ﯾﻘﻀﻲ‬ ‫ھﺎﻟﻲ‬ ‫ﻣﺬﻧﺐ‬ ‫ﻓﺈن‬ ‫ﻟﺬﻟﻚ‬ ‫ﻧﺘﯿﺠﺔ‬
‫اﻷرض‬ ‫ﻣﻦ‬ ‫ﻣﺮﺋﻲ‬ ‫وﻏﯿﺮ‬ ‫اﻟﺸﻤﺲ‬ ‫ﻋﻦ‬ ‫ﺑﻌﯿﺪا‬ ‫ﺳﻨﺔ‬.‫و‬‫ﻣﺪاره‬ ‫ﻣﻦ‬ ‫ﺻﻐﯿﺮ‬ ‫ﺟﺰء‬ ‫ﻓﻲ‬ ‫اﻟﻤﺠﺮدة‬ ‫ﺑﺎﻟﻌﯿﻦ‬ ‫رؤﯾﺘﮫ‬ ‫ﯾﻤﻜﻨﻨﺎ‬
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‫داﺋﺮة‬ ‫اﺟﻞ‬ ‫ﻣﻦ‬c = 0‫اﻟﻤﺮﻛﺰي‬ ‫ﻓﺎﻻﻧﺤﺮاف‬
‫اﻻﺗﺠﺎه‬ ‫ﻓﻲ‬y‫اﻻﺗﺠﺎه‬ ‫ﻓﻲ‬ ‫ﻣﺪاه‬ ‫ﻣﻊ‬ ‫ﻣﻘﺎرﻧﺔ‬x
‫ﯾﺘﺰاﯾﺪ‬.‫ﻋﺮﺿﺎ‬ ‫واﻷﻗﻞ‬ ‫اﻷطﻮل‬ ‫اﻟﻨﺎﻗﺼﺔ‬ ‫اﻟﻘﻄﻮع‬ ‫ﻣﻊ‬ ‫ﺗﺘﻮاﻓﻖ‬ ‫اﻟﻤﺮﻛﺰي‬ ‫ﻟﻼﻧﺤﺮاف‬ ‫اﻟﻌﻈﻤﻰ‬ ‫اﻟﻘﯿﻢ‬ ‫ﻓﺈن‬ ‫وھﻜﺬا‬
‫اﻟﻨﺎﻗﺺ‬ ‫ﻟﻠﻘﻄﻊ‬ ‫اﻟﻤﺮﻛﺰي‬ ‫اﻻﻧﺤﺮاف‬ ‫ﻗﯿﻢ‬ ‫ﻧﻄﺎق‬ ‫أن‬ ‫ﺣﯿﺚ‬
‫ﯾﺘﻔ‬ ‫اﻟﻜﻮاﻛﺐ‬ ‫ﻣﺪارات‬ ‫ﻓﻲ‬ ‫اﻻﻧﺤﺮاف‬ ‫إن‬‫ﺎ‬‫و‬‫ﺗﻔ‬ ‫ت‬
‫اﻷرض‬ ‫ﻟﻤﺪار‬0.017‫أﺧﺮى‬ ‫ﻧﺎﺣﯿﺔ‬ ‫ﻣﻦ‬ ،‫داﺋﺮي‬ ‫ﺷﺒﮫ‬ ‫ﻣﻨﮫ‬ ‫ﯾﺠﻌﻞ‬ ‫اﻟﺬي‬
‫اﻟﺘﺴﻌﺔ‬ ‫اﻟﻜﻮاﻛﺐ‬ ‫ﻓﻲ‬ ‫اﻷﻋﻠﻰ‬ ‫اﻻﻧﺤﺮاف‬.‫اﻟﺸﻜﻞ‬
‫اﻟﺪاﺋﺮة‬ ‫ﻋﻦ‬ ‫ﺗﻤﯿﯿﺰه‬ ‫اﻟﺼﻌﺐ‬ ‫ﻣﻦ‬ ‫ﯾﻜﻮن‬ ‫ﻟﻠﻤﺪار‬ ‫ﻣﺮﻛﺰي‬ ‫اﻧﺤﺮاف‬ ‫درﺟﺔ‬ ‫أﻋﻠﻰ‬ ‫ﻋﻨﺪ‬ ‫ﺣﺘﻰ‬
‫اﻟﺸﻜﻞ‬6.13)a(‫ﺑﻠﻮﺗ‬ ‫ﻣﺪار‬ ‫ﺗﻤﺜﻞ‬‫ﻮ‬‫ﻣﺮﻛﺰي‬ ‫اﻧﺤﺮاف‬ ‫درﺟﺔ‬ ‫أﻋﻠﻰ‬ ‫ﯾﻤﻠﻚ‬ ‫واﻟﺬي‬
‫اﻟﺰرﻗﺎء‬ ‫اﻟﻨﻘﻄﺔ‬ ‫ھﻲ‬ ‫اﻷﺧﺮى‬ ‫واﻟﺒﺆرة‬ ‫اﻟﻨﺎﻗﺺ‬ ‫اﻟﻘﻄﻊ‬ ‫ﺑﺆرة‬ ‫ھﻲ‬ ‫واﻟﺘﻲ‬ ‫اﻟﻜﺒﯿﺮة‬ ‫اﻟﺼﻔﺮاء‬ ‫اﻟﻨﻘﻄﺔ‬ ‫ﻓﻲ‬ ‫ﺗﻘﻊ‬ ‫اﻟﺸﻤﺲ‬
‫ھﺎﻟﻲ‬.
‫ﻓﻠﮭﺬا‬‫ﻛﺒﻠﺮ‬ ‫ﻗﺎﻧﻮن‬‫اﻷول‬‫ھﻮ‬‫ﺑﺎﻹﻋﺠﺎب‬ ‫ﺟﺪﯾﺮ‬ ‫اﻧﺠﺎز‬
‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫ﻛﻤﺎ‬ ‫اﻟﺜﺎﻧﻮي‬ ‫اﻟﻤﺤﻮر‬ ‫ﻣﻦ‬ ‫ﺑﻜﺜﯿﺮ‬ ‫أطﻮل‬ ‫رﺋﯿﺴﻲ‬ ‫ﺑﻤﺤﻮر‬ ‫ﯾﻮﺻﻒ‬
‫ﺣﻮاﻟﻲ‬ ‫طﻮﯾﻠﺔ‬76‫اﻷرض‬ ‫ﻣﻦ‬ ‫ﻣﺮﺋﻲ‬ ‫وﻏﯿﺮ‬ ‫اﻟﺸﻤﺲ‬ ‫ﻋﻦ‬ ‫ﺑﻌﯿﺪا‬ ‫ﺳﻨﺔ‬
‫اﻟﺸﻤﺲ‬ ‫ﻣﻦ‬ ‫ﯾﻘﺘﺮب‬ ‫ﻋﻨﺪﻣﺎ‬.
‫اﻟﺸﻜﻞ‬6.13

22
‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫ﺗﺮاه‬ ‫ﻛﺎﻟﺬي‬ ‫اھﻠﯿﻠﺠﻲ‬ ‫ﻣﺪار‬ ‫ﻓﻲ‬ ‫ﻛﻮﻛﺐ‬ ‫اﻵن‬ ‫ﺗﺨﯿﻞ‬5.13‫اﻟﺒﺆرة‬ ‫ﻓﻲ‬ ‫اﻟﺸﻤﺲ‬ ‫ﺗﻘﻊ‬ ،F2‫ﻓﻲ‬ ‫اﻟﻜﻮﻛﺐ‬ ‫ﯾﻜﻮن‬ ‫ﻋﻨﺪﻣﺎ‬ ،
‫ھﻲ‬ ‫واﻟﺸﻤﺲ‬ ‫اﻟﻜﻮﻛﺐ‬ ‫ﺑﯿﻦ‬ ‫واﻟﻤﺴﺎﻓﺔ‬ ،‫اﻟﺮﺳﻢ‬ ‫ﻣﻦ‬ ‫اﻟﯿﺴﺎر‬ ‫أﻗﺼﻰ‬a + c.‫اﻷ‬ ‫ﺗﺴﻤﻰ‬ ‫اﻟﻨﻘﻄﺔ‬ ‫ھﺬه‬‫و‬‫ج‬aphelion.‫ﯾﻜﻮن‬ ‫ﺣﯿﺚ‬
‫اﻟﻤﺪار‬ ‫ھﺬا‬ ‫وﺿﻤﻦ‬ ‫اﻟﺸﻤﺲ‬ ‫ﻋﻦ‬ ‫ﯾﻤﻜﻦ‬ ‫ﻣﺎ‬ ‫أﺑﻌﺪ‬ ‫اﻟﻜﻮﻛﺐ‬) .‫ﻓﻲ‬ ‫ﻟﺠﺴﻢ‬‫اﻷ‬ ‫اﻟﻨﻘﻄﺔ‬ ‫ھﺬه‬ ‫ﺗﺴﻤﻰ‬ ،‫اﻷرض‬ ‫ﺣﻮل‬ ‫ﻣﺪاره‬‫و‬‫ج‬.(
،‫اﻟﻨﺎﻗﺺ‬ ‫اﻟﻘﻄﻊ‬ ‫ﻣﻦ‬ ‫اﻟﻌﻤﻮدي‬ ‫اﻻﺗﺠﺎه‬ ‫أﻗﺼﻰ‬ ‫ﻓﻲ‬ ‫اﻟﻜﻮﻛﺐ‬ ‫ﯾﻜﻮن‬ ‫ﻋﻨﺪﻣﺎ‬ ،‫ﺑﺎﻟﻤﻘﺎﺑﻞ‬‫ﺑﺎﻟﺤﻀﯿﺾ‬ ‫اﻟﻨﻘﻄﺔ‬ ‫ھﺬه‬ ‫ﺗﺴﻤﻰ‬perihelion
‫اﻟﻜﻮﻛﺐ‬ ‫ﺑﯿﻦ‬ ‫اﻟﻤﺴﺎﻓﺔ‬‫و‬‫ھﻲ‬ ‫اﻟﺸﻤﺲ‬a - c..
‫ﻟﻄﺒﯿﻌﺔ‬ ‫ﻣﺒﺎﺷﺮة‬ ‫ﻧﺘﯿﺠﺔ‬‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫ﻓﻲ‬ ‫اﻟﻌﻜﺴﻲ‬ ‫اﻟﺘﺮﺑﯿﻊ‬،‫واﻟﺪاﺋﺮﯾﺔ‬ ‫اﻻھﻠﯿﻠﺠﯿﺔ‬ ‫اﻟﻤﺪارات‬ ‫ﻧﺎﻗﺸﻨﺎ‬ ‫وﻗﺪ‬.
‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫ﺑﻤﺮﻛﺰ‬ ‫ﻣﻘﯿﺪة‬ ‫ﺗﻜﻮن‬ ‫أن‬ ‫ﻻﺑﺪ‬ ‫اﻟﻤﺪارات‬ ‫ﻣﻦ‬ ‫اﻷﺷﻜﺎل‬ ‫ھﺬه‬ ‫ﻋﻠﻰ‬ ‫اﻷﺟﺴﺎم‬ ‫أن‬ ‫ووﺟﺪﻧﺎ‬.‫و‬‫اﻟﻜﻮاﻛﺐ‬ ‫اﻷﺟﺴﺎم‬ ‫ھﺬه‬ ‫ﺗﺸﻤﻞ‬،
‫اﻟﻜﻮﻛﺐ‬ ‫ﺣﻮل‬ ‫ﺗﺪور‬ ‫اﻟﺘﻲ‬ ‫اﻷﻗﻤﺎر‬ ‫وﻛﺬﻟﻚ‬ ،‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬.‫ھﻨﺎك‬ ‫ﯾﻜﻮن‬ ‫أن‬ ‫ﯾﻤﻜﻦ‬ ‫ﻛﻤﺎ‬
‫أﺑﺪا‬ ‫ﯾﻌﻮد‬ ‫ﻻ‬ ‫ﺛﻢ‬ ‫واﺣﺪة‬ ‫ﻟﻤﺮة‬ ‫اﻟﺸﻤﺲ‬ ‫أﻣﺎم‬ ‫ﻣﻦ‬ ‫ﯾﻤﺮ‬ ‫ﻗﺪ‬ ‫اﻟﻔﻀﺎء‬ ‫أﻋﻤﺎق‬ ‫ﻣﻦ‬ ‫ﻗﺎدم‬ ‫ﻧﯿﺰك‬ ‫ﻣﺜﻞ‬ ‫ﻣﻘﯿﺪة‬ ‫ﻏﯿﺮ‬ ‫أﺟﺴﺎم‬ ‫أﯾﻀﺎ‬.‫ﻗﻮة‬ ‫إن‬
،‫ﺑﯿﻨﮭﻤﺎ‬ ‫اﻟﻤﺴﺎﻓﺔ‬ ‫ﻣﺮﺑﻊ‬ ‫ﻣﻊ‬ ‫ﻋﻜﺴﯿﺎ‬ ‫أﯾﻀﺎ‬ ‫ﺗﺘﻨﺎﺳﺐ‬ ‫اﻷﺟﺴﺎم‬ ‫ھﺬه‬ ‫ﺑﯿﻦ‬‫و‬‫ﺗ‬‫طﺮﯾﻘﯿﻦ‬ ‫وﻓﻖ‬ ‫اﻷﺟﺴﺎم‬ ‫ھﺬه‬ ‫ﺴﯿﺮ‬
‫اﻟﺰاﺋﺪة‬ ‫اﻟﻘﻄﻮع‬(e > 1).
‫اﻟﺰ‬ ‫اﻻﻧﺪﻓﺎع‬ ‫ﻟﺤﻔﻆ‬ ‫ﻛﻨﺘﯿﺠﺔ‬ ‫اﻟﺜﺎﻧﻲ‬ ‫ﻛﺒﻠﺮ‬ ‫ﻗﺎﻧﻮن‬ ‫ﻧﺮى‬ ‫أن‬ ‫ﻧﺴﺘﻄﯿﻊ‬‫ا‬‫و‬‫اﻟ‬ ‫ﻋﻠﻰ‬ ‫ي‬‫ﻨﺤﻮ‬‫ﻛﺘﻠﺘﮫ‬ ‫ﻛﻮﻛﺒﺎ‬ ‫اﻻﻋﺘﺒﺎر‬ ‫ﺑﻌﯿﻦ‬ ‫ﻧﺄﺧﺬ‬ ،‫اﻟﺘﺎﻟﻲ‬MP
‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫ﻛﻤﺎ‬ ‫اھﻠﯿﻠﺠﻲ‬ ‫ﻣﺪار‬ ‫ﻓﻲ‬ ‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫وﯾﺘﺤﺮك‬7.13a.‫ﻛﻨﻈﺎم‬ ‫اﻟﻜﻮﻛﺐ‬ ‫ﻧﻌﺘﺒﺮ‬ ‫دﻋﻨﺎ‬.‫اﻟﺸﻤﺲ‬ ‫ﻧﻤﻮذج‬ ‫وﺳﯿﻜﻮن‬‫ذو‬
‫ﺗﺘﺤﺮك‬ ‫ﻻ‬ ‫اﻟﺸﻤﺲ‬ ‫ﻣﻦ‬ ‫ﯾﺠﻌﻞ‬ ‫اﻟﺬي‬ ‫اﻷﻣﺮ‬ ‫اﻟﻜﻮﻛﺐ‬ ‫ﻛﺘﻠﺔ‬ ‫ﻣﻦ‬ ‫ﺑﻜﺜﯿﺮ‬ ‫أﻛﺒﺮ‬ ‫ھﺎﺋﻠﺔ‬ ‫ﻛﺘﻠﺔ‬.‫ھﻲ‬ ‫اﻟﻜﻮﻛﺐ‬ ‫ﻓﻲ‬ ‫ﺗﺆﺛﺮ‬ ‫اﻟﺘﻲ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫إن‬
‫اﻟﺸﻤﺲ‬ ‫وﺑﺎﺗﺠﺎه‬ ‫اﻟﻘﻄﺮ‬ ‫ﻧﺼﻒ‬ ‫ﻋﻠﻰ‬ ‫داﺋﻤﺎ‬ ‫ﻣﺤﻤﻮﻟﺔ‬‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫ﻛﻤﺎ‬7.13a.‫ﻟﻠﻜ‬ ‫اﻟﺪوران‬ ‫ﻋﺰم‬ ‫أن‬ ‫ﻛﻤﺎ‬‫ﺑﺴﺒﺐ‬ ‫ﻮﻛﺐ‬
‫ﻷن‬ ‫وذﻟﻚ‬ ‫ﺻﻔﺮ‬ ‫أﻧﮫ‬ ‫اﻟﻮاﺿﺢ‬ ‫ﻣﻦ‬ ‫اﻟﻤﺮﻛﺰﯾﺔ‬ ‫اﻟﻘﻮة‬F‫ﻋﻠﻰ‬ ‫ﺗﻨﻄﺒﻖ‬r‫إذا‬:
www.trgma.com
‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫ﺗﺮاه‬ ‫ﻛﺎﻟﺬي‬ ‫اھﻠﯿﻠﺠﻲ‬ ‫ﻣﺪار‬ ‫ﻓﻲ‬ ‫ﻛﻮﻛﺐ‬ ‫اﻵن‬ ‫ﺗﺨﯿﻞ‬
‫ھﻲ‬ ‫واﻟﺸﻤﺲ‬ ‫اﻟﻜﻮﻛﺐ‬ ‫ﺑﯿﻦ‬ ‫واﻟﻤﺴﺎﻓﺔ‬ ،‫اﻟﺮﺳﻢ‬ ‫ﻣﻦ‬ ‫اﻟﯿﺴﺎر‬ ‫أﻗﺼﻰ‬
‫اﻟﻤﺪار‬ ‫ھﺬا‬ ‫وﺿﻤﻦ‬ ‫اﻟﺸﻤﺲ‬ ‫ﻋﻦ‬ ‫ﯾﻤﻜﻦ‬ ‫ﻣﺎ‬ ‫أﺑﻌﺪ‬ ‫اﻟﻜﻮﻛﺐ‬
،‫اﻟﻨﺎﻗﺺ‬ ‫اﻟﻘﻄﻊ‬ ‫ﻣﻦ‬ ‫اﻟﻌﻤﻮدي‬ ‫اﻻﺗﺠﺎه‬ ‫أﻗﺼﻰ‬ ‫ﻓﻲ‬ ‫اﻟﻜﻮﻛﺐ‬ ‫ﯾﻜﻮن‬ ‫ﻋﻨﺪﻣﺎ‬ ،‫ﺑﺎﻟﻤﻘﺎﺑﻞ‬
)‫اﻷرض‬ ‫ﻟﻤﺪار‬،‫اﻟﺤﻀﯿﺾ‬(،‫و‬‫اﻟﻜﻮﻛﺐ‬ ‫ﺑﯿﻦ‬ ‫اﻟﻤﺴﺎﻓﺔ‬
‫ﻛﺒﻠﺮ‬ ‫ﻗﺎﻧﻮن‬‫اﻷول‬‫ھﻮ‬‫ﻟﻄﺒﯿﻌﺔ‬ ‫ﻣﺒﺎﺷﺮة‬ ‫ﻧﺘﯿﺠﺔ‬
‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫ﺑﻤﺮﻛﺰ‬ ‫ﻣﻘﯿﺪة‬ ‫ﺗﻜﻮن‬ ‫أن‬ ‫ﻻﺑﺪ‬ ‫اﻟﻤﺪارات‬ ‫ﻣﻦ‬ ‫اﻷﺷﻜﺎل‬ ‫ھﺬه‬ ‫ﻋﻠﻰ‬ ‫اﻷﺟﺴﺎم‬ ‫أن‬ ‫ووﺟﺪﻧﺎ‬
،‫اﻟﻜﻮﯾﻜﺒﺎت‬‫ﻣﺮارا‬ ‫ﺗﺘﺤﺮك‬ ‫اﻟﺘﻲ‬ ‫واﻟﻤﺬﻧﺒﺎت‬‫اﻟﻜﻮﻛﺐ‬ ‫ﺣﻮل‬ ‫ﺗﺪور‬ ‫اﻟﺘﻲ‬ ‫اﻷﻗﻤﺎر‬ ‫وﻛﺬﻟﻚ‬ ،‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬
‫أﺑﺪا‬ ‫ﯾﻌﻮد‬ ‫ﻻ‬ ‫ﺛﻢ‬ ‫واﺣﺪة‬ ‫ﻟﻤﺮة‬ ‫اﻟﺸﻤﺲ‬ ‫أﻣﺎم‬ ‫ﻣﻦ‬ ‫ﯾﻤﺮ‬ ‫ﻗﺪ‬ ‫اﻟﻔﻀﺎء‬ ‫أﻋﻤﺎق‬ ‫ﻣﻦ‬ ‫ﻗﺎدم‬ ‫ﻧﯿﺰك‬ ‫ﻣﺜﻞ‬ ‫ﻣﻘﯿﺪة‬ ‫ﻏﯿﺮ‬ ‫أﺟﺴﺎم‬ ‫أﯾﻀﺎ‬
‫اﻟﺸﻤﺲ‬ ‫ﺑﯿﻦ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬‫و‬،‫ﺑﯿﻨﮭﻤﺎ‬ ‫اﻟﻤﺴﺎﻓﺔ‬ ‫ﻣﺮﺑﻊ‬ ‫ﻣﻊ‬ ‫ﻋﻜﺴﯿﺎ‬ ‫أﯾﻀﺎ‬ ‫ﺗﺘﻨﺎﺳﺐ‬ ‫اﻷﺟﺴﺎم‬ ‫ھﺬه‬ ‫ﺑﯿﻦ‬
‫ﯾﺸﻤﻼن‬:‫اﻟﻤﻜﺎﻓﺌﺔ‬ ‫اﻟﻘﻄﻮع‬(e = 1)‫و‬‫اﻟﺰاﺋﺪة‬ ‫اﻟﻘﻄﻮع‬
‫اﻟﺜﺎﻧﻲ‬ ‫ﻛﺒﻠﺮ‬ ‫ﻗﺎﻧﻮن‬
‫اﻟﺰ‬ ‫اﻻﻧﺪﻓﺎع‬ ‫ﻟﺤﻔﻆ‬ ‫ﻛﻨﺘﯿﺠﺔ‬ ‫اﻟﺜﺎﻧﻲ‬ ‫ﻛﺒﻠﺮ‬ ‫ﻗﺎﻧﻮن‬ ‫ﻧﺮى‬ ‫أن‬ ‫ﻧﺴﺘﻄﯿﻊ‬
‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫ﻛﻤﺎ‬ ‫اھﻠﯿﻠﺠﻲ‬ ‫ﻣﺪار‬ ‫ﻓﻲ‬ ‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫وﯾﺘﺤﺮك‬
‫ﺗﺘﺤﺮك‬ ‫ﻻ‬ ‫اﻟﺸﻤﺲ‬ ‫ﻣﻦ‬ ‫ﯾﺠﻌﻞ‬ ‫اﻟﺬي‬ ‫اﻷﻣﺮ‬ ‫اﻟﻜﻮﻛﺐ‬ ‫ﻛﺘﻠﺔ‬ ‫ﻣﻦ‬ ‫ﺑﻜﺜﯿﺮ‬ ‫أﻛﺒﺮ‬ ‫ھﺎﺋﻠﺔ‬ ‫ﻛﺘﻠﺔ‬
،‫ﻣﺮﻛﺰﯾﺔ‬ ‫ﻗﻮة‬‫اﻟﺸﻤﺲ‬ ‫وﺑﺎﺗﺠﺎه‬ ‫اﻟﻘﻄﺮ‬ ‫ﻧﺼﻒ‬ ‫ﻋﻠﻰ‬ ‫داﺋﻤﺎ‬ ‫ﻣﺤﻤﻮﻟﺔ‬
‫ﻷن‬ ‫وذﻟﻚ‬ ‫ﺻﻔﺮ‬ ‫أﻧﮫ‬ ‫اﻟﻮاﺿﺢ‬ ‫ﻣﻦ‬ ‫اﻟﻤﺮﻛﺰﯾﺔ‬ ‫اﻟﻘﻮة‬
‫اﻟﺸﻜﻞ‬7.13)a(‫ﻗ‬‫ﻋﻠﻰ‬ ‫اﻟﻤﺆﺛﺮة‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻮة‬
‫اﻟﺸﻤﺲ‬ ‫وﺑﺎﺗﺠﺎه‬ ‫اﻟﻜﻮﻛﺐ‬.(b)‫اﻟﻜﻮﻛﺐ‬ ‫ﻣﺪار‬
‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬.‫اﻟﺰﻣﻦ‬ ‫ﺧﻼل‬ ‫ﯾﻤﺴﺢ‬ ‫اﻟﻘﻄﺮ‬ ‫ﻧﺼﻒ‬
‫ﺗﺴﺎوي‬ ‫ﻣﺴﺎﺣﺔ‬‫إﻟﻰ‬‫ﻣﺘﻮازي‬ ‫ﻣﺴﺎﺣﺔ‬ ‫ﻧﺼﻒ‬
‫ﺑﻌﺪاه‬ ‫اﻟﺬي‬ ‫اﻷﺿﻼع‬r‫و‬drv dt..
‫اﻟﺸﻤﺲ‬ ‫وﺑﺎﺗﺠﺎه‬ ‫اﻟﻜﻮﻛﺐ‬
‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬
dt‫ﺗﺴﺎوي‬ ‫ﻣﺴﺎﺣﺔ‬
‫ﺑﻌﺪاه‬ ‫اﻟﺬي‬ ‫اﻷﺿﻼع‬

23
‫ﯾﺴﺎوي‬‫أي‬ ‫اﻟﺰﻣﻦ‬ ‫ﻣﻊ‬ ‫ﻟﻠﻨﻈﺎم‬ ‫اﻻﻧﺪﻓﺎع‬ ‫ﻋﺰم‬ ‫ﺗﻐﯿﺮ‬ ‫ﻣﻌﺪل‬ ‫إﻟﻰ‬:
‫اﻻﻧﺪﻓﺎع‬ ‫ﻋﺰم‬ ‫ﻓﺈن‬L‫اﻟﺤﺮﻛﺔ‬ ‫ﻓﻲ‬ ‫ﺛﺎﺑﺖ‬ ‫ﻟﻠﻜﻮﻛﺐ‬:
‫اﻟﺰﻣﻨﯿﺔ‬ ‫اﻟﻔﺘﺮة‬ ‫ﻓﻲ‬ ،‫اﻟﺘﺎﻟﻲ‬ ‫اﻟﮭﻨﺪﺳﻲ‬ ‫اﻻﻋﺘﺒﺎر‬ ‫إﻟﻰ‬ ‫اﻟﻨﺘﯿﺠﺔ‬ ‫ھﺬه‬ ‫ﻧﺮد‬ ‫أن‬ ‫ﺑﻮﺳﻌﻨﺎ‬dt‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫اﻟﻘﻄﺮ‬ ‫ﻧﺼﻒ‬ ‫ﯾﻤﺴﺢ‬7.13b.
‫اﻷﺿﻼع‬ ‫ﻣﺘﻮازي‬ ‫ﻣﺴﺎﺣﺔ‬ ‫ﻧﺼﻒ‬│r × d r│‫ﺑﻌﺪاه‬ ‫اﻟﺬي‬r‫و‬dr.‫ﺧﻼل‬ ‫اﻟﻜﻮﻛﺐ‬ ‫إزاﺣﺔ‬ ‫أن‬ ‫وﺑﻤﺎ‬
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‫ﻟﻠﻨﻈﺎم‬ ‫اﻟﺨﺎرﺟﻲ‬ ‫اﻟﺪوران‬ ‫ﻋﺰم‬ ‫ﺑﺄن‬ ‫ﻧﺘﺬﻛﺮ‬‫ﯾﺴﺎوي‬
τ = dL/dt‫و‬ ،‫أن‬ ‫ﺑﻤﺎ‬τ = 0‫اﻻﻧﺪﻓﺎع‬ ‫ﻋﺰم‬ ‫ﻓﺈن‬
‫اﻟﺰﻣﻨﯿﺔ‬ ‫اﻟﻔﺘﺮة‬ ‫ﻓﻲ‬ ،‫اﻟﺘﺎﻟﻲ‬ ‫اﻟﮭﻨﺪﺳﻲ‬ ‫اﻻﻋﺘﺒﺎر‬ ‫إﻟﻰ‬ ‫اﻟﻨﺘﯿﺠﺔ‬ ‫ھﺬه‬ ‫ﻧﺮد‬ ‫أن‬ ‫ﺑﻮﺳﻌﻨﺎ‬
‫ﻣﺴﺎﺣﺔ‬dA‫واﻟﺬي‬ ،‫ﯾﺴﺎوي‬‫اﻷﺿﻼع‬ ‫ﻣﺘﻮازي‬ ‫ﻣﺴﺎﺣﺔ‬ ‫ﻧﺼﻒ‬
‫اﻟﺰﻣﻨﯿﺔ‬ ‫اﻟﻔﺘﺮة‬dt‫ﺑﺎﻟﻌﻼﻗﺔ‬ ‫ﺗﻌﻄﻰ‬:dr = v dt
‫إذا‬‫ﻟﺪﯾﻨﺎ‬ ‫أﺻﺒﺢ‬:

24
‫اﻟﺸﻤﺲ‬ ‫ﺑﯿﻦ‬ ‫اﻟﻘﻄﺮ‬ ‫ﻧﺼﻒ‬ ‫أن‬ ‫ﻧﺴﺘﻨﺘﺞ‬ ‫ھﻜﺬا‬‫و‬‫ﻣ‬ ‫ﻣﺴﺎﺣﺎت‬ ‫ﯾﻤﺴﺢ‬ ‫اﻟﻜﻮﻛﺐ‬‫ﺘﺴﺎوﯾ‬‫أزﻣﻨﺔ‬ ‫ﻓﻲ‬ ‫ﺔ‬
‫ﻣﺮﻛﺰﯾﺔ‬ ‫ﻗﻮة‬ ‫ھﻲ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫أن‬ ‫ﻟﺤﻘﯿﻘﺔ‬ ‫ﻧﺘﯿﺠﺔ‬ ‫ھﻲ‬ ‫اﻟﻨﺘﯿﺠﺔ‬ ‫ھﺬه‬ ‫أن‬ ‫إدراك‬ ‫اﻟﻤﮭﻢ‬ ‫وﻣﻦ‬‫و‬‫اﻻﻧﺪﻓﺎع‬ ‫أن‬ ‫ﯾﻌﻨﻲ‬ ‫ﺑﺪوره‬ ‫ھﺬا‬
‫ﻋﻠﻰ‬ ‫ﯾﻨﻄﺒﻖ‬ ‫اﻟﻘﺎﻧﻮن‬ ‫ﻓﺈن‬ ‫ﺑﺎﻟﺘﺎﻟﻲ‬‫أي‬‫ﻣﺮﻛﺰﯾﺔ‬ ‫ﻗﻮة‬ ‫ﺗﺘﻀﻤﻦ‬ ‫ﺣﺎﻟﺔ‬‫ﻻ‬ ‫أم‬ ‫ﻋﻜﺴﻲ‬ ‫ﺗﺮﺑﯿﻊ‬ ‫ﻛﺎﻧﺖ‬ ‫ﺳﻮاء‬ ،.
‫ا‬ ‫ﻗﺎﻧﻮن‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬ ‫ﺑﮫ‬ ‫اﻟﺘﻨﺒﺆ‬ ‫ﯾﻤﻜﻦ‬ ‫اﻟﺜﺎﻟﺚ‬ ‫ﻛﺒﻠﺮ‬ ‫ﻗﺎﻧﻮن‬ ‫ﺑﺄن‬ ‫ﻧﺮى‬ ‫أن‬ ‫اﻟﻤﻔﯿﺪ‬ ‫ﻣﻦ‬‫اﻟﺪاﺋﺮﯾﺔ‬ ‫ﻟﻠﻤﺪارات‬ ‫اﻟﻌﻜﺴﻲ‬ ‫ﻟﺘﺮﺑﯿﻊ‬.‫إﻟﻰ‬ ‫ﺑﺎﻟﻨﻈﺮ‬
‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫اﻟﺤﺮﻛﺔ‬ ‫ﺑﮫ‬ ‫ﯾﻔﺘﺮض‬)‫ﻛﺘﻠﺘﮭﺎ‬MS(‫اﻟﺸﻜﻞ‬ ‫ﻓﻲ‬ ‫ﻛﻤﺎ‬ ،‫داﺋﺮي‬ ‫ﻣﺪار‬ ‫ﻓﻲ‬8.13.‫ﻗﻮة‬ ‫أن‬ ‫وﺑﻤﺎ‬
‫ﻣﺪاره‬ ‫ﻓﻲ‬ ‫اﻟﻤﺘﺤﺮك‬ ‫ﻟﻠﻜﻮﻛﺐ‬ ‫ﻣﺮﻛﺰﯾﺎ‬ ‫ﺗﺴﺎرﻋﺎ‬ ‫ﺗﻮﻟﺪ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬،‫ﻣﻨﺘﻈﻤﺔ‬ ‫داﺋﺮﯾﺔ‬ ‫ﺣﺮﻛﺔ‬ ‫ﻓﻲ‬ ‫ﻟﺠﺴﯿﻢ‬ ‫اﻟﺜﺎﻧﻲ‬ ‫ﻧﯿﻮﺗﻦ‬ ‫ﻗﺎﻧﻮن‬ ‫ﻧﺴﺘﺨﺪم‬.
T‫اﻟﺪور‬ ‫ھﻲ‬‫ﺗﺼﺒﺢ‬ ‫اﻟﺴﺎﺑﻘﺔ‬ ‫ﻓﺎﻟﻤﻌﺎدﻟﺔ‬ ‫ﺑﺎﻟﺘﺎﻟﻲ‬ ،:
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‫ﻣﻦ‬ ‫ﻛﻞ‬ ‫أن‬ ‫ﺣﯿﺚ‬L‫و‬MP‫ﺛﺎﺑﺘﯿﻦ‬،‫و‬‫اﻟﺸﻤﺲ‬ ‫ﺑﯿﻦ‬ ‫اﻟﻘﻄﺮ‬ ‫ﻧﺼﻒ‬ ‫أن‬ ‫ﻧﺴﺘﻨﺘﺞ‬ ‫ھﻜﺬا‬
‫ﻣ‬‫ﺘﺴﺎوﯾ‬‫ﺔ‬.‫ﻣﺮﻛﺰﯾﺔ‬ ‫ﻗﻮة‬ ‫ھﻲ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬ ‫ﻗﻮة‬ ‫أن‬ ‫ﻟﺤﻘﯿﻘﺔ‬ ‫ﻧﺘﯿﺠﺔ‬ ‫ھﻲ‬ ‫اﻟﻨﺘﯿﺠﺔ‬ ‫ھﺬه‬ ‫أن‬ ‫إدراك‬ ‫اﻟﻤﮭﻢ‬ ‫وﻣﻦ‬
‫اﻟﺰاوي‬‫ﺛﺎﺑﺖ‬ ‫ﻟﻠﻜﻮﻛﺐ‬‫و‬‫ﻋﻠﻰ‬ ‫ﯾﻨﻄﺒﻖ‬ ‫اﻟﻘﺎﻧﻮن‬ ‫ﻓﺈن‬ ‫ﺑﺎﻟﺘﺎﻟﻲ‬
‫اﻟﺜﺎﻟﺚ‬ ‫ﻛﺒﻠﺮ‬ ‫ﻗﺎﻧﻮن‬
‫ا‬ ‫ﻗﺎﻧﻮن‬ ‫ﺧﻼل‬ ‫ﻣﻦ‬ ‫ﺑﮫ‬ ‫اﻟﺘﻨﺒﺆ‬ ‫ﯾﻤﻜﻦ‬ ‫اﻟﺜﺎﻟﺚ‬ ‫ﻛﺒﻠﺮ‬ ‫ﻗﺎﻧﻮن‬ ‫ﺑﺄن‬ ‫ﻧﺮى‬ ‫أن‬ ‫اﻟﻤﻔﯿﺪ‬ ‫ﻣﻦ‬
‫ﻛﺘﻠﺘﮫ‬ ‫ﻛﻮﻛﺐ‬MP‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫اﻟﺤﺮﻛﺔ‬ ‫ﺑﮫ‬ ‫ﯾﻔﺘﺮض‬
‫ﻣﺪاره‬ ‫ﻓﻲ‬ ‫اﻟﻤﺘﺤﺮك‬ ‫ﻟﻠﻜﻮﻛﺐ‬ ‫ﻣﺮﻛﺰﯾﺎ‬ ‫ﺗﺴﺎرﻋﺎ‬ ‫ﺗﻮﻟﺪ‬ ‫اﻟﺠﺎذﺑﯿﺔ‬
‫ھﻲ‬ ‫ﻟﻠﻜﻮﻛﺐ‬ ‫اﻟﻤﺪارﯾﺔ‬ ‫اﻟﺴﺮﻋﺔ‬2πr/T‫ﺣﯿﺚ‬ ،T
‫ﺣﯿﺚ‬KS‫ﺑــ‬ ‫ﯾﻌﻄﻰ‬ ‫ﺛﺎﺑﺖ‬:4π2
/GMS=KS
‫اﻟﺸﻜﻞ‬8.13‫ﻛﺘﻠﺘﮫ‬ ‫ﻛﻮﻛﺐ‬MP‫ﯾﺘﺤﺮك‬
‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫داﺋﺮي‬ ‫ﻣﺪار‬ ‫ﻓﻲ‬.‫ان‬
‫ﻋﻄﺎرد‬ ‫ﻋﺪا‬ ‫اﻟﻜﻮاﻛﺐ‬ ‫ﺟﻤﯿﻊ‬ ‫ﻣﺪارات‬
‫داﺋﺮﯾﺔ‬ ‫ﺗﻘﺮﯾﺒﺎ‬ ‫وﺑﻠﻮﺗﻮ‬.
‫اﻟﺸﻤﺲ‬ ‫ﺣﻮل‬ ‫داﺋﺮي‬ ‫ﻣﺪار‬ ‫ﻓﻲ‬
‫ﻋﻄﺎرد‬ ‫ﻋﺪا‬ ‫اﻟﻜﻮاﻛﺐ‬ ‫ﺟﻤﯿﻊ‬ ‫ﻣﺪارات‬

25
‫اﻷﺧﯿﺮ‬ ‫اﻟﻌﻤﻮد‬ ‫ﯾﺤﻘﻖ‬ ،‫ﻣﻔﯿﺪة‬ ‫ﻛﻮﻛﺒﯿﺔ‬ ‫ﻣﻌﻠﻮﻣﺎت‬ ‫ﻣﺠﻤﻮﻋﺔ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬T2
/r3
‫ﺛﺎﺑﺘﺔ‬ ‫ﻧﺴﺒﺔ‬.‫اﻟﺼﻐﯿﺮة‬ ‫اﻻﺧﺘﻼﻓﺎت‬ ‫إن‬
‫ﻟﻠﻜﻮاﻛﺐ‬ ‫اﻟﺮﺋﯿﺴﻲ‬ ‫اﻟﻤﺤﻮر‬ ‫وﻧﺼﻒ‬ ‫اﻟﺪور‬ ‫ﻋﻦ‬ ‫اﻟﻤﻘﺎﺳﺔ‬ ‫اﻟﻤﻌﻠﻮﻣﺎت‬ ‫ﻓﻲ‬ ‫اﻟﺘﺸﻜﯿﻚ‬ ‫إﻟﻰ‬ ‫أدت‬ ‫اﻟﻌﻤﻮد‬ ‫ھﺬا‬ ‫ﻓﻲ‬.
‫اﺳﺘﺒﺪﻟﻨﺎ‬ ‫إذا‬ ‫اﻻھﻠﯿﻠﺠﯿﺔ‬ ‫اﻟﻤﺪارات‬ ‫أﺟﻞ‬ ‫ﻣﻦ‬ ‫أﯾﻀﺎ‬ ‫ﺻﺤﯿﺤﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ھﺬه‬r‫ﺑﺎﻟﻤﺴﺎﻓﺔ‬a)‫اﻟﺮﺋﯿﺴﻲ‬ ‫اﻟﻤﺤﻮر‬ ‫ﻧﺼﻒ‬(‫اﻟﺸﻜﻞ‬5.13
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‫اﻟﺠﺪول‬2.13‫اﻷﺧﯿﺮ‬ ‫اﻟﻌﻤﻮد‬ ‫ﯾﺤﻘﻖ‬ ،‫ﻣﻔﯿﺪة‬ ‫ﻛﻮﻛﺒﯿﺔ‬ ‫ﻣﻌﻠﻮﻣﺎت‬ ‫ﻣﺠﻤﻮﻋﺔ‬ ‫ﻋﻦ‬ ‫ﻋﺒﺎرة‬
‫ﻟﻠﻜﻮاﻛﺐ‬ ‫اﻟﺮﺋﯿﺴﻲ‬ ‫اﻟﻤﺤﻮر‬ ‫وﻧﺼﻒ‬ ‫اﻟﺪور‬ ‫ﻋﻦ‬ ‫اﻟﻤﻘﺎﺳﺔ‬ ‫اﻟﻤﻌﻠﻮﻣﺎت‬ ‫ﻓﻲ‬ ‫اﻟﺘﺸﻜﯿﻚ‬ ‫إﻟﻰ‬ ‫أدت‬ ‫اﻟﻌﻤﻮد‬ ‫ھﺬا‬ ‫ﻓﻲ‬
‫اﺳﺘﺒﺪﻟﻨﺎ‬ ‫إذا‬ ‫اﻻھﻠﯿﻠﺠﯿﺔ‬ ‫اﻟﻤﺪارات‬ ‫أﺟﻞ‬ ‫ﻣﻦ‬ ‫أﯾﻀﺎ‬ ‫ﺻﺤﯿﺤﺔ‬ ‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ھﺬه‬

Ch13 universal gravitation

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Ch13 universal gravitation