Ch15 oscillatory motion

‫ﻋﺸﺭ‬ ‫ﺍﻟﺨﺎﻤﺱ‬ ‫ﺍﻟﺠﺯﺀ‬:‫ﺍﻹﻫﺘﺯﺍﺯﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬
2
‫ﺘﻭﺜﻴــﻕ‬
‫اﻟﺒﯿﺎن‬
Physics for Scientists and Engineers
By
Raymond A. Serway & John W. Jewett
6th Edition
‫اﻟﻮﺣﺪة‬‫اﻟﺜﺎﻧﯿﺔ‬:‫واﻷﻣﻮاج‬ ‫اﻻھﺘﺰازات‬‫اﻟﻤﯿﻜﺎﻧﯿﻜﯿﺔ‬
‫ﻋﺸﺮ‬ ‫اﻟﺨﺎﻣﺲ‬ ‫اﻟﺠﺰء‬:‫اﻻھﺘﺰازﯾﺔ‬ ‫اﻟﺤﺮﻛﺔ‬
‫د‬/.‫ﺳﻜﯿﻚ‬ ‫ﻓﻼح‬ ‫ﺣﺎزم‬
‫ﻓﯿﺰﯾﺎء‬
‫ﻛﺘﺎب‬
25-12-2008
‫ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ‬ ‫ﻭﺍﻷﻤﻭﺍﺝ‬ ‫ﺍﻻﻫﺘﺯﺍﺯﺍﺕ‬
‫د‬.‫ﺳﻛﯾك‬ ‫ﻓﻼح‬ ‫ﺣﺎزم‬
www.trgma.com
‫م‬‫اﻟﺒﻨﺪ‬
1‫اﻟﻤﺼﺪر‬Physics for Scientists and Engineers
Raymond A. Serway & John W. Jewett
2‫اﻟﻤﻮﺿﻮع‬
3‫اﻟﻤﺘﺮﺟﻢ‬
4‫اﻟﻤ‬‫اﻟﻌﻠﻤﯿﺔ‬ ‫ﺮاﺟﻌﺔ‬
5‫اﻟﻠﻐﻮﯾﺔ‬ ‫اﻟﻤﺮاﺟﻌﺔ‬
6‫اﻟﺘﺼﻨﯿﻒ‬
7‫اﻟﻔﺌﺔ‬
11‫اﻟﺘﺎرﯾﺦ‬

3
‫ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ‬ ‫ﻭﺍﻻﻤﻭﺍﺝ‬ ‫ﺍﻻﻫﺘﺯﺍﺯﺍﺕ‬
Oscillations and Mechanical Waves
‫ﺍﻟﺩﻭﺭﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻴﺴﻤﻰ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻤﻥ‬ ‫ﺨﺎﺹ‬ ‫ﻨﻭﻉ‬ ‫ﺒﺩﺭﺍﺴﺔ‬ ‫ﺍﻟﻭﺤﺩﺓ‬ ‫ﻫﺫﻩ‬ ‫ﻓﻲ‬ ‫ﺴﻨﺒﺩﺃ‬periodic motion.‫ﺤﺭﻜﺔ‬ ‫ﻫﺫﻩ‬
‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻥ‬ ‫ﻤﺤﺩﺩﺓ‬ ‫ﻓﺘﺭﺓ‬ ‫ﺒﻌﺩ‬ ‫ﻤﺭﺓ‬ ‫ﻜل‬ ‫ﻓﻲ‬ ‫ﻤﺤﺩﺩ‬ ‫ﻤﻭﻀﻊ‬ ‫ﺇﻟﻰ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻓﻴﻬﺎ‬ ‫ﻴﻌﻭﺩ‬ ‫ﻟﻠﺠﺴﻡ‬ ‫ﺭﺍﺭﻴﺔ‬.‫ﺘﺘﺤﺭﻙ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻷﺠﺴﺎﻡ‬ ‫ﻭﻤﻥ‬
‫ﺯﻨﺒﺭﻙ‬ ‫ﻓﻲ‬ ‫ﺍﻟﻤﻌﻠﻘﺔ‬ ‫ﻭﺍﻟﻜﺘﻠﺔ‬ ‫ﺍﻟﺒﻨﺩﻭل‬ ‫ﻫﻭ‬ ‫ﺍﻟﻁﺭﻴﻘﺔ‬ ‫ﺒﻬﺫﻩ‬ ‫ﺩﻭﺭﻴﺔ‬ ‫ﺤﺭﻜﺔ‬.‫ﺘﺴﻤﻰ‬ ‫ﺍﻷﺠﺴﺎﻡ‬ ‫ﻟﻬﺫﻩ‬ ‫ﻭﺍﻹﻴﺎﺏ‬ ‫ﺍﻟﺫﻫﺎﺏ‬ ‫ﺤﺭﻜﺔ‬ ‫ﺇﻥ‬
‫ﻋ‬ ‫ﺍﻟﺠﺯﺀ‬ ‫ﻫﺫﺍ‬ ‫ﻓﻲ‬ ‫ﻨﺭﻜﺯ‬ ‫ﻭﺴﻭﻑ‬‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻠﻰ‬simple harmonic
‫ﺍﻟﺩﻭﺭﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻤﻥ‬ ‫ﺨﺎﺼﺔ‬ ‫ﺤﺎﻟﺔ‬ ‫ﻫﻲ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻭﻫﺫﻩ‬.‫ﺃﻥ‬ ‫ﻴﻤﻜﻥ‬ ‫ﺍﻟﺩﻭﺭﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺎﺕ‬ ‫ﻜل‬ ‫ﺍﻥ‬ ‫ﻨﺠﺩ‬ ‫ﺴﻭﻑ‬ ‫ﻜﻤﺎ‬
‫ﺍﻟﺤﺭﻜﺎﺕ‬ ‫ﻟﺩﺭﺍﺴﺔ‬ ‫ﹰ‬‫ﺎ‬‫ﺒﺴﻴﻁ‬ ‫ﻨﻤﻭﺫﺠﺎ‬ ‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺘﻌﺘﺒﺭ‬ ‫ﺤﻴﺙ‬ ‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬ ‫ﻨﺩﺭﺴﻬﺎ‬
‫ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ‬ ‫ﺍﻷﻤﻭﺍﺝ‬ ‫ﻭﺩﺭﺍﺴﺔ‬ ‫ﻟﻔﻬﻡ‬ ‫ﺍﻟﻨﻅﺭﻱ‬ ‫ﺍﻷﺴﺎﺱ‬ ‫ﺘﺸﻜل‬ ‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺔ‬mechanical waves.
‫ﺍﻟﻤﺎﺀ‬ ‫ﻭﺃﻤﻭﺍﺝ‬ ‫ﻤﺸﺩﻭﺩ‬ ‫ﺤﺒل‬ ‫ﻓﻲ‬ ‫ﺍﻟﻤﻨﺘﺸﺭﺓ‬ ‫ﻭﺍﻷﻤﻭﺍﺝ‬ ‫ﺍﻟﺯﻻﺯل‬ ‫ﻭﺃﻤﻭﺍﺝ‬ ‫ﺍﻟﺼﻭﺘﻴﺔ‬ ‫ﺍﻷﻤﻭﺍﺝ‬.‫ﺍﻷﻤﻭﺍﺝ‬ ‫ﻤﻥ‬ ‫ﺍﻷﻨﻭﺍﻉ‬ ‫ﻫﺫﻩ‬ ‫ﻜل‬
‫ﺍﻟ‬ ‫ﻋﺒﺭ‬ ‫ﺘﻨﺘﻘل‬ ‫ﺍﻟﺼﻭﺘﻴﺔ‬ ‫ﺍﻷﻤﻭﺍﺝ‬ ‫ﻓﻤﺜﻼ‬‫ﻬﻭﺍﺀ‬‫ﻤﻥ‬ ‫ﺘﻨﺘﺞ‬‫ﻟﻸﻤﺎﻡ‬ ‫ﺍﻟﻬﻭﺍﺀ‬ ‫ﺠﺯﻴﺌﺎﺕ‬ ‫ﺍﻫﺘﺯﺍﺯ‬
‫ﻭﺍﻷﺴﻔل‬ ‫ﻟﻸﻋﻠﻰ‬ ‫ﺍﻟﻤﺎﺀ‬ ‫ﺠﺯﻴﺌﺎﺕ‬ ‫ﻻﻫﺘﺯﺍﺯ‬ ‫ﺍﻟﺤﺎل‬ ‫ﻭﻜﺫﻟﻙ‬ ،‫ﺍﻟﺼﻭﺘﻴﺔ‬ ‫ﺍﻷﻤﻭﺍﺝ‬ ‫ﺘﺴﺒﺏ‬ ‫ﺍﻫﺘﺯﺍﺯﻴﺔ‬ ‫ﺤﺭﻜﺔ‬‫ﺘ‬‫ﺴﺒﺏ‬
‫ﺭﺍﻜﺩﺓ‬ ‫ﻤﺎﺀ‬ ‫ﺒﺭﻜﺔ‬ ‫ﻓﻲ‬ ‫ﺤﺠﺭ‬ ‫ﻴﺴﻘﻁ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﻨﺭﺍﻫﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻷﻤﻭﺍﺝ‬.‫ﻤﺎﺩﻱ‬ ‫ﻭﺴﻁ‬ ‫ﻋﺒﺭ‬ ‫ﺘﻨﺘﻘل‬ ‫ﺍﻷﻤﻭﺍﺝ‬ ‫ﻓﺎﻥ‬ ‫ﻋﺎﻤﺔ‬ ‫ﻭﺒﺼﻭﺭﺓ‬
www.trgma.com
‫ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ‬ ‫ﻭﺍﻻﻤﻭﺍﺝ‬ ‫ﺍﻻﻫﺘﺯﺍﺯﺍﺕ‬
Oscillations and Mechanical Waves
‫ﺍﻟﺩﻭﺭﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻴﺴﻤﻰ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻤﻥ‬ ‫ﺨﺎﺹ‬ ‫ﻨﻭﻉ‬ ‫ﺒﺩﺭﺍﺴﺔ‬ ‫ﺍﻟﻭﺤﺩﺓ‬ ‫ﻫﺫﻩ‬ ‫ﻓﻲ‬ ‫ﺴﻨﺒﺩﺃ‬
‫ﺘﻜ‬‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻥ‬ ‫ﻤﺤﺩﺩﺓ‬ ‫ﻓﺘﺭﺓ‬ ‫ﺒﻌﺩ‬ ‫ﻤﺭﺓ‬ ‫ﻜل‬ ‫ﻓﻲ‬ ‫ﻤﺤﺩﺩ‬ ‫ﻤﻭﻀﻊ‬ ‫ﺇﻟﻰ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻓﻴﻬﺎ‬ ‫ﻴﻌﻭﺩ‬ ‫ﻟﻠﺠﺴﻡ‬ ‫ﺭﺍﺭﻴﺔ‬
‫ﺯﻨﺒﺭﻙ‬ ‫ﻓﻲ‬ ‫ﺍﻟﻤﻌﻠﻘﺔ‬ ‫ﻭﺍﻟﻜﺘﻠﺔ‬ ‫ﺍﻟﺒﻨﺩﻭل‬ ‫ﻫﻭ‬ ‫ﺍﻟﻁﺭﻴﻘﺔ‬ ‫ﺒﻬﺫﻩ‬ ‫ﺩﻭﺭﻴﺔ‬ ‫ﺤﺭﻜﺔ‬
‫ﺍﻫﺘﺯﺍﺯﻴﺔ‬ ‫ﺤﺭﻜﺔ‬oscillations.‫ﻋ‬ ‫ﺍﻟﺠﺯﺀ‬ ‫ﻫﺫﺍ‬ ‫ﻓﻲ‬ ‫ﻨﺭﻜﺯ‬ ‫ﻭﺴﻭﻑ‬
motion‫ﺍﻟﺩﻭﺭﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻤﻥ‬ ‫ﺨﺎﺼﺔ‬ ‫ﺤﺎﻟﺔ‬ ‫ﻫﻲ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻭﻫﺫﻩ‬
‫ﺍﻟﺤﺭﻜﺎﺕ‬ ‫ﻟﺩﺭﺍﺴﺔ‬ ‫ﹰ‬‫ﺎ‬‫ﺒﺴﻴﻁ‬ ‫ﻨﻤﻭﺫﺠﺎ‬ ‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺘﻌﺘﺒﺭ‬ ‫ﺤﻴﺙ‬ ‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬ ‫ﻨﺩﺭﺴﻬﺎ‬
‫ﺍﻟﺩﻭﺭﻴﺔ‬.
‫ﺍﻟﺤﺭﻜ‬‫ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ‬ ‫ﺍﻷﻤﻭﺍﺝ‬ ‫ﻭﺩﺭﺍﺴﺔ‬ ‫ﻟﻔﻬﻡ‬ ‫ﺍﻟﻨﻅﺭﻱ‬ ‫ﺍﻷﺴﺎﺱ‬ ‫ﺘﺸﻜل‬ ‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺔ‬
‫ﺍﻟﻤﺎﺀ‬ ‫ﻭﺃﻤﻭﺍﺝ‬ ‫ﻤﺸﺩﻭﺩ‬ ‫ﺤﺒل‬ ‫ﻓﻲ‬ ‫ﺍﻟﻤﻨﺘﺸﺭﺓ‬ ‫ﻭﺍﻷﻤﻭﺍﺝ‬ ‫ﺍﻟﺯﻻﺯل‬ ‫ﻭﺃﻤﻭﺍﺝ‬ ‫ﺍﻟﺼﻭﺘﻴﺔ‬ ‫ﺍﻷﻤﻭﺍﺝ‬
‫ﺒﺴﺒﺏ‬ ‫ﺘﺤﺩﺙ‬‫ﺍﻟ‬‫ﺤﺭﻜﺔ‬‫ﺍﻻ‬‫ﻫﺘﺯﺍﺯﻴﺔ‬.‫ﺍﻟ‬ ‫ﻋﺒﺭ‬ ‫ﺘﻨﺘﻘل‬ ‫ﺍﻟﺼﻭﺘﻴﺔ‬ ‫ﺍﻷﻤﻭﺍﺝ‬ ‫ﻓﻤﺜﻼ‬
‫ﻭﺍﻟﺨﻠﻑ‬‫ﻓﻲ‬‫ﻭﺍﻷﺴﻔل‬ ‫ﻟﻸﻋﻠﻰ‬ ‫ﺍﻟﻤﺎﺀ‬ ‫ﺠﺯﻴﺌﺎﺕ‬ ‫ﻻﻫﺘﺯﺍﺯ‬ ‫ﺍﻟﺤﺎل‬ ‫ﻭﻜﺫﻟﻙ‬ ،‫ﺍﻟﺼﻭﺘﻴﺔ‬ ‫ﺍﻷﻤﻭﺍﺝ‬ ‫ﺘﺴﺒﺏ‬ ‫ﺍﻫﺘﺯﺍﺯﻴﺔ‬ ‫ﺤﺭﻜﺔ‬
‫ﺭﺍﻜﺩﺓ‬ ‫ﻤﺎﺀ‬ ‫ﺒﺭﻜﺔ‬ ‫ﻓﻲ‬ ‫ﺤﺠﺭ‬ ‫ﻴﺴﻘﻁ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﻨﺭﺍﻫﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻷﻤﻭﺍﺝ‬

4
‫ﺍﻟﺠﺴﻡ‬ ‫ﺘﺠﻌل‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻁﺭﻴﻘﺔ‬ ‫ﺒﻨﻔﺱ‬ ‫ﺘﺅﺜﺭ‬ ‫ﺍﻟﻭﺴﻁ‬ ‫ﺠﺯﺌﻴﺎﺕ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻓﺎﻥ‬ ‫ﻭﻟﻬﺫﺍ‬
‫ﻭﺍﻷﻤﻭﺍﺝ‬ ‫ﺍﻻﻫﺘﺯﺍﺯﺍﺕ‬ ‫ﻨﻅﺭﻴﺔ‬ ‫ﻓﻬﻡ‬ ‫ﺇﻟﻰ‬ ‫ﻨﺤﺘﺎﺝ‬ ،‫ﺍﻟﻁﺒﻴﻌﻴﺔ‬ ‫ﺍﻟﻅﻭﺍﻫﺭ‬ ‫ﻤﻥ‬ ‫ﺍﻟﻜﺜﻴﺭ‬ ‫ﻟﺸﺭﺡ‬.‫ﺍﻟﺴﺤﺎﺏ‬ ‫ﻨﺎﻁﺤﺎﺕ‬ ‫ﺇﻟﻰ‬ ‫ﻨﻅﺭﻨﺎ‬ ‫ﻟﻭ‬ ‫ﻓﻤﺜﻼ‬
‫ﺍﻟﻤﻌﻤﺎﺭﻴﻴﻥ‬ ‫ﺍﻟﻤﺼﻤﻤﻴﻥ‬ ‫ﻓﺎﻥ‬ ‫ﻭﻟﻬﺫﺍ‬ ،‫ﺍﻫﺘﺯﺍﺯﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﺘﺘﺤﺭﻙ‬ ‫ﺍﻟﺤﻘﻴﻘﺔ‬ ‫ﻓﻲ‬ ‫ﻫﻲ‬ ‫ﻭﻟﻜﻥ‬
‫ﺍﻟﺘﺼﻤﻴﻡ‬ ‫ﻋﻨﺩ‬ ‫ﺍﻟﺤﺴﺒﺎﻥ‬ ‫ﻓﻲ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻫﺫﻩ‬ ‫ﻴﺄﺨﺫﻭﺍ‬ ‫ﻭﺍﻟﻤﻬﻨﺩﺴﻴﻥ‬.‫ﻨﻌﻠﻡ‬ ‫ﺃﻥ‬ ‫ﻴﺠﺏ‬ ‫ﻓﺎﻨﻪ‬ ‫ﻭﺍﻟﺘﻠﻔﺯﻴﻭﻥ‬ ‫ﺍﻟﺭﺍﺩﻴﻭ‬ ‫ﻴﻌﻤل‬ ‫ﻜﻴﻑ‬ ‫ﻭﻟﻔﻬﻡ‬
‫ﺍﻟﻔﺭﺍﻍ‬ ‫ﻓﻲ‬ ‫ﺘﻨﺘﺸﺭ‬ ‫ﻭﻜﻴﻑ‬ ‫ﺘﻨﺸﺄ‬ ‫ﻭﻜﻴﻑ‬ ‫ﺃﺴﺎﺴﻬﺎ‬ ‫ﻫﻭ‬ ‫ﻭﻤﺎ‬ ‫ﺍﻟﻜﻬﺭﻭﻤﻐﻨﺎﻁﻴﺴﻴﺔ‬ ‫ﺍﻷﻤﻭﺍﺝ‬ ‫ﺘﻌﻤل‬ ‫ﻜﻴﻑ‬.‫ﻭﺃ‬‫ﻤﻤﺎ‬ ‫ﺍﻟﻜﺜﻴﺭ‬ ‫ﻓﺎﻥ‬ ‫ﺨﻴﺭﺍ‬
‫ﺃﻤﻭﺍﺝ‬ ‫ﺼﻭﺭﺓ‬ ‫ﻓﻲ‬ ‫ﻋﻠﻴﻬﺎ‬ ‫ﺤﺼﻠﻭﺍ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻤﻌﻠﻭﻤﺎﺕ‬ ‫ﺃﺴﺎﺴﻪ‬ ‫ﻜﺎﻥ‬ ‫ﺍﻟﺫﺭﻱ‬ ‫ﺍﻟﺘﺭﻜﻴﺏ‬ ‫ﻋﻥ‬ ‫ﺍﻟﻌﻠﻤﺎﺀ‬ ‫ﻋﺭﻓﻪ‬.‫ﺩﺭﺍﺴﺔ‬ ‫ﻓﺎﻥ‬ ‫ﻭﻟﻬﺫﺍ‬
‫ﺍﻟﺫﺭﻴﺔ‬ ‫ﺍﻟﻔﻴﺯﻴﺎﺀ‬ ‫ﻨﻅﺭﻴﺎﺕ‬ ‫ﺫﻟﻙ‬ ‫ﻓﻲ‬ ‫ﺒﻤﺎ‬ ‫ﺍﻟﻔﻴﺯﻴﺎﺌﻴﺔ‬ ‫ﺍﻟﻅﻭﺍﻫﺭ‬ ‫ﻤﻥ‬ ‫ﺍﻟﻜﺜﻴﺭ‬ ‫ﻟﻔﻬﻡ‬ ‫ﻀﺭﻭﺭﻴﺔ‬ ‫ﻭﺍﻷﻤﻭﺍﺝ‬ ‫ﺍﻻﻫﺘﺯﺍﺯﺍﺕ‬ ‫ﻓﻴﺯﻴﺎﺀ‬.
‫ﻓ‬ ‫ﺸﺠﺭ‬ ‫ﻭﺭﻗﺔ‬ ‫ﻤﻥ‬ ‫ﺘﺴﻘﻁ‬ ‫ﺍﻟﻤﺎﺀ‬ ‫ﻤﻥ‬ ‫ﻗﻁﺭﺍﺕ‬‫ﻲ‬‫ﺤ‬‫ﻤﺎﺀ‬ ‫ﻭﺽ‬.‫ﺤﺭﻜﺔ‬ ‫ﻴﺴﺒﺏ‬ ‫ﺍﻟﻤﺎﺀ‬ ‫ﻗﻁﺭﺍﺕ‬ ‫ﺴﻘﻭﻁ‬ ‫ﻋﻥ‬ ‫ﺍﻟﻨﺎﺘﺞ‬ ‫ﺍﻻﻀﻁﺭﺍﺏ‬
‫ﺍﻟﻤﺎﺀ‬ ‫ﺴﻘﻭﻁ‬ ‫ﻨﻘﻁﺔ‬ ‫ﻤﻥ‬ ‫ﺍﻟﻤﺒﺘﻌﺩﺓ‬ ‫ﺍﻷﻤﻭﺍﺝ‬ ‫ﻤﻊ‬ ‫ﻤﺭﺘﺒﻁﺔ‬ ‫ﺍﻻﻫﺘﺯﺍﺯﺍﺕ‬ ‫ﻫﺫﻩ‬.‫ﻨﺩﺭﺴﻪ‬ ‫ﺴﻭﻑ‬ ‫ﻤﺎ‬ ‫ﻭﻫﺫﺍ‬
‫ﻭﺍﻷﻤﻭﺍﺝ‬ ‫ﺒﺎﻻﻫﺘﺯﺍﺯﺍﺕ‬ ‫ﺍﻟﻤﺭﺘﺒﻁﺔ‬ ‫ﺍﻟﻔﻴﺯﻴﺎﺀ‬ ‫ﻟﻨﻜﺘﺸﻑ‬ ‫ﺍﻟﺜﺎﻨﻴﺔ‬ ‫ﺍﻟﻭﺤﺩﺓ‬ ‫ﻓﻲ‬.
www.trgma.com
‫ﺤﺭﻜﺔ‬ ‫ﺘﺘﺤﺭﻙ‬ ‫ﺠﺯﺌﻴﺎﺘﻪ‬‫ﺩﻭﺭﻴﺔ‬ ‫ﺍﻫﺘﺯﺍﺯﻴﺔ‬.‫ﺍﻟﺠﺴﻡ‬ ‫ﺘﺠﻌل‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻁﺭﻴﻘﺔ‬ ‫ﺒﻨﻔﺱ‬ ‫ﺘﺅﺜﺭ‬ ‫ﺍﻟﻭﺴﻁ‬ ‫ﺠﺯﺌﻴﺎﺕ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻓﺎﻥ‬ ‫ﻭﻟﻬﺫﺍ‬
‫ﺍﻫﺘﺯﺍﺯﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻴﺘﺤﺭﻙ‬ ‫ﺯﻨﺒﺭﻙ‬ ‫ﻓﻲ‬ ‫ﺍﻟﻤﻌﻠﻕ‬.
‫ﻭﺍﻷﻤﻭﺍﺝ‬ ‫ﺍﻻﻫﺘﺯﺍﺯﺍﺕ‬ ‫ﻨﻅﺭﻴﺔ‬ ‫ﻓﻬﻡ‬ ‫ﺇﻟﻰ‬ ‫ﻨﺤﺘﺎﺝ‬ ،‫ﺍﻟﻁﺒﻴﻌﻴﺔ‬ ‫ﺍﻟﻅﻭﺍﻫﺭ‬ ‫ﻤﻥ‬ ‫ﺍﻟﻜﺜﻴﺭ‬ ‫ﻟﺸﺭﺡ‬
‫ﺴﺎﻜﻨﺔ‬ ‫ﺇﻨﻬﺎ‬ ‫ﻨﻌﺘﻘﺩ‬ ‫ﻗﺩ‬ ‫ﺍﻟﺠﺴﻭﺭ‬ ‫ﻭﺍﻟﻰ‬‫ﺍﻟﻤﻌﻤﺎﺭﻴﻴﻥ‬ ‫ﺍﻟﻤﺼﻤﻤﻴﻥ‬ ‫ﻓﺎﻥ‬ ‫ﻭﻟﻬﺫﺍ‬ ،‫ﺍﻫﺘﺯﺍﺯﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﺘﺘﺤﺭﻙ‬ ‫ﺍﻟﺤﻘﻴﻘﺔ‬ ‫ﻓﻲ‬ ‫ﻫﻲ‬ ‫ﻭﻟﻜﻥ‬
‫ﺍﻟﺘﺼﻤﻴﻡ‬ ‫ﻋﻨﺩ‬ ‫ﺍﻟﺤﺴﺒﺎﻥ‬ ‫ﻓﻲ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻫﺫﻩ‬ ‫ﻴﺄﺨﺫﻭﺍ‬ ‫ﻭﺍﻟﻤﻬﻨﺩﺴﻴﻥ‬
‫ﺍﻟﻔﺭﺍﻍ‬ ‫ﻓﻲ‬ ‫ﺘﻨﺘﺸﺭ‬ ‫ﻭﻜﻴﻑ‬ ‫ﺘﻨﺸﺄ‬ ‫ﻭﻜﻴﻑ‬ ‫ﺃﺴﺎﺴﻬﺎ‬ ‫ﻫﻭ‬ ‫ﻭﻤﺎ‬ ‫ﺍﻟﻜﻬﺭﻭﻤﻐﻨﺎﻁﻴﺴﻴﺔ‬ ‫ﺍﻷﻤﻭﺍﺝ‬ ‫ﺘﻌﻤل‬ ‫ﻜﻴﻑ‬
‫ﺃﻤﻭﺍﺝ‬ ‫ﺼﻭﺭﺓ‬ ‫ﻓﻲ‬ ‫ﻋﻠﻴﻬﺎ‬ ‫ﺤﺼﻠﻭﺍ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻤﻌﻠﻭﻤﺎﺕ‬ ‫ﺃﺴﺎﺴﻪ‬ ‫ﻜﺎﻥ‬ ‫ﺍﻟﺫﺭﻱ‬ ‫ﺍﻟﺘﺭﻜﻴﺏ‬ ‫ﻋﻥ‬ ‫ﺍﻟﻌﻠﻤﺎﺀ‬ ‫ﻋﺭﻓﻪ‬
‫ﺍﻟﺫﺭﻴﺔ‬ ‫ﺍﻟﻔﻴﺯﻴﺎﺀ‬ ‫ﻨﻅﺭﻴﺎﺕ‬ ‫ﺫﻟﻙ‬ ‫ﻓﻲ‬ ‫ﺒﻤﺎ‬ ‫ﺍﻟﻔﻴﺯﻴﺎﺌﻴﺔ‬ ‫ﺍﻟﻅﻭﺍﻫﺭ‬ ‫ﻤﻥ‬ ‫ﺍﻟﻜﺜﻴﺭ‬ ‫ﻟﻔﻬﻡ‬ ‫ﻀﺭﻭﺭﻴﺔ‬ ‫ﻭﺍﻷﻤﻭﺍﺝ‬ ‫ﺍﻻﻫﺘﺯﺍﺯﺍﺕ‬ ‫ﻓﻴﺯﻴﺎﺀ‬
‫ﻓ‬ ‫ﺸﺠﺭ‬ ‫ﻭﺭﻗﺔ‬ ‫ﻤﻥ‬ ‫ﺘﺴﻘﻁ‬ ‫ﺍﻟﻤﺎﺀ‬ ‫ﻤﻥ‬ ‫ﻗﻁﺭﺍﺕ‬
‫ﺍﻟﻤﺎﺀ‬ ‫ﻟﺴﻁﺢ‬ ‫ﺍﻫﺘﺯﺍﺯﻴﺔ‬.‫ﺍﻟﻤﺎﺀ‬ ‫ﺴﻘﻭﻁ‬ ‫ﻨﻘﻁﺔ‬ ‫ﻤﻥ‬ ‫ﺍﻟﻤﺒﺘﻌﺩﺓ‬ ‫ﺍﻷﻤﻭﺍﺝ‬ ‫ﻤﻊ‬ ‫ﻤﺭﺘﺒﻁﺔ‬ ‫ﺍﻻﻫﺘﺯﺍﺯﺍﺕ‬ ‫ﻫﺫﻩ‬
‫ﻭﺍﻷﻤﻭﺍﺝ‬ ‫ﺒﺎﻻﻫﺘﺯﺍﺯﺍﺕ‬ ‫ﺍﻟﻤﺭﺘﺒﻁﺔ‬ ‫ﺍﻟﻔﻴﺯﻴﺎﺀ‬ ‫ﻟﻨﻜﺘﺸﻑ‬ ‫ﺍﻟﺜﺎﻨﻴﺔ‬ ‫ﺍﻟﻭﺤﺩﺓ‬ ‫ﻓﻲ‬

5
:‫ﻭﺍ‬ ‫ﺍﻻﻫﺘﺯﺍﺍﺕ‬‫ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ‬ ‫ﻷﻤﻭﺍﺝ‬
‫ﻋﺸﺭ‬ ‫ﺍﻟﺨﺎﻤﺱ‬ ‫ﺍﻟﺠﺯﺀ‬:‫ﺍﻻﻫﺘﺯﺍﺯﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬
Motion of an Object Attached to a
Spring
‫ﺯﻨﺒﺭﻙ‬ ‫ﻓﻲ‬ ‫ﻤﻌﻠﻘﺔ‬ ‫ﺠﺴﻡ‬ ‫ﺤﺭﻜﺔ‬
Mathematical Representation of Simple
Harmonic Motion
‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺍﻟﺭﻴﺎﻀﻲ‬ ‫ﺍﻟﺘﻤﺜﻴل‬
Energy of the Simple Harmonic
Oscillator
‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﺍﻟﻤﺘﺤﺭﻙ‬ ‫ﻟﻠﺠﺴﻡ‬ ‫ﺍﻟﻁﺎﻗﺔ‬
Comparing Simple Harmonic Motion
with Uniform Circular Motion
‫ﻭﺍﻟﺤﺭﻜﺔ‬ ‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺒﻴﻥ‬ ‫ﻤﻘﺎﺭﻨﺔ‬
The Pendulum
Damped Oscillations
Forced Oscillations
www.trgma.com
‫ﺍﻟﺜﺎﻨﻴﺔ‬ ‫ﺍﻟﻭﺤﺩﺓ‬:
‫ﻋﺸﺭ‬ ‫ﺍﻟﺨﺎﻤﺱ‬ ‫ﺍﻟﺠﺯﺀ‬
‫ﺯﻨﺒﺭﻙ‬ ‫ﻓﻲ‬ ‫ﻤﻌﻠﻘﺔ‬ ‫ﺠﺴﻡ‬ ‫ﺤﺭﻜﺔ‬
‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺍﻟﺭﻴﺎﻀﻲ‬ ‫ﺍﻟﺘﻤﺜﻴل‬
‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﺍﻟﻤﺘﺤﺭﻙ‬ ‫ﻟﻠﺠﺴﻡ‬ ‫ﺍﻟﻁﺎﻗﺔ‬
‫ﻭﺍﻟﺤﺭﻜﺔ‬ ‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺒﻴﻥ‬ ‫ﻤﻘﺎﺭﻨﺔ‬
‫ﺍﻟﻤﻨﺘﻅﻤﺔ‬ ‫ﺍﻟﺩﺍﺌﺭﻴﺔ‬
‫ﺍﻟﺒﻨﺩﻭل‬
‫ﺍﻟﻤﺨﻤﺩﺓ‬ ‫ﺍﻟﺤﺭﻜﺔ‬
‫ﺍﻟ‬ ‫ﺍﻟﺤﺭﻜﺔ‬‫ﻘﺴﺭﻴﺔ‬

6
 ‫ﺜﺎﺒﺘﺔ‬ ‫ﺯﻤﻨﻴﺔ‬ ‫ﻓﺘﺭﺓ‬ ‫ﺒﻌﺩ‬ ‫ﻤﺤﺩﺩ‬ ‫ﻤﻭﻗﻊ‬ ‫ﺇﻟﻰ‬ ‫ﺍﻟﺠﺴﻡ‬.‫ﺍﻟﺸﻤﺱ‬ ‫ﺤﻭل‬ ‫ﺍﻷﺭﺽ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻓﻤﺜﻼ‬
‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻥ‬ ‫ﻤﺤﺩﺩﺓ‬ ‫ﻓﺘﺭﺓ‬ ‫ﻜل‬ ‫ﻤﺤﺩﺩ‬ ‫ﻤﻭﻗﻊ‬ ‫ﺇﻟﻰ‬ ‫ﺘﻌﻭﺩ‬ ‫ﺍﻷﺭﺽ‬ ‫ﺇﻥ‬ ‫ﺤﻴﺙ‬ ‫ﺩﻭﺭﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻫﻲ‬.‫ﻴﻌﻭﺩ‬ ‫ﺍﻟﻘﻤﺭ‬ ‫ﺍﻥ‬ ‫ﻜﻤﺎ‬‫ﺇﻟﻰ‬‫ﻨﻔﺱ‬
‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻥ‬ ‫ﻤﺤﺩﺩﺓ‬ ‫ﻓﺘﺭﺓ‬ ‫ﻜل‬.
‫ﺤﻭل‬ ‫ﺘﺘﺫﺒﺫﺏ‬ ‫ﺍﻟﺼﻠﺒﺔ‬ ‫ﺍﻟﻤﻭﺍﺩ‬ ‫ﻓﻲ‬ ‫ﺍﻟﺠﺯﺌﻴﺎﺕ‬ ‫ﻓﺤﺭﻜﺔ‬ ،‫ﺩﻭﺭﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﺘﺘﺤﺭﻙ‬ ‫ﻨﻅﻤﺔ‬
‫ﻤﺜل‬ ‫ﺍﻟﻜﻬﺭﻭﻤﻐﻨﺎﻁﻴﺴﻴﺔ‬ ‫ﺍﻻﻤﻭﺍﺝ‬ ‫ﻜﺫﻟﻙ‬ ،‫ﻤﺴﺘﻤﺭﺓ‬ ‫ﺩﻭﺭﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻓﻲ‬ ‫ﺍﺘﺯﺍﻨﻬﺎ‬ ‫ﻤﻭﻀﻊ‬‫ﺃ‬‫ﻭ‬ ‫ﺍﻟﻀﻭﺀ‬ ‫ﻤﻭﺍﺝ‬‫ﺃ‬‫ﻭ‬ ‫ﺍﻟﺭﺍﺩﺍﺭ‬ ‫ﻤﻭﺍﺝ‬‫ﺃ‬‫ﻤﻭﺍﺝ‬
‫ﻤﺠﺎﻟﻬﺎ‬ ‫ﺘﺫﺒﺫﺏ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬ ‫ﺍﻟﻔﺭﺍﻍ‬ ‫ﻓﻲ‬ ‫ﺘﻨﺘﺘﺸﺭ‬ ‫ﺍﻟﺭﺍﺩﻴﻭ‬‫ﺍﻟﻜ‬‫ﻬﺭﻭﻤﻐﻨﺎﻁﻴﺴﻲ‬‫ﺍﻟﻜﻬﺭﺒﻲ‬ ‫ﻭﺍﻟﺠﻬﺩ‬ ‫ﺍﻟﻤﺘﺭﺩﺩ‬ ‫ﺍﻟﻜﻬﺭﺒﻲ‬ ‫ﺍﻟﺘﻴﺎﺭ‬ ‫ﻜﺫﻟﻙ‬ ،
‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻊ‬ ‫ﺩﻭﺭﻴﺔ‬ ‫ﺒﺼﻔﺔ‬ ‫ﺘﺘﻐﻴﺭ‬ ‫ﺍﻟﻜﻬﺭﺒﻴﺔ‬ ‫ﻭﺍﻟﺸﺤﻨﺔ‬.
‫ﻟﻸ‬ ‫ﺘﺤﺩﺙ‬ ‫ﺍﻟﺩﻭﺭﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻤﻥ‬ ‫ﺨﺎﺼﺔ‬ ‫ﺤﺎﻟﺔ‬ ‫ﻫﻨﺎﻙ‬‫ﻤﻭﻀﻊ‬ ‫ﻤﻊ‬ ‫ﻁﺭﺩﻴﺎ‬ ‫ﺘﺘﻨﺎﺴﺏ‬ ‫ﺍﻟﻤﻴﻜﺎﻨﻴﻜﻴﺔ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻓﻴﻬﺎ‬ ‫ﺘﻜﻭﻥ‬ ‫ﺍﻟﻤﻜﻴﺎﻨﻴﻜﻴﺔ‬ ‫ﻨﻅﻤﺔ‬
‫ﺍﻟﻘﻭﺓ‬ ‫ﻫﺫﻩ‬ ‫ﻜﺎﻨﺕ‬ ‫ﺍﺫﺍ‬‫ﺍﻹ‬ ‫ﻨﻘﻁﺔ‬ ‫ﺍﺘﺠﺎﻩ‬ ‫ﻓﻲ‬ ‫ﺩﺍﺌﻤﺎ‬‫ﻓ‬ ‫ﺘﺯﺍﻥ‬‫ﺈ‬‫ﺒ‬ ‫ﺘﻌﺭﻑ‬ ‫ﺍﻟﺤﺎﻟﺔ‬ ‫ﻫﺫﻩ‬ ‫ﻓﻲ‬ ‫ﻨﻪ‬‫ﺎﺴﻡ‬‫ﺍﻟﺤﺭﻜﺔ‬
simple harmonic motion‫ﺍﻟﺠﺯﺀ‬ ‫ﻫﺫﺍ‬ ‫ﻓﻲ‬ ‫ﻋﻠﻴﻪ‬ ‫ﺍﻟﺩﺭﺍﺴﺔ‬ ‫ﻨﺭﻜﺯ‬ ‫ﺴﻭﻑ‬ ‫ﻤﺎ‬ ‫ﻭﻫﺫﺍ‬.
www.trgma.com
‫ﺍﻟﺩﻭﺭﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬‫ﻓﻴﻬ‬ ‫ﻴﻌﻭﺩ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻫﻲ‬‫ﺎ‬‫ﺜﺎﺒﺘﺔ‬ ‫ﺯﻤﻨﻴﺔ‬ ‫ﻓﺘﺭﺓ‬ ‫ﺒﻌﺩ‬ ‫ﻤﺤﺩﺩ‬ ‫ﻤﻭﻗﻊ‬ ‫ﺇﻟﻰ‬ ‫ﺍﻟﺠﺴﻡ‬
‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻥ‬ ‫ﻤﺤﺩﺩﺓ‬ ‫ﻓﺘﺭﺓ‬ ‫ﻜل‬ ‫ﻤﺤﺩﺩ‬ ‫ﻤﻭﻗﻊ‬ ‫ﺇﻟﻰ‬ ‫ﺘﻌﻭﺩ‬ ‫ﺍﻷﺭﺽ‬ ‫ﺇﻥ‬ ‫ﺤﻴﺙ‬ ‫ﺩﻭﺭﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻫﻲ‬
‫ﻟﻸﺭﺽ‬ ‫ﺒﺎﻟﻨﺴﺒﺔ‬ ‫ﻤﻭﻗﻌﻪ‬‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻥ‬ ‫ﻤﺤﺩﺩﺓ‬ ‫ﻓﺘﺭﺓ‬ ‫ﻜل‬
‫ﺇ‬‫ﻀﺎﻓﺔ‬‫ﺇ‬‫ﺍﻷ‬ ‫ﻤﻥ‬ ‫ﺍﻟﻜﺜﻴﺭ‬ ‫ﻫﻨﺎﻙ‬ ‫ﺫﻟﻙ‬ ‫ﻟﻰ‬‫ﻷ‬ ‫ﻤﺜﻠﺔ‬‫ﺤﻭل‬ ‫ﺘﺘﺫﺒﺫﺏ‬ ‫ﺍﻟﺼﻠﺒﺔ‬ ‫ﺍﻟﻤﻭﺍﺩ‬ ‫ﻓﻲ‬ ‫ﺍﻟﺠﺯﺌﻴﺎﺕ‬ ‫ﻓﺤﺭﻜﺔ‬ ،‫ﺩﻭﺭﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﺘﺘﺤﺭﻙ‬ ‫ﻨﻅﻤﺔ‬
‫ﻤﺜل‬ ‫ﺍﻟﻜﻬﺭﻭﻤﻐﻨﺎﻁﻴﺴﻴﺔ‬ ‫ﺍﻻﻤﻭﺍﺝ‬ ‫ﻜﺫﻟﻙ‬ ،‫ﻤﺴﺘﻤﺭﺓ‬ ‫ﺩﻭﺭﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻓﻲ‬ ‫ﺍﺘﺯﺍﻨﻬﺎ‬ ‫ﻤﻭﻀﻊ‬
‫ﻤﺠﺎﻟﻬﺎ‬ ‫ﺘﺫﺒﺫﺏ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬ ‫ﺍﻟﻔﺭﺍﻍ‬ ‫ﻓﻲ‬ ‫ﺘﻨﺘﺘﺸﺭ‬ ‫ﺍﻟﺭﺍﺩﻴﻭ‬
‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻊ‬ ‫ﺩﻭﺭﻴﺔ‬ ‫ﺒﺼﻔﺔ‬ ‫ﺘﺘﻐﻴﺭ‬ ‫ﺍﻟﻜﻬﺭﺒﻴﺔ‬ ‫ﻭﺍﻟﺸﺤﻨﺔ‬
‫ﻟﻸ‬ ‫ﺘﺤﺩﺙ‬ ‫ﺍﻟﺩﻭﺭﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻤﻥ‬ ‫ﺨﺎﺼﺔ‬ ‫ﺤﺎﻟﺔ‬ ‫ﻫﻨﺎﻙ‬
‫ﻤﺎ‬ ‫ﺍﺘﺯﺍﻥ‬ ‫ﻟﻨﻘﻁﺔ‬ ‫ﺒﺎﻟﻨﺴﺒﺔ‬ ‫ﺍﻟﺠﺴﻡ‬.‫ﺍﻟﻘﻭﺓ‬ ‫ﻫﺫﻩ‬ ‫ﻜﺎﻨﺕ‬ ‫ﺍﺫﺍ‬
‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬simple harmonic motion

7
Motion of an object attached to a spring
‫ﻜﺘﻠﺘﻪ‬ ‫ﺴﻡ‬m‫ﻜﻤﺎ‬ ‫ﺍﻻﺤﺘﻜﺎﻙ‬ ‫ﻋﺩﻴﻡ‬ ‫ﺃﻓﻘﻲ‬ ‫ﺴﻁﺢ‬ ‫ﻋﻠﻰ‬ ‫ﻴﺘﺤﺭﻙ‬ ‫ﺯﻨﺒﺭﻙ‬ ‫ﻤﻊ‬ ‫ﻤﺘﺼل‬
‫ﻴﻜﻭﻥ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﻓﺎﻥ‬ ،‫ﺼﻔﺭ‬ ‫ﺘﺴﺎﻭﻱ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﺍﺴﺘﻁﺎﻟﺔ‬ ‫ﺇﻥ‬ ‫ﺃﻱ‬ ‫ﺍﻻﺴﺘﻘﺭﺍﺭ‬ ‫ﺤﺎﻟﺔ‬ ‫ﻓﻲ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﻴﻜﻭﻥ‬ ‫ﻋﻨﺩﻤﺎ‬
‫ﺍﺴﺘ‬ ‫ﺇﻟﻰ‬ ‫ﻴﺅﺩﻱ‬ ‫ﺍﻀﻁﺭﺍﺏ‬ ‫ﺃﻱ‬ ‫ﺇﻟﻰ‬ ‫ﺘﻌﺭﺽ‬ ‫ﺇﺫﺍ‬ ‫ﻭﻟﻠﻴﺴﺎﺭ‬ ‫ﻟﻠﻴﻤﻴﻥ‬ ‫ﻴﺘﺫﺒﺫﺏ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﺇﻥ‬ ‫ﻨﻌﻠﻡ‬‫ﺃﻭ‬ ‫ﻁﺎﻟﺘﻪ‬
‫ﺘﺘﻭﻟﺩ‬ ‫ﻤﻌﻴﻨﺔ‬ ‫ﻗﻭﺓ‬ ‫ﻓﺎﻥ‬ ‫ﺨﺎﺭﺠﻴﺔ‬ ‫ﻟﻘﻭﺓ‬ ‫ﻨﺘﻴﺠﺔ‬ ‫ﻟﻠﻴﺴﺎﺭ‬ ‫ﺃﻭ‬ ‫ﻟﻠﻴﻤﻴﻥ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺃﺯﻴﺢ‬ ‫ﺇﺫﺍ‬ ‫ﺍﻨﻪ‬ ‫ﻨﻔﻬﻡ‬ ‫ﺃﻥ‬ ‫ﻨﺴﺘﻁﻴﻊ‬
‫ﺍﻻﺴﺘﺭﺠﺎﻋﺔ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﺘﺴﻤﻰ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﻓﻲ‬restoring force‫ﺍﻟﺨﺎﺭﺠﻴﺔ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﺍﺘﺠﺎﻩ‬ ‫ﻋﻜﺱ‬ ‫ﺍﺘﺠﺎﻫﻬﺎ‬.‫ﺤﺴﺎﺏ‬ ‫ﻭﻴﻤﻜﻥ‬
‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﻓﻲ‬ ‫ﺘﺘﻭﻟﺩ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻻﺴﺘﺭﺠﺎﻋﻴﺔ‬ ‫ﺍﻟﻘﻭﺓ‬‫ﻫﻭﻙ‬ ‫ﻗﺎﻨﻭﻥ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬Hooke’s law.
‫اﻻﺣﺘﻜﺎك‬ ‫ﻋﺪﯾﻢ‬ ‫ﺳﻄﺢ‬ ‫ﻋﻠﻰ‬ ‫ﯾﺘﺤﺮك‬ ‫زﻧﺒﺮك‬ ‫ﻓﻲ‬ ‫ﻣﻌﻠﻖ‬ ‫ﺟﺴﻢ‬) .a(‫ﻋﻦ‬ ‫ﻟﻠﯿﻤﯿﻦ‬ ‫إزاﺣﺔ‬ ‫اﻟﺠﺴﻢ‬ ‫ﯾﺘﺤﺮك‬ ‫ﻋﻨﺪﻣﺎ‬
‫ﻟﻠﯿﺴﺎر‬ ‫ﺗﻜﻮن‬ ‫اﻟﺰﻧﺒﺮك‬ ‫ﻓﻲ‬ ‫اﻟﻤﺘﻮﻟﺪة‬ ‫اﻟﻘﻮة‬ ‫ﻓﺎن‬ ،) .b(‫ﻋﻨﺪ‬ ‫اﺗﺰان‬ ‫ﺣﺎﻟﺔ‬ ‫ﻓﻲ‬ ‫اﻟﺠﺴﻢ‬ ‫ﯾﻜﻮن‬ ‫ﻋﻨﺪﻣﺎ‬)x=0(،
‫ﺻﻔﺮ‬ ‫ﺗﻜﻮن‬ ‫اﻟﺰﻧﺒﺮك‬ ‫ﻓﻲ‬ ‫ﺘﻮﻟﺪة‬) .c(‫ﺑﺤﯿﺚ‬ ‫ﻟﻠﯿﺴﺎر‬ ‫اﻟﺠﺴﻢ‬ ‫إزاﺣﺔ‬ ‫ﺗﻜﻮن‬ ‫ﻋﻨﺪﻣﺎ‬)x<0(‫ﻓﻲ‬ ‫اﻟﻤﺘﻮﻟﺪة‬ ‫اﻟﻘﻮة‬ ‫ﻓﺎن‬ ،
www.trgma.com
1.15‫ﺯﻨﺒﺭﻙ‬ ‫ﻓﻲ‬ ‫ﻤﻌﻠﻕ‬ ‫ﺠﺴﻡ‬ ‫ﺤﺭﻜﺔ‬Motion of an object attached to a spring
‫ﺠ‬ ‫ﻨﻔﺭﺽ‬ ‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﻜﻨﻤﻭﺫﺝ‬‫ﻜﺘﻠﺘﻪ‬ ‫ﺴﻡ‬
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬1.15.‫ﻴﻜﻭﻥ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﻓﺎﻥ‬ ،‫ﺼﻔﺭ‬ ‫ﺘﺴﺎﻭﻱ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﺍﺴﺘﻁﺎﻟﺔ‬ ‫ﺇﻥ‬ ‫ﺃﻱ‬ ‫ﺍﻻﺴﺘﻘﺭﺍﺭ‬ ‫ﺤﺎﻟﺔ‬ ‫ﻓﻲ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﻴﻜﻭﻥ‬ ‫ﻋﻨﺩﻤﺎ‬
‫ﺍﺘﺯﺍﻥ‬ ‫ﺤﺎﻟﺔ‬ ‫ﻓﻲ‬x=0.‫ﺍﺴﺘ‬ ‫ﺇﻟﻰ‬ ‫ﻴﺅﺩﻱ‬ ‫ﺍﻀﻁﺭﺍﺏ‬ ‫ﺃﻱ‬ ‫ﺇﻟﻰ‬ ‫ﺘﻌﺭﺽ‬ ‫ﺇﺫﺍ‬ ‫ﻭﻟﻠﻴﺴﺎﺭ‬ ‫ﻟﻠﻴﻤﻴﻥ‬ ‫ﻴﺘﺫﺒﺫﺏ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﺇﻥ‬ ‫ﻨﻌﻠﻡ‬
‫ﺍﻨﻀﻐﺎﻁﻪ‬.
‫ﺍﻟﺸﻜل‬ ‫ﻤﻥ‬1.15‫ﺘﺘﻭﻟﺩ‬ ‫ﻤﻌﻴﻨﺔ‬ ‫ﻗﻭﺓ‬ ‫ﻓﺎﻥ‬ ‫ﺨﺎﺭﺠﻴﺔ‬ ‫ﻟﻘﻭﺓ‬ ‫ﻨﺘﻴﺠﺔ‬ ‫ﻟﻠﻴﺴﺎﺭ‬ ‫ﺃﻭ‬ ‫ﻟﻠﻴﻤﻴﻥ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺃﺯﻴﺢ‬ ‫ﺇﺫﺍ‬ ‫ﺍﻨﻪ‬ ‫ﻨﻔﻬﻡ‬ ‫ﺃﻥ‬ ‫ﻨﺴﺘﻁﻴﻊ‬
‫ﺍﻻﺴﺘﺭﺠﺎﻋﺔ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﺘﺴﻤﻰ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﻓﻲ‬
‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﻓﻲ‬ ‫ﺘﺘﻭﻟﺩ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻻﺴﺘﺭﺠﺎﻋﻴﺔ‬ ‫ﺍﻟﻘﻭﺓ‬
‫ﺷﻜﻞ‬1.15‫ﯾﻮﺿﺢ‬‫اﻻﺣﺘﻜﺎك‬ ‫ﻋﺪﯾﻢ‬ ‫ﺳﻄﺢ‬ ‫ﻋﻠﻰ‬ ‫ﯾﺘﺤﺮك‬ ‫زﻧﺒﺮك‬ ‫ﻓﻲ‬ ‫ﻣﻌﻠﻖ‬ ‫ﺟﺴﻢ‬
‫اﻻﺗﺰان‬ ‫ﻧﻘﻄﺔ‬)x<0(‫ﻟﻠﯿﺴﺎر‬ ‫ﺗﻜﻮن‬ ‫اﻟﺰﻧﺒﺮك‬ ‫ﻓﻲ‬ ‫اﻟﻤﺘﻮﻟﺪة‬ ‫اﻟﻘﻮة‬ ‫ﻓﺎن‬ ،
‫اﻟﻤ‬ ‫اﻟﻘﻮة‬ ‫ﻓﺎن‬‫ﺻﻔﺮ‬ ‫ﺗﻜﻮن‬ ‫اﻟﺰﻧﺒﺮك‬ ‫ﻓﻲ‬ ‫ﺘﻮﻟﺪة‬
‫اﻟﯿﻤﯿﻦ‬ ‫اﺗﺠﺎه‬ ‫ﻓﻲ‬ ‫ﺗﻜﻮن‬ ‫اﻟﺰﻧﺒﺮك‬.
‫ﺸﻜل‬1.15

8
Fs = - kx (15.1)
‫ﺍﻻﺴﺘﺭﺠﺎﻋﻴﺔ‬ ‫ﺒﺎﻟﻘﻭﺓ‬restoring force‫ﺍﺘﺠﺎﻩ‬ ‫ﻓﻲ‬ ‫ﺩﺍﺌﻤﺎ‬ ‫ﺘﻜﻭﻥ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻫﺫﻩ‬ ‫ﻻﻥ‬‫ﻨﻘﻁﺔ‬
‫ﻭﻟﻬﺫﺍ‬‫ﺃﺯﻴﺢ‬ ‫ﺇﺫﺍ‬‫ﺍﻟﻨﻘﻁﺔ‬ ‫ﻤﻥ‬ ‫ﺍﻟﻴﻤﻴﻥ‬ ‫ﺇﻟﻰ‬ ‫ﺍﻟﺠﺴﻡ‬x=0‫ﻜﻤﺎ‬ ‫ﺍﻟﻤﻭﺠﺏ‬ ‫ﺍﻻﺘﺠﺎﻩ‬ ‫ﻓﻲ‬ ‫ﺃﻱ‬
‫ﺍﻻﺴﺘﺭﺠﺎﻋﻴﺔ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﺍﺘﺠﺎﻩ‬ ‫ﻓﺎﻥ‬‫ﺍ‬ ‫ﺇﻟﻰ‬‫ﻟﻴﺴﺎﺭ‬.‫ﺍﻟﻨﻘﻁﺔ‬ ‫ﻤﻥ‬ ‫ﺍﻟﻴﺴﺎﺭ‬ ‫ﺇﻟﻰ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺇﺯﺍﺤﺔ‬ ‫ﻭﻋﻨﺩ‬x=0‫ﻓﻲ‬ ‫ﺃﻱ‬ ،
‫ﺍﻟ‬ ‫ﻓﺎﻥ‬‫ﺘﻜﻭﻥ‬ ‫ﺴﻭﻑ‬ ‫ﺍﻻﺴﺘﺭﺠﺎﻋﻴﺔ‬ ‫ﻘﻭﺓ‬‫ﺍ‬ ‫ﺇﻟﻰ‬‫ﻟﻴﻤﻴﻥ‬.
‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻤﻊ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻋﻠﻰ‬1.15‫ﺍﻻﺘﺠﺎﻩ‬ ‫ﻓﻲ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻤﺤﺼﻠﺔ‬ ‫ﻓﺎﻥ‬x‫ﺘﻜﻭﻥ‬
xmakx =−
x
m
k
ax −=
(15.2)
‫ﻭﻓﻲ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻭﻗﻊ‬ ‫ﻤﻊ‬ ‫ﻁﺭﺩﻴﺎ‬ ‫ﺘﺘﻨﺎﺴﺏ‬ ‫ﺍﻟﻌﺠﻠﺔ‬ ‫ﻓﺎﻥ‬ ‫ﻭﻟﻬﺫﺍ‬‫ﻴﺘﺼﺭﻑ‬ ‫ﺠﺴﻡ‬ ‫ﻭﻜل‬ ‫ﺍﻻﺘﺯﺍﻥ‬ ‫ﻨﻘﻁﺔ‬ ‫ﻤﻥ‬ ‫ﻟﻺﺯﺍﺤﺔ‬ ‫ﺍﻟﻤﻌﺎﻜﺱ‬ ‫ﺍﻻﺘﺠﺎﻩ‬
‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺒﺎﻟﺤﺭﻜﺔ‬ ‫ﺘﺴﻤﻰ‬ ‫ﺤﺭﻜﺘﻪ‬ ‫ﻥ‬simple harmonic motion.‫ﺠﺴﻡ‬ ‫ﺃﻱ‬ ‫ﻓﺎﻥ‬ ‫ﻭﻋﻠﻴﻪ‬
‫ﻟﻤﻭﻀﻊ‬ ‫ﺍﻟﻤﻌﺎﻜﺱ‬ ‫ﺍﻻﺘﺠﺎﻩ‬ ‫ﻭﻓﻲ‬ ‫ﺇﺯﺍﺤﺘﻪ‬ ‫ﻤﻊ‬ ‫ﻁﺭﺩﻴﺎ‬ ‫ﹰ‬‫ﺎ‬‫ﺘﻨﺎﺴﺒ‬ ‫ﺘﺘﻨﺎﺴﺏ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻫﺫﺍ‬ ‫ﻋﺠﻠﺔ‬ ‫ﺘﻜﻭﻥ‬ ‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻴﺘﺤﺭﻙ‬
www.trgma.com
‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﻓﻲ‬ ‫ﺍﻟﻤﺘﻭﻟﺩﺓ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﺴﻤﻴﺕ‬‫ﺍﻻﺴﺘﺭﺠﺎﻋﻴﺔ‬ ‫ﺒﺎﻟﻘﻭﺓ‬
‫ﺍﻻ‬‫ﺍﺘﺠﺎﻩ‬ ‫ﻋﻜﺱ‬ ‫ﻭﻓﻲ‬ ‫ﺘﺯﺍﻥ‬‫ﺍﻟﺠﺴﻡ‬ ‫ﺍﺯﺍﺤﺔ‬.‫ﻭﻟﻬﺫﺍ‬
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬1.15)a(‫ﺍﻻﺴﺘﺭﺠﺎﻋﻴﺔ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﺍﺘﺠﺎﻩ‬ ‫ﻓﺎﻥ‬
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﻜﻤﺎ‬ ‫ﺍﻟﺴﺎﻟﺏ‬ ‫ﺍﻻﺘﺠﺎﻩ‬1.15)c(‫ﺍﻟ‬ ‫ﻓﺎﻥ‬
‫ﺍﻟﺜﺎﻨﻲ‬ ‫ﻨﻴﻭﺘﻥ‬ ‫ﻗﺎﻨﻭﻥ‬ ‫ﺒﺘﻁﺒﻴﻕ‬ΣFx=max‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻤﻊ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻋﻠﻰ‬
‫ﺍﻟﺘﺎﻟﻲ‬ ‫ﺍﻟﻨﺤﻭ‬ ‫ﻋﻠﻰ‬:
‫ﻭﻓﻲ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻭﻗﻊ‬ ‫ﻤﻊ‬ ‫ﻁﺭﺩﻴﺎ‬ ‫ﺘﺘﻨﺎﺴﺏ‬ ‫ﺍﻟﻌﺠﻠﺔ‬ ‫ﻓﺎﻥ‬ ‫ﻭﻟﻬﺫﺍ‬
‫ﻓ‬ ‫ﺍﻟﻨﺤﻭ‬ ‫ﻫﺫﺍ‬ ‫ﻋﻠﻰ‬‫ﺈ‬‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺒﺎﻟﺤﺭﻜﺔ‬ ‫ﺘﺴﻤﻰ‬ ‫ﺤﺭﻜﺘﻪ‬ ‫ﻥ‬
‫ﻟﻤﻭﻀﻊ‬ ‫ﺍﻟﻤﻌﺎﻜﺱ‬ ‫ﺍﻻﺘﺠﺎﻩ‬ ‫ﻭﻓﻲ‬ ‫ﺇﺯﺍﺤﺘﻪ‬ ‫ﻤﻊ‬ ‫ﻁﺭﺩﻴﺎ‬ ‫ﹰ‬‫ﺎ‬‫ﺘﻨﺎﺴﺒ‬ ‫ﺘﺘﻨﺎﺴﺏ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻫﺫﺍ‬ ‫ﻋﺠﻠﺔ‬ ‫ﺘﻜﻭﻥ‬ ‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻴﺘﺤﺭﻙ‬
‫ﺍﻻ‬‫ﺘﺯﺍﻥ‬.

9
‫ﺍﻟﻤﻭﻀﻊ‬ ‫ﺇﻟﻰ‬A‫ﺘﺭﻙ‬ ‫ﺜﻡ‬ ‫ﻭﻤﻥ‬،‫ﻫﻲ‬ ‫ﻟﻪ‬ ‫ﺍﻻﺒﺘﺩﺍﺌﻴﺔ‬ ‫ﺍﻟﻌﺠﻠﺔ‬ ‫ﻓﺎﻥ‬–kA/m.‫ﻴﻤﺭ‬ ‫ﻭﻋﻨﺩﻤﺎ‬
‫ﺴﺭﻋﺔ‬ ‫ﻓﺎﻥ‬ ‫ﺍﻟﻠﺤﻅﺔ‬ ‫ﻫﺫﻩ‬ ‫ﻋﻨﺩ‬ ،‫ﻟﻠﺼﻔﺭ‬ ‫ﻤﺴﺎﻭﻴﺔ‬ ‫ﺘﻜﻭﻥ‬ ‫ﺍﻟﻌﺠﻠﺔ‬ ‫ﻓﺎﻥ‬‫ﺍﻟﺠﺴﻡ‬‫ﻤﺎ‬ ‫ﺍﻜﺒﺭ‬ ‫ﺘﻜﻭﻥ‬
‫ﺍﻟﺠ‬ ‫ﻴﺴﺘﻤﺭ‬‫ﻟﺘﺼل‬ ‫ﻤﻭﺠﺒﺔ‬ ‫ﺒﻌﺠﻠﺔ‬ ‫ﺍﻻﺘﺯﺍﻥ‬ ‫ﻨﻘﻁﺔ‬ ‫ﻤﻥ‬ ‫ﺍﻟﻴﺴﺎﺭ‬ ‫ﺍﻟﻰ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻓﻲ‬ ‫ﺴﻡ‬
‫ﻭﺘﺴﺎﻭﻱ‬ ‫ﻤﻭﺠﺒﺔ‬ ‫ﺍﻟﻌﺠﻠﺔ‬ ‫ﺘﻜﻭﻥ‬ ‫ﻭﻋﻨﺩﻫﺎ‬ ،+kA/m‫ﻟﻠﺼﻔﺭ‬ ‫ﻤﺴﺎﻭﻴﺔ‬ ‫ﺍﻟﺴﺭﻋﺔ‬ ‫ﻭﺘﻜﻭﻥ‬.
‫ﻨﻘﻁﺔ‬ ‫ﺇﻟﻰ‬ ‫ﻴﻌﻭﺩ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﺤﺭﻜﺘﻪ‬ ‫ﻤﻥ‬ ‫ﻜﺎﻤﻠﺔ‬ ‫ﺩﻭﺭﺓ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻴﻜﻤل‬‫ﺍﻟﺒﺩﺀ‬‫ﺍﺨﺭﻯ‬ ‫ﻤﺭﺓ‬ ‫ﻴﻤﺭ‬ ‫ﺜﻡ‬ ،‫ﺍ‬‫ﻟﻨﻘﻁﺔ‬x=0‫ﻤﻤﻜﻨﺔ‬ ‫ﺴﺭﻋﺔ‬ ‫ﺒﺎﻜﺒﺭ‬.
‫ﻫﻤﺎ‬ ‫ﻨﻘﻁﺘﻴﻥ‬ ‫ﺒﻴﻥ‬ ‫ﻴﺘﺫﺒﺫﺏ‬ ‫ﺍﻟﺠﺴﻡ‬x=±A‫ﺒﻭﺍﺴﻁﺔ‬ ‫ﺍﻟﻤﺒﺫﻭﻟﺔ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻭﻻﻥ‬ ،‫ﺍﺤﺘﻜﺎﻙ‬ ‫ﻭﺠﻭﺩ‬ ‫ﻋﺩﻡ‬ ‫ﺤﺎﻟﺔ‬ ‫ﻓﻲ‬
‫ﻤﺎﻻﻨﻬﺎﻴﺔ‬ ‫ﺇﻟﻰ‬ ‫ﺘﺴﺘﻤﺭ‬ ‫ﺴﻭﻑ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻓﺈﻥ‬ ،‫ﻤﺤﺎﻓﻅﺔ‬ ‫ﻗﻭﺓ‬.‫ﻓﻲ‬ ‫ﺍﺤﺘﻜﺎﻙ‬ ‫ﻭﺠﻭﺩ‬ ‫ﻓﻲ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺤﺎﻟﺔ‬ ‫ﻨﺩﺭﺱ‬ ‫ﻭﺴﻭﻑ‬
‫ﺍﻟﻤﻌﻠﻕ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻋﻠﻰ‬ ‫ﺘﻨﻁﺒﻕ‬ ‫ﺍﻟﺴﺎﺒﻕ‬ ‫ﺍﻟﺠﺯﺀ‬ ‫ﻓﻲ‬ ‫ﻨﺎﻗﺸﻨﺎﻫﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﺤﺎﻟﺔ‬‫ﺍﺴﺘﻁﺎﻟﺔ‬ ‫ﻴﺴﺒﺏ‬ ‫ﺴﻭﻑ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺍﻥ‬ ‫ﺤﻴﺙ‬ ‫ﺯﻨﺒﺭﻙ‬ ‫ﻓﻲ‬
‫ﻋﻨﺩ‬ ‫ﺠﺩﻴﺩﺓ‬ ‫ﻨﻘﻁﺔ‬ ‫ﺍﻻﺘﺯﺍﻥ‬ ‫ﻨﻘﻁﺔ‬ ‫ﻭﺘﺼﺒﺢ‬ ‫ﻤﻌﻴﻨﺔ‬ ‫ﻤﺴﺎﻓﺔ‬ ‫ﺍﻻﺘﺯﺍﻥ‬ ‫ﻨﻘﻁﺔ‬ ‫ﺘﺯﺍﺡ‬ ‫ﺒﺤﻴﺙ‬ ‫ﻟﻠﺯﻨﺒﺭﻙ‬x=0.‫ﺍﻥ‬ ‫ﻨﻔﺭﺽ‬ ‫ﺫﻟﻙ‬ ‫ﻭﻟﺘﻭﻀﻴﺢ‬
‫ﺒﻪ‬ ‫ﹰ‬‫ﺎ‬‫ﻤﻌﻠﻘ‬ ‫ﺠﺴﻡ‬ ‫ﺃﻱ‬ ‫ﻫﻨﺎﻙ‬ ‫ﻴﻜﻭﻥ‬ ‫ﻻ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﺍﻻﺘﺯﺍﻥ‬ ‫ﻨﻘﻁﺔ‬ ‫ﻤﻥ‬ ‫ﻟﻠﺯﻨﺒﺭﻙ‬ ‫ﺍﻟﻜﻠﻴﺔ‬ ‫ﺍﻻﺴﺘﻁﺎﻟﺔ‬ ‫ﺘﻤﺜل‬.‫ﻓﺎﻥ‬ ‫ﻭﻋﻠﻴﻪ‬
‫ﻭ‬ ‫ﺒﻪ‬ ‫ﺍﻟﻤﻌﻠﻕ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻟﻭﺯﻥ‬ ‫ﻨﺘﻴﺠﺔ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﺍﺴﺘﻁﺎﻟﺔ‬ ‫ﺘﻤﺜل‬x‫ﺍﻟﺤﺎﺩﺜﺔ‬ ‫ﺍﻻﺴﺘﻁﺎﻟﺔ‬ ‫ﻫﻲ‬
‫ﻫﻲ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻋﻠﻰ‬ ‫ﺍﻟﻤﺤﺼﻠﺔ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻤﻘﺩﺍﺭ‬
Fs-Fg=-k(-(mg/k)+x)-mg=-kx
www.trgma.com
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺃﺯﻴﺢ‬ ‫ﺇﺫﺍ‬1.15‫ﺍﻟﻤﻭﻀﻊ‬ ‫ﺇﻟﻰ‬
‫ﺍﻻﺘﺯﺍﻥ‬ ‫ﻨﻘﻁﺔ‬ ‫ﺨﻼل‬ ‫ﺍﻟﺠﺴﻡ‬x=0‫ﺴﺭﻋﺔ‬ ‫ﻓﺎﻥ‬ ‫ﺍﻟﻠﺤﻅﺔ‬ ‫ﻫﺫﻩ‬ ‫ﻋﻨﺩ‬ ،‫ﻟﻠﺼﻔﺭ‬ ‫ﻤﺴﺎﻭﻴﺔ‬ ‫ﺘﻜﻭﻥ‬ ‫ﺍﻟﻌﺠﻠﺔ‬ ‫ﻓﺎﻥ‬
‫ﺘﻐﻴﺭ‬ ‫ﺍﻟﻌﺠﻠﺔ‬ ‫ﻻﻥ‬ ‫ﻴﻤﻜﻥ‬‫ﺍﺘﺠﺎﻫﻬﺎ‬)‫ﺇﺸﺎﺭﺘﻬﺎ‬(.‫ﺍﻟﺠ‬ ‫ﻴﺴﺘﻤﺭ‬
‫ﺍﻟﻤﻭﻀﻊ‬ ‫ﺇﻟﻰ‬x=-A‫ﻭﺘﺴﺎﻭﻱ‬ ‫ﻤﻭﺠﺒﺔ‬ ‫ﺍﻟﻌﺠﻠﺔ‬ ‫ﺘﻜﻭﻥ‬ ‫ﻭﻋﻨﺩﻫﺎ‬ ،
‫ﻨﻘﻁﺔ‬ ‫ﺇﻟﻰ‬ ‫ﻴﻌﻭﺩ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﺤﺭﻜﺘﻪ‬ ‫ﻤﻥ‬ ‫ﻜﺎﻤﻠﺔ‬ ‫ﺩﻭﺭﺓ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻴﻜﻤل‬
‫ﺍﻥ‬ ‫ﻨﺭﻯ‬ ‫ﻭﺒﻬﺫﺍ‬‫ﻫﻤﺎ‬ ‫ﻨﻘﻁﺘﻴﻥ‬ ‫ﺒﻴﻥ‬ ‫ﻴﺘﺫﺒﺫﺏ‬ ‫ﺍﻟﺠﺴﻡ‬
‫ﺍﻟﺯﻨﺒﺭﻙ‬‫ﻫﻲ‬‫ﻤﺎﻻﻨﻬﺎﻴﺔ‬ ‫ﺇﻟﻰ‬ ‫ﺘﺴﺘﻤﺭ‬ ‫ﺴﻭﻑ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻓﺈﻥ‬ ،‫ﻤﺤﺎﻓﻅﺔ‬ ‫ﻗﻭﺓ‬
‫ﺍﻟﺠﺯﺀ‬6.15.
‫ﺍﻟﻤﻌﻠﻕ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻋﻠﻰ‬ ‫ﺘﻨﻁﺒﻕ‬ ‫ﺍﻟﺴﺎﺒﻕ‬ ‫ﺍﻟﺠﺯﺀ‬ ‫ﻓﻲ‬ ‫ﻨﺎﻗﺸﻨﺎﻫﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﺤﺎﻟﺔ‬
‫ﻋﻨﺩ‬ ‫ﺠﺩﻴﺩﺓ‬ ‫ﻨﻘﻁﺔ‬ ‫ﺍﻻﺘﺯﺍﻥ‬ ‫ﻨﻘﻁﺔ‬ ‫ﻭﺘﺼﺒﺢ‬ ‫ﻤﻌﻴﻨﺔ‬ ‫ﻤﺴﺎﻓﺔ‬ ‫ﺍﻻﺘﺯﺍﻥ‬ ‫ﻨﻘﻁﺔ‬ ‫ﺘﺯﺍﺡ‬ ‫ﺒﺤﻴﺙ‬ ‫ﻟﻠﺯﻨﺒﺭﻙ‬
xs‫ﺒﻪ‬ ‫ﹰ‬‫ﺎ‬‫ﻤﻌﻠﻘ‬ ‫ﺠﺴﻡ‬ ‫ﺃﻱ‬ ‫ﻫﻨﺎﻙ‬ ‫ﻴﻜﻭﻥ‬ ‫ﻻ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﺍﻻﺘﺯﺍﻥ‬ ‫ﻨﻘﻁﺔ‬ ‫ﻤﻥ‬ ‫ﻟﻠﺯﻨﺒﺭﻙ‬ ‫ﺍﻟﻜﻠﻴﺔ‬ ‫ﺍﻻﺴﺘﻁﺎﻟﺔ‬ ‫ﺘﻤﺜل‬
xs=-(mg/k)+x‫ﺤﻴﺙ‬‫ﺇﻥ‬–(mg/k)‫ﻭ‬ ‫ﺒﻪ‬ ‫ﺍﻟﻤﻌﻠﻕ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻟﻭﺯﻥ‬ ‫ﻨﺘﻴﺠﺔ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﺍﺴﺘﻁﺎﻟﺔ‬ ‫ﺘﻤﺜل‬
‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﻨﺘﻴﺠﺔ‬ ‫ﻟﻠﺯﻨﺒﺭﻙ‬.‫ﻫﻲ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻋﻠﻰ‬ ‫ﺍﻟﻤﺤﺼﻠﺔ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻤﻘﺩﺍﺭ‬
kx.

10
‫ﺒﺎﻟﺯﻨﺒ‬ ‫ﺍﻟﻤﺘﺼل‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻋﻠﻰ‬ ‫ﺍﻟﻤﺅﺜﺭﺓ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻨﻔﺴﻬﺎ‬ ‫ﻫﻲ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻋﻠﻰ‬ ‫ﺍﻟﻤﺤﺼﻠﺔ‬ ‫ﺍﻟﻘﻭﺓ‬‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﻜﻤﺎ‬ ‫ﺍﻻﻓﻘﻲ‬ ‫ﺍﻟﻭﻀﻊ‬ ‫ﻓﻲ‬ ‫ﺭﻙ‬
‫ﺍﻟﻤﻭﻀﻊ‬ ‫ﺇﻟﻰ‬ ‫ﺯﻨﺒﺭﻙ‬ ‫ﺒﻨﻬﺎﻴﺔ‬ ‫ﻤﺘﺼل‬ ‫ﺠﺴﻡ‬ ‫ﺴﺤﺏ‬x=A‫ﺘﺭﻙ‬ ‫ﺜﻡ‬ ‫ﻭﻤﻥ‬.‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻤﺴﺎﻓﺔ‬ ‫ﻫﻲ‬ ‫ﻤﺎ‬ ‫ﺤﺭﻜﺘﻪ‬ ‫ﻤﻥ‬ ‫ﻜﺎﻤﻠﺔ‬ ‫ﺩﻭﺭﺓ‬ ‫ﻓﻲ‬
(a) A/2, (b) A, (c) 2A, (d) 4A.
‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺍﻟﺭﻴﺎﻀﻲ‬ ‫ﺍﻟﺘﻤﺜﻴل‬Mathematical representation of simple
‫ﻋ‬ ‫ﺘﺤﺩﺜﻨﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺘﺼﻑ‬ ‫ﺭﻴﺎﻀﻴﺔ‬ ‫ﻤﻌﺎﺩﻟﺔ‬ ‫ﺒﺎﺸﺘﻘﺎﻕ‬ ‫ﺍﻵﻥ‬ ‫ﺴﻨﻘﻭﻡ‬‫ﻨ‬‫ﹰ‬‫ﺎ‬‫ﻤﺴﺒﻘ‬ ‫ﻬﺎ‬.‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬1.15‫ﺘﺤﺩﺩ‬‫ﺍﻟﻘﻭﺓ‬
‫ﻤﺤﻭﺭ‬ ‫ﻋﻠﻰ‬ ‫ﻴﺘﺫﺒﺫﺏ‬ ‫ﻭﺍﻟﺫﻱ‬ ‫ﺒﺎﻟﺯﻨﺒﺭﻙ‬ ‫ﺍﻟﻤﺘﺼل‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻋﻠﻰ‬ ‫ﺍﻟﻤﺅﺜﺭﺓ‬x.‫ﺒ‬ ‫ﺍﻟﺴﺭﻋﺔ‬ ‫ﺘﻐﻴﺭ‬ ‫ﻤﻌﺩل‬ ‫ﻫﻲ‬ ‫ﺍﻟﻌﺠﻠﺔ‬ ‫ﺇﻥ‬ ‫ﻭﺤﻴﺙ‬‫ﺎﻟﻨﺴﺒﺔ‬
2
2
dt
xd
dt
dv
a ==
www.trgma.com
‫ﺒﺎﻟﺯﻨﺒ‬ ‫ﺍﻟﻤﺘﺼل‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻋﻠﻰ‬ ‫ﺍﻟﻤﺅﺜﺭﺓ‬ ‫ﺍﻟﻘﻭﺓ‬ ‫ﻨﻔﺴﻬﺎ‬ ‫ﻫﻲ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻋﻠﻰ‬ ‫ﺍﻟﻤﺤﺼﻠﺔ‬ ‫ﺍﻟﻘﻭﺓ‬
1.15.
‫ﻟﻠﺘﻔﻜﻴﺭ‬ ‫ﺴﺅﺍل‬1.15
‫ﺍﻟﻤﻭﻀﻊ‬ ‫ﺇﻟﻰ‬ ‫ﺯﻨﺒﺭﻙ‬ ‫ﺒﻨﻬﺎﻴﺔ‬ ‫ﻤﺘﺼل‬ ‫ﺠﺴﻡ‬ ‫ﺴﺤﺏ‬
‫ﺍﻟﺠﺴﻡ‬ ‫ﻴﻘﻁﻌﻬﺎ‬.(a) A/2, (b) A, (c) 2A, (d) 4A
2.15‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺍﻟﺭﻴﺎﻀﻲ‬ ‫ﺍﻟﺘﻤﺜﻴل‬
harmonic oscillator
‫ﻋ‬ ‫ﺘﺤﺩﺜﻨﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺘﺼﻑ‬ ‫ﺭﻴﺎﻀﻴﺔ‬ ‫ﻤﻌﺎﺩﻟﺔ‬ ‫ﺒﺎﺸﺘﻘﺎﻕ‬ ‫ﺍﻵﻥ‬ ‫ﺴﻨﻘﻭﻡ‬
‫ﻤﺤﻭﺭ‬ ‫ﻋﻠﻰ‬ ‫ﻴﺘﺫﺒﺫﺏ‬ ‫ﻭﺍﻟﺫﻱ‬ ‫ﺒﺎﻟﺯﻨﺒﺭﻙ‬ ‫ﺍﻟﻤﺘﺼل‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻋﻠﻰ‬ ‫ﺍﻟﻤﺅﺜﺭﺓ‬
‫ﺍﻟﻌﺠﻠﺔ‬ ‫ﻓﺎﻥ‬ ‫ﻟﻠﺯﻤﻥ‬a‫ﺘﻌﻁﻰ‬‫ﺍﻟﺘﺎﻟﻲ‬ ‫ﺍﻟﻨﺤﻭ‬ ‫ﻋﻠﻰ‬

11
x
m
k
dt
xd
−=2
2
(15.3)
‫ﺍﻟﺭﻤﺯ‬ ‫ﺍﺨﺘﺭﻨﺎ‬ω2
‫ﻋﻥ‬ ‫ﺒﺩﻻ‬ω‫ﺍﻟﺫﻱ‬ ‫ﺍﻟﺤل‬ ‫ﻨﺠﻌل‬ ‫ﺤﺘﻰ‬ ‫ﻭﺫﻟﻙ‬‫ﺃﻜﺜﺭ‬ ‫ﻟﻪ‬ ‫ﻨﺼل‬ ‫ﺴﻭﻑ‬
m
k
=2
ω (15.4)
‫ﺍﻟﺘﺎﻟﻲ‬ ‫ﺍﻟﻨﺤﻭ‬ ‫ﻋﻠﻰ‬ ‫ﻨﻜﺘﺒﻬﺎ‬ ‫ﺃﻥ‬ ‫ﻴﻤﻜﻥ‬:
x
dt
xd 2
2
2
ω−= (15.5)
‫ﺩﺍﻟﺔ‬ ‫ﺴﻴﻜﻭﻥ‬ ‫ﺍﻟﺤل‬ ‫ﻫﺫﺍ‬ ‫ﺭﻴﺎﻀﻲ‬ ‫ﺤل‬ ‫ﺍﻟﻰ‬ ‫ﻭﺘﺤﺘﺎﺝ‬ ‫ﺍﻟﺜﺎﻨﻴﺔ‬ ‫ﺍﻟﺩﺭﺠﺔ‬ ‫ﻤﻥ‬ ‫ﺘﻔﺎﻀﻠﻴﺔ‬ ‫ﻤﻌﺎﺩﻟﺔ‬ ‫ﻫﻲ‬x(t)‫ﺘﺤﻘﻕ‬
‫ﻟﻠﺯﻤﻥ‬ ‫ﺒﺎﻟﻨﺴﺒﺔ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻭﻗﻊ‬ ‫ﺘﻐﻴﺭ‬ ‫ﻭﺘﻤﺜل‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬.‫ﻤﻌﺎﺩﻟﺔ‬ ‫ﻋﻥ‬ ‫ﻨﺒﺤﺙ‬ ‫ﺴﻭﻑ‬x(t)‫ﻟﻬﺎ‬ ‫ﺍﻟﺜﺎﻨﻴﺔ‬ ‫ﺍﻟﻤﺸﺘﻘﺔ‬ ‫ﻨﺘﻴﺠﺔ‬ ‫ﺘﻜﻭﻥ‬ ‫ﺒﺤﻴﺙ‬
‫ﻓﻲ‬ ‫ﻭﻤﻀﺭﻭﺒﺔ‬ ‫ﺍﻹﺸﺎﺭﺓ‬ ‫ﺴﺎﻟﺒﺔ‬ ‫ﺍﻷﺼﻠﻴﺔ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻨﻔﺴﻬﺎ‬ ‫ﻫﻲ‬ω2
.‫ﺍﻟﺠﻴﺏ‬ ‫ﺩﺍﻟﺔ‬ ‫ﻤﺜل‬ ‫ﺍﻟﺠﻴﺒﻴﺔ‬ ‫ﺍﻟﺩﻭﺍل‬ ‫ﻓﻲ‬ ‫ﻤﺘﺤﻘﻕ‬ ‫ﻭﻫﺫﺍ‬sin‫ﺃﻭ‬
www.trgma.com
‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻋﻥ‬ ‫ﻨﻌﺒﺭ‬ ‫ﺍﻥ‬ ‫ﻴﻤﻜﻥ‬ ‫ﻭﺒﺎﻟﺘﺎﻟﻲ‬2.15‫ﺍﻟﺘﺎﻟﻲ‬ ‫ﺍﻟﻨﺤﻭ‬ ‫ﻋﻠﻰ‬
(15.3)
‫ﺍﻟﻨﺴﺒﺔ‬ ‫ﻋﻥ‬ ‫ﻟﻨﻌﺒﺭ‬
m
k
‫ﺒـ‬ω2
)‫ﺍﻟﺭﻤﺯ‬ ‫ﺍﺨﺘﺭﻨﺎ‬
‫ﺴﻬﻭﻟﺔ‬(،‫ﻭﻋﻠﻴﻪ‬ ،
(15.4)
‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻓﺎﻥ‬ ‫ﻭﺒﺎﻟﺘﺎﻟﻲ‬3.15‫ﺍﻟﺘﺎﻟﻲ‬ ‫ﺍﻟﻨﺤﻭ‬ ‫ﻋﻠﻰ‬ ‫ﻨﻜﺘﺒﻬﺎ‬ ‫ﺃﻥ‬ ‫ﻴﻤﻜﻥ‬
(15.5)
‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬)5.15(‫ﺩﺍﻟﺔ‬ ‫ﺴﻴﻜﻭﻥ‬ ‫ﺍﻟﺤل‬ ‫ﻫﺫﺍ‬ ‫ﺭﻴﺎﻀﻲ‬ ‫ﺤل‬ ‫ﺍﻟﻰ‬ ‫ﻭﺘﺤﺘﺎﺝ‬ ‫ﺍﻟﺜﺎﻨﻴﺔ‬ ‫ﺍﻟﺩﺭﺠﺔ‬ ‫ﻤﻥ‬ ‫ﺘﻔﺎﻀﻠﻴﺔ‬ ‫ﻤﻌﺎﺩﻟﺔ‬ ‫ﻫﻲ‬
‫ﻟﻠﺯﻤﻥ‬ ‫ﺒﺎﻟﻨﺴﺒﺔ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻭﻗﻊ‬ ‫ﺘﻐﻴﺭ‬ ‫ﻭﺘﻤﺜل‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬
‫ﻓﻲ‬ ‫ﻭﻤﻀﺭﻭﺒﺔ‬ ‫ﺍﻹﺸﺎﺭﺓ‬ ‫ﺴﺎﻟﺒﺔ‬ ‫ﺍﻷﺼﻠﻴﺔ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻨﻔﺴﻬﺎ‬ ‫ﻫﻲ‬

12
‫ﺘﻜﻭﻥ‬ ‫ﺍﻥ‬ ‫ﺍﻟﻤﻤﻜﻥ‬ ‫ﻤﻥ‬ ‫ﻭﺒﺎﻟﺘﺎﻟﻲ‬‫ﻟﻠﻤﻌﺎﺩﻟﺔ‬ ‫ﺤﻼ‬ ‫ﻤﻨﻬﻤﺎ‬ ‫ﺃﻱ‬)5.15.(‫ﻟﻠﻤﻌﺎﺩﻟﺔ‬ ‫ﻜﺤل‬ ‫ﺍﻟﺘﺎﻟﻴﺔ‬ ‫ﺍﻟﺩﺍﻟﺔ‬ ‫ﻨﺨﺘﺎﺭ‬ ‫ﺴﻭﻑ‬
x(t) = A cos (ωt+ϕ) (15.6)
‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﺇﻥ‬ ‫ﻤﻥ‬ ‫ﻨﺘﺄﻜﺩ‬ ‫ﻭﻟﻜﻲ‬)6.15(‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﺘﺤﻘﻕ‬)5.15(‫ﺍﻻﺸﺘﻘﺎﻕ‬ ‫ﻫﺫﺍ‬ ‫ﺘﺘﺒﻊ‬
)sin()cos( ϕωωϕω +−=+= tAt
dt
d
A
dt
dx
)cos()sin( 2
2
2
ϕωωϕωω +−=+−= tAt
dt
d
A
dt
xd
‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﺇﻟﻰ‬ ‫ﺘﺅﻭل‬)5.15(‫ﺍﻟﺩﺍﻟﺔ‬ ‫ﺃﻥ‬ ‫ﻴﻌﻨﻲ‬ ‫ﻭﻫﺫﺍ‬)6.15(‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺍﻟﺘﻔﺎﻀﻠﻴﺔ‬ ‫ﻟﻠﻤﻌﺎﺩﻟﺔ‬ ‫ﺤل‬ ‫ﻫﻲ‬
.‫ﹰ‬‫ﺎ‬‫ﺒﻴﺎﻨﻴ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺘﻤﺜﻴل‬ ‫ﻤﻥ‬ ‫ﺍﻟﻤﻨﺎﺴﺏ‬ ‫ﻤﻥ‬ ،‫ﻟﻬﺎ‬ ‫ﺍﻟﻔﻴﺯﻴﺎﺌﻲ‬ ‫ﺍﻟﻤﻌﻨﻰ‬ ‫ﻨﻌﺭﻑ‬ ‫ﻭﻟﻜﻲ‬
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﻜﻤﺎ‬ ،‫ﺍﻟﺯﻤﻥ‬ ‫ﻓﻲ‬ ‫ﻜﺩﺍﻟﺔ‬2.15a.‫ﺇﻥ‬ ‫ﻻﺤﻅ‬ ،‫ﺃﻭﻻ‬A‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺒﺴﻌﺔ‬ ‫ﺘﻌﺭﻑ‬amplitude‫ﻭﻫﻲ‬
‫ﺍﻟﺴﺎﻟﺏ‬ ‫ﺍﻭ‬ ‫ﺍﻟﻤﻭﺠﺏ‬ ‫ﺍﻻﺘﺠﺎﻩ‬ ‫ﻓﻲ‬ ‫ﺴﻭﺍﺀ‬ ‫ﻟﻠﺠﺴﻡ‬ ‫ﺇﺯﺍﺤﺔ‬ ‫ﺃﻗﺼﻰ‬.‫ﺍﻟﺜﺎﺒﺕ‬ ‫ﺃﻤﺎ‬ω‫ﻴ‬‫ﺍﻟﺯﺍﻭﻱ‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﺒﺎﺴﻡ‬ ‫ﻌﺭﻑ‬angular
www.trgma.com
‫ﺍﻟﺘﻤﺎﻡ‬ ‫ﺠﻴﺏ‬cos،‫ﺘﻜﻭﻥ‬ ‫ﺍﻥ‬ ‫ﺍﻟﻤﻤﻜﻥ‬ ‫ﻤﻥ‬ ‫ﻭﺒﺎﻟﺘﺎﻟﻲ‬
‫ﺍﻟﺘﻔﺎﻀﻠﻴﺔ‬.
(15.6)
‫ﺇﻥ‬ ‫ﺤﻴﺙ‬A‫ﻭ‬ ،ω‫ﻭ‬ ،ϕ‫ﺜﻭﺍﺒﺕ‬ ‫ﻋﻥ‬ ‫ﻋﺒﺎﺭﺓ‬.‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﺇﻥ‬ ‫ﻤﻥ‬ ‫ﻨﺘﺄﻜﺩ‬ ‫ﻭﻟﻜﻲ‬
(15.7)
(15.8)
‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﺃﻥ‬ ‫ﻨﺠﺩ‬)8.15(‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﺇﻟﻰ‬ ‫ﺘﺅﻭل‬
‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬.
‫ﺍﻟﺜﻭﺍﺒﺕ‬A‫ﻭ‬ω‫ﻭ‬ϕ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺜﻭﺍﺒﺕ‬ ‫ﺘﻤﺜل‬.
‫ﻨﺭﺴ‬ ‫ﺤﻴﺙ‬‫ﻡ‬x‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﻜﻤﺎ‬ ،‫ﺍﻟﺯﻤﻥ‬ ‫ﻓﻲ‬ ‫ﻜﺩﺍﻟﺔ‬
‫ﺍﻟﺴﺎﻟﺏ‬ ‫ﺍﻭ‬ ‫ﺍﻟﻤﻭﺠﺏ‬ ‫ﺍﻻﺘﺠﺎﻩ‬ ‫ﻓﻲ‬ ‫ﺴﻭﺍﺀ‬ ‫ﻟﻠﺠﺴﻡ‬ ‫ﺇﺯﺍﺤﺔ‬ ‫ﺃﻗﺼﻰ‬

13
‫ﺍﻟﺘﺫﺒﺫﺏ‬ ‫ﺴﺭﻋﺔ‬ ‫ﻤﻘﺩﺍﺭ‬ ‫ﺘﺤﺩﺩ‬ ‫ﻭﻫﻲ‬–‫ﻤﻘﺩﺍﺭ‬ ‫ﺯﺍﺩ‬ ‫ﻓﻜﻠﻤﺎ‬‫ﺯﺍﺩﺕ‬ ‫ﻜﻠﻤﺎ‬ ‫ﺯﻤﻥ‬ ‫ﻭﺤﺩﺓ‬ ‫ﻟﻜل‬ ‫ﺍﻟﺘﺫﺒﺫﺏ‬
‫ﺍﻟﺘﺎﻟﻴﺔ‬ ‫ﺒﺎﻟﻌﻼﻗﺔ‬ ‫ﻴﻌﻁﻰ‬ ‫ﺍﻟﺯﺍﻭﻱ‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﺇﻥ‬ ‫ﻨﺠﺩ‬:
m
k
=ω (15.9)
‫ﺍﻟﻁﻭﺭ‬ ‫ﺜﺎﺒﺕ‬ ‫ﻴﺴﻤﻰ‬phase constant‫ﺍﻻﺒﺘﺩﺍﺌﻴﺔ‬ ‫ﺍﻟﻁﻭﺭ‬ ‫ﺯﺍﻭﻴﺔ‬ ‫ﺃﻭ‬initial phase
‫ﻭﺴﺭﻋ‬ ‫ﻤﻭﻗﻊ‬ ‫ﻴﺤﺩﺩﺍﻥ‬‫ﺘﻪ‬‫ﺍﻟﺯﻤﻥ‬ ‫ﻋﻨﺩ‬t=0.‫ﻟﻪ‬ ‫ﺇﺯﺍﺤﺔ‬ ‫ﺃﻗﺼﻰ‬ ‫ﻋﻨﺩ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻜﺎﻥ‬ ‫ﺇﺫﺍ‬
ϕ=0‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﻤﻭﻀﺢ‬ ‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺍﻟﺒﻴﺎﻨﻲ‬ ‫ﻭﺍﻟﺘﻤﺜﻴل‬2.15b.‫ﻭﺍﻟﻜﻤﻴﺔ‬)ωt+ϕ(
phase of motion.‫ﺍﻟﺩﺍﺭﺓ‬ ‫ﺍﻥ‬ ‫ﻻﺤﻅ‬x(t)‫ﻓﺘﺭﺓ‬ ‫ﻜل‬ ‫ﺘﺘﻜﺭﺭ‬ ‫ﻭﻗﻴﻤﺘﻬﺎ‬ ‫ﺩﻭﺭﻴﺎ‬ ‫ﺘﺘﻐﻴﺭ‬
‫ﻟﻼزاﺣﺔ‬ ‫ﻣﻨﺤﻨﻰ‬ ‫ﯾﻮﺿﺢ‬x‫اﻟﺰﻣﻦ‬ ‫ﻣﻊ‬t‫ﺑﺴﯿﻄﺔ‬ ‫ﺗﻮاﻓﻘﯿﺔ‬ ‫ﺣﺮﻛﺔ‬ ‫ﯾﺘﺤﺮك‬ ‫ﻟﺠﺴﻢ‬.‫ھﻲ‬ ‫اﻟﺤﺮﻛﺔ‬ ‫ﺳﻌﺔ‬A،
) .b(‫ﻟﻼزاﺣﺔ‬ ‫ﻣﻨﺤﻨﻰ‬ ‫ﯾﻮﺿﺢ‬x‫اﻟﺰﻣﻦ‬ ‫ﻣﻊ‬t‫ﺗﻜﻮن‬ ‫ﻋﻨﺪﻣﺎ‬ ‫ﺧﺎﺻﺔ‬ ‫ﻟﺤﺎﻟﺔ‬x=A
www.trgma.com
frequency‫ﻭﺤﺩﺓ‬ ‫ﻭﻟﻪ‬rad/s.‫ﺍﻟﺘﺫﺒﺫﺏ‬ ‫ﺴﺭﻋﺔ‬ ‫ﻤﻘﺩﺍﺭ‬ ‫ﺘﺤﺩﺩ‬ ‫ﻭﻫﻲ‬
‫ﻗﻴﻤﺔ‬ω.‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻤﻥ‬4.15‫ﺍﻟﺘﺎﻟﻴﺔ‬ ‫ﺒﺎﻟﻌﻼﻗﺔ‬ ‫ﻴﻌﻁﻰ‬ ‫ﺍﻟﺯﺍﻭﻱ‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﺇﻥ‬ ‫ﻨﺠﺩ‬
‫ﺒﺎﻟﺯﺍﻭﻴﺔ‬ ‫ﻴﺤﺩﺩ‬ ‫ﺍﻟﺫﻱ‬ ‫ﺍﻟﺜﺎﺒﺕ‬ϕ‫ﺍﻟﻁﻭﺭ‬ ‫ﺜﺎﺒﺕ‬ ‫ﻴﺴﻤﻰ‬
angle‫ﺍﻟﺴ‬ ‫ﻤﻊ‬ ‫ﺍﻟﺜﺎﺒﺕ‬ ‫ﻭﻫﺫﺍ‬‫ﻌﺔ‬A‫ﻭﺴﺭﻋ‬ ‫ﻤﻭﻗﻊ‬ ‫ﻴﺤﺩﺩﺍﻥ‬
x=A‫ﻭﻋﻨﺩ‬t=0‫ﺍﻟﻁﻭﺭ‬ ‫ﺜﺎﺒﺕ‬ ‫ﻓﺎﻥ‬ ،=0
‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻁﻭﺭ‬ ‫ﺒﺎﺴﻡ‬ ‫ﺘﻌﺭﻑ‬phase of motion
‫ﺯﻤﻨﻴﺔ‬ωt+2π.
‫اﻟﺸﻜﻞ‬2.15)a(‫ﻟﻼزاﺣﺔ‬ ‫ﻣﻨﺤﻨﻰ‬ ‫ﯾﻮﺿﺢ‬
‫اﻟﺪوري‬ ‫واﻟﺰﻣﻦ‬T‫اﻟﻄﻮر‬ ‫وﺛﺎﺑﺖ‬ ،ϕ) .
‫و‬ϕ=0.
‫ﺸﻜل‬2.15

14
‫ﺍﻟﺭﻴ‬ ‫ﺍﻟﺘﻤﺜﻴل‬ ‫ﺘﺸﻜل‬‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺎﻀﻲ‬.‫ﻴﻤﻜﻥ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻋﻠﻰ‬ ‫ﺍﻟﻤﺅﺜﺭﺓ‬ ‫ﻓﺎﻟﻘﻭﺓ‬
‫ﺒﺎﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻴﻭﺼﻑ‬ ‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻴﺘﺤﺭﻙ‬ ‫ﺍﻟﺫﻱ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻭﻤﻭﻀﻊ‬ ،6.15‫ﻗﻤﻨﺎ‬ ‫ﻭﺍﺫﺍ‬ ،
‫ﻨﺼﻔﻬ‬ ‫ﺃﻥ‬ ‫ﺍﻟﻤﻤﻜﻥ‬ ‫ﻤﻥ‬ ‫ﺍﻨﻪ‬ ‫ﻭﻭﺠﺩﻨﺎ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺤﺭﻜﺔ‬ ‫ﺒﺘﺤﻠﻴل‬‫ﺎ‬‫ﺍﻟﺘﻔﺎﻀﻠﻴﺔ‬ ‫ﺒﺎﻟﻤﻌﺎﺩﻟﺔ‬5.15‫ﺘﻭﺍ‬ ‫ﺤﺭﻜﺔ‬ ‫ﺘﻜﻭﻥ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻓﺎﻥ‬‫ﻓﻘﻴﺔ‬
‫ﺒﺎﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻴﻭﺼﻑ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻭﻗﻊ‬ ‫ﻜﺎﻥ‬ ‫ﻭﺇﺫﺍ‬6.15‫ﺃﻴﻀﺎ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻓﺎﻥ‬‫ﻴﺨﻀﻊ‬‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﻟﺤﺭﻜﺔ‬.
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﺍﻟﻤﻭﻀﺢ‬ ‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﻟﺤﺭﻜﺔ‬ ‫ﺍﻟﺒﻴﺎﻨﻲ‬ ‫ﺍﻟﺘﻤﺜﻴل‬ ‫ﻟﻨﻔﺘﺭﺽ‬4.15‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﺘﺼﻔﻬﺎ‬ ‫ﻜﻤﺎ‬6.15.‫ﻴﻜﻭﻥ‬ ‫ﻋﻨﺩﻤﺎ‬
‫ﻓﺎﻥ‬ ،‫ﺍﻟﻤﻨﺤﻨﻰ‬ ‫ﻋﻠﻰ‬)a(‫ﺍﻟ‬‫ﻤﻭﺠﺒﺔ‬ ‫ﺘﻜﻭﻥ‬ ‫ﻭﺍﻟﺴﺭﻋﺔ‬ ‫ﻤﻭﻀﻊ‬)b(‫ﻜﻼ‬ ‫ﻭﺍﻟﺴﺭﻋﺔ‬ ‫ﺍﻟﻤﻭﻀﻊ‬،‫ﹰ‬‫ﺎ‬‫ﺴﺎﻟﺒ‬ ‫ﻫﻤﺎ‬
)d(،‫ﺼﻔﺭ‬ ‫ﻭﺍﻟﺴﺭﻋﺔ‬ ‫ﺴﺎﻟﺏ‬ ‫ﺍﻟﻤﻭﻀﻊ‬)e(،‫ﺴﺎﻟﺒﺔ‬ ‫ﻭﺍﻟﺴﺭﻋﺔ‬ ‫ﻤﻭﺠﺏ‬ ‫ﺍﻟﻤﻭﻀﻊ‬
www.trgma.com
‫ﺍﻟﻤﻌﺎﻻﺕ‬1.15‫ﻭ‬5.15‫ﻭ‬6.15‫ﺍﻟﺭﻴ‬ ‫ﺍﻟﺘﻤﺜﻴل‬ ‫ﺘﺸﻜل‬
‫ﺒﺎﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻨﻤﺜﻠﻬﺎ‬ ‫ﺃﻥ‬1.15‫ﺒﺎﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻴﻭﺼﻑ‬ ‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻴﺘﺤﺭﻙ‬ ‫ﺍﻟﺫﻱ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻭﻤﻭﻀﻊ‬ ،
‫ﻨﺼﻔﻬ‬ ‫ﺃﻥ‬ ‫ﺍﻟﻤﻤﻜﻥ‬ ‫ﻤﻥ‬ ‫ﺍﻨﻪ‬ ‫ﻭﻭﺠﺩﻨﺎ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺤﺭﻜﺔ‬ ‫ﺒﺘﺤﻠﻴل‬
‫ﺒﺴﻴﻁﺔ‬.‫ﺒﺎﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻴﻭﺼﻑ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻭﻗﻊ‬ ‫ﻜﺎﻥ‬ ‫ﻭﺇﺫﺍ‬
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﺍﻟﻤﻭﻀﺢ‬ ‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﻟﺤﺭﻜﺔ‬ ‫ﺍﻟﺒﻴﺎﻨﻲ‬ ‫ﺍﻟﺘﻤﺜﻴل‬ ‫ﻟﻨﻔﺘﺭﺽ‬
‫ﺍﻟﻨﻘﻁﺔ‬ ‫ﻋﻨﺩ‬ ‫ﺍﻟﺠﺴﻡ‬A‫ﻓﺎﻥ‬ ،‫ﺍﻟﻤﻨﺤﻨﻰ‬ ‫ﻋﻠﻰ‬
)c(،‫ﺴﺎﻟﺒﺔ‬ ‫ﻭﺍﻟﺴﺭﻋﺔ‬ ‫ﻤﻭﺠﺏ‬ ‫ﺍﻟﻤﻭﻀﻊ‬)
)f(‫ﻤﻭﺠﺒﺔ؟‬ ‫ﻭﺍﻟﺴﺭﻋﺔ‬ ‫ﺴﺎﻟﺏ‬ ‫ﺍﻟﻤﻭﻀﻊ‬
‫اﻟﺸﻜﻞ‬15.3‫ﺗﻌﺮض‬ ‫ﻋﻤﻠﯿﺔ‬ ‫ﺗﺠﺮﺑﺔ‬ ‫ﯾﻮﺿﺢ‬
‫اﻟﺒﺴﯿﻄﺔ‬ ‫اﻟﺘﻮاﻓﻘﯿﺔ‬ ‫اﻟﺤﺮﻛﺔ‬.‫ﻛﺘﻠﺘﮫ‬ ‫ﺟﺴﻢ‬m
‫ﻣﺘﺼﻞ‬‫اﻟﺠﺴﻢ‬ ‫ﻋﻠﻰ‬ ‫وﻣﺜﺒﺖ‬ ‫رأﺳﯿﺎ‬ ‫ﯾﺘﺬﺑﺬب‬ ‫ﺑﺰﻧﺒﺮك‬
‫ﻗﻠﻢ‬.‫ﺗﺘﺤﺮك‬ ‫ورﻗﺔ‬ ‫ﺑﻤﺤﺎذاة‬ ،‫اﻟﺠﺴﻢ‬ ‫ﯾﺘﺬﺑﺬب‬ ‫ﻋﻨﺪﻣﺎ‬
‫ﺳﻮف‬ ‫اﻟﻘﻠﻢ‬ ‫ﻓﺎن‬ ‫اﻟﺘﺬﺑﺬب‬ ‫اﺗﺠﺎه‬ ‫ﻋﻠﻰ‬ ‫ﻋﻤﻮدﯾﺎ‬
‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ﻓﻲ‬ ‫ﻛﻤﺎ‬ ‫ﺟﯿﺒﻲ‬ ‫ﻣﻨﺤﻨﻰ‬ ‫ﯾﺮﺳﻢ‬15.6.
‫اﻟﺸﻜﻞ‬
‫اﻟﺒﺴﯿﻄﺔ‬ ‫اﻟﺘﻮاﻓﻘﯿﺔ‬ ‫اﻟﺤﺮﻛﺔ‬
‫ﻣﺘﺼﻞ‬
‫ﻗﻠﻢ‬
‫ﺳﻮف‬ ‫اﻟﻘﻠﻢ‬ ‫ﻓﺎن‬ ‫اﻟﺘﺬﺑﺬب‬ ‫اﺗﺠﺎه‬ ‫ﻋﻠﻰ‬ ‫ﻋﻤﻮدﯾﺎ‬
‫اﻟﻤﻌﺎدﻟﺔ‬ ‫ﻓﻲ‬ ‫ﻛﻤﺎ‬ ‫ﺟﯿﺒﻲ‬ ‫ﻣﻨﺤﻨﻰ‬ ‫ﯾﺮﺳﻢ‬

15
‫ﺑﺴﯿﻄﺔ‬ ‫ﺗﻮاﻓﻘﯿﺔ‬ ‫ﺣﺮﻛﺔ‬ ‫ﯾﺘﺤﺮك‬ ‫ﻟﺠﺴﻢ‬ ‫ﻣﻨﺤﻨﻰ‬ ‫ﯾﻤﺜﻞ‬.‫اﻟﺠﺴ‬ ‫ﻓﺎن‬ ‫ﻣﻌﯿﻦ‬ ‫زﻣﻦ‬ ‫ﻋﻨﺪ‬‫اﻟﻤﻮﺿﻊ‬ ‫ﻋﻦ‬ ‫ﯾﻜﻮن‬ ‫ﻢ‬A‫ﻋﻠﻰ‬
‫اﻟﻤﻨﺤﻨﻰ‬.
‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻴﺘﺤﺭﻙ‬ ‫ﺠﺴﻡ‬ ‫ﻴﻤﺜﻼﻥ‬ ‫ﻤﻨﺤﻨﻴﻴﻥ‬ ‫ﻴﻭﻀﺢ‬.‫ﺇﻥ‬ ‫ﻫﻲ‬ ‫ﺍﻟﺤﺭﻜﺘﻴﻥ‬ ‫ﻟﻬﺎﺘﻴﻥ‬ ‫ﺍﻟﺼﺤﻴﺢ‬ ‫ﺍﻟﻭﺼﻑ‬
‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻥ‬ ‫ﺍﻜﺒﺭ‬ ‫ﻭﺴﻌﺔ‬ ‫ﺯﺍﻭﻱ‬ ‫ﺘﺭﺩﺩ‬ ‫ﻟﻪ‬A،)b(‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻥ‬ ‫ﺍﺼﻐﺭ‬ ‫ﺴﻌﺔ‬ ‫ﻭﻟﻜﻥ‬ ‫ﺍﻜﺒﺭ‬ ‫ﺯﺍﻭﻱ‬ ‫ﺘﺭﺩﺩ‬A،
‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻥ‬ ‫ﺍﻜﺒﺭ‬ ‫ﻭﺴﻌﺔ‬ ‫ﺍﺼﻐﺭ‬A،)d(‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻥ‬ ‫ﺍﻜﺒﺭ‬ ‫ﻭﺴﻌﺔ‬ ‫ﺯﺍﻭﻱ‬ ‫ﺘﺭﺩﺩ‬A.
www.trgma.com
‫اﻟﺸﻜﻞ‬4.15‫ﺑﺴﯿﻄﺔ‬ ‫ﺗﻮاﻓﻘﯿﺔ‬ ‫ﺣﺮﻛﺔ‬ ‫ﯾﺘﺤﺮك‬ ‫ﻟﺠﺴﻢ‬ ‫ﻣﻨﺤﻨﻰ‬ ‫ﯾﻤﺜﻞ‬
‫ﺍﻟﺸﻜل‬5.15‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻴﺘﺤﺭﻙ‬ ‫ﺠﺴﻡ‬ ‫ﻴﻤﺜﻼﻥ‬ ‫ﻤﻨﺤﻨﻴﻴﻥ‬ ‫ﻴﻭﻀﺢ‬
‫ﺍﻟﺠﺴﻡ‬B‫ﻴﻜﻭﻥ‬)a(‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻥ‬ ‫ﺍﻜﺒﺭ‬ ‫ﻭﺴﻌﺔ‬ ‫ﺯﺍﻭﻱ‬ ‫ﺘﺭﺩﺩ‬ ‫ﻟﻪ‬
)c(‫ﺯﺍﻭﻱ‬ ‫ﺘﺭﺩﺩ‬‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻥ‬ ‫ﺍﻜﺒﺭ‬ ‫ﻭﺴﻌﺔ‬ ‫ﺍﺼﻐﺭ‬

16
‫ﻤﺨﺘﻠﻔﺔ‬ ‫ﻭﺴﻌﺔ‬ ‫ﺯﺍﻭﻱ‬ ‫ﺘﺭﺩﺩ‬ ‫ﻟﻬﻤﺎ‬ ،‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻴﺘﺤﺭﻜﺎﻥ‬ ‫ﻟﺠﺴﻤﻴﻥ‬.
‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺍﻟﺭﻴﺎﻀﻲ‬ ‫ﺎﻟﺘﻤﺜﻴل‬.‫ﻓ‬‫ﺍﻟﺩﻭﺭ‬ ‫ﺎﻟﺯﻤﻥ‬‫ﻱ‬T‫ﻫﻭ‬ ‫ﻟﻠﺤﺭﻜﺔ‬
‫ﻜﺎﻤﻠﺔ‬ ‫ﺩﻭﺭﺓ‬ ‫ﻟﻴﺘﺤﺭﻙ‬ ‫ﻟﻠﺠﺴﻡ‬ ‫ﺍﻟﻼﺯﻤﺔ‬ ‫ﺍﻟﺯﻤﻨﻴﺔ‬ ‫ﺍﻟﻔﺘﺭﺓ‬)‫ﺍﻟﺸﻜل‬2.15a.(‫ﺍﻹﺯﺍﺤﺔ‬ ‫ﻗﻴﻤﺔ‬x‫ﻭﺍﻟﺴﺭﻋﺔ‬v‫ﺍﻟﺯﻤﻥ‬ ‫ﻋﻨﺩ‬ ‫ﻟﻠﺠﺴﻡ‬t
‫ﺍﻟﻁﻭﺭ‬ ‫ﺃﻥ‬ ‫ﻤﻥ‬ ‫ﺍﻻﺴﺘﻔﺎﺩﺓ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬ ‫ﺍﻟﺯﺍﻭﻱ‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﻤﻊ‬ ‫ﺍﻟﺩﻭﺭﻱ‬ ‫ﺍﻟﺯﻤﻥ‬ ‫ﻨﺭﺒﻁ‬ ‫ﺃﻥ‬ ‫ﻴﻤﻜﻨﻨﺎ‬
www.trgma.com
‫ﺍﻟﺸﻜل‬5.15‫ﺒﻴﺎﻨﻴﻴﻥ‬ ‫ﻤﻨﺤﻨﻴﻴﻥ‬x-t‫ﻤﺨﺘﻠﻔﺔ‬ ‫ﻭﺴﻌﺔ‬ ‫ﺯﺍﻭﻱ‬ ‫ﺘﺭﺩﺩ‬ ‫ﻟﻬﻤﺎ‬ ،‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻴﺘﺤﺭﻜﺎﻥ‬ ‫ﻟﺠﺴﻤﻴﻥ‬
‫ﺍﻟﺘﻭﻀﻴﺢ‬ ‫ﻤﻥ‬ ‫ﺒﻤﺯﻴﺩ‬ ‫ﺍﻵﻥ‬ ‫ﻟﻨﻘﻭﻡ‬‫ﺒ‬ ‫ﺒﺎﻻﺴﺘﻌﺎﻨﺔ‬‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺍﻟﺭﻴﺎﻀﻲ‬ ‫ﺎﻟﺘﻤﺜﻴل‬
‫ﻜﺎﻤﻠﺔ‬ ‫ﺩﻭﺭﺓ‬ ‫ﻟﻴﺘﺤﺭﻙ‬ ‫ﻟﻠﺠﺴﻡ‬ ‫ﺍﻟﻼﺯﻤﺔ‬ ‫ﺍﻟﺯﻤﻨﻴﺔ‬ ‫ﺍﻟﻔﺘﺭﺓ‬
‫ﺘﺴﺎﻭﻱ‬x‫ﻭ‬v‫ﺯﻤﻥ‬ ‫ﻋﻨﺩ‬t+T.‫ﺍﻟﻁﻭﺭ‬ ‫ﺃﻥ‬ ‫ﻤﻥ‬ ‫ﺍﻻﺴﺘﻔﺎﺩﺓ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬ ‫ﺍﻟﺯﺍﻭﻱ‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﻤﻊ‬ ‫ﺍﻟﺩﻭﺭﻱ‬ ‫ﺍﻟﺯﻤﻥ‬ ‫ﻨﺭﺒﻁ‬ ‫ﺃﻥ‬ ‫ﻴﻤﻜﻨﻨﺎ‬
‫ﺒﻤﻘﺩﺍﺭ‬ ‫ﻴﺯﺩﺍﺩ‬2π‫ﺯﻤﻨﻴﺔ‬ ‫ﻓﺘﺭﺓ‬ ‫ﻜل‬T:

17
[ω (t + T) + ϕ] – (ωt + ϕ) = 2
ωT = 2π,
ω
π2
=T (15.10)
frequency‫ﻟﻠﺤﺭﻜﺔ‬.‫ﻜﺎﻤﻠﺔ‬ ‫ﻻﻫﺘﺯﺍﺯﺓ‬ ‫ﺍﻟﺯﻤﻨﻴﺔ‬ ‫ﺍﻟﻔﺘﺭﺓ‬ ‫ﻫﻭ‬ ‫ﺍﻟﺩﻭﺭﻱ‬ ‫ﺍﻟﺯﻤﻥ‬ ‫ﺇﻥ‬ ‫ﺤﻴﺙ‬
‫ﻓﺘﺭﺓ‬ ‫ﻓﻲ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺒﻬﺎ‬ ‫ﻴﻘﻭﻡ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻻﻫﺘﺯﺍﺯﺍﺕ‬ ‫ﻋﺩﺩ‬ ‫ﻴﻤﺜل‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﻓﺎﻥ‬‫ﺍﻟﺯﻤﻥ‬.
π
ω
2
1
==
T
f (15.11)
‫ﺍﻟﻬﺭﺘﺯ‬ ‫ﺒﺎﺴﻡ‬ ‫ﻭﺘﻌﺭﻑ‬ ‫ﺜﺎﻨﻴﺔ‬ ‫ﻟﻜل‬ ‫ﺩﻭﺭﺓ‬ ‫ﻫﻲ‬hertz‫ﺒﺎﻟﺭﻤﺯ‬ ‫ﻟﻬﺎ‬ ‫ﻭﻴﺭﻤﺯ‬Hz.‫ﻟﻠﻤﻌﺎﺩﻟﺔ‬ ‫ﻭﺒﺎﻟﻨﺴﺒﺔ‬11.15‫ﻓﺎﻥ‬
T
f
π
πω
2
2 == (15.12)
‫ﻭ‬10.15‫ﻭ‬11.15‫ﻭﺍﻟ‬ ‫ﺍﻟﺩﻭﺭﻱ‬ ‫ﺍﻟﺯﻤﻥ‬ ‫ﻋﻥ‬ ‫ﻟﻠﺘﻌﺒﻴﺭ‬‫ﻤﺘﺼل‬ ‫ﺠﺴﻡ‬ ‫ﻟﺤﺭﻜﺔ‬ ‫ﺘﺭﺩﺩ‬
‫ﻟ‬ ‫ﺍﻟﺫﺍﺘﻴﺔ‬ ‫ﺍﻟﺨﺼﺎﺌﺹ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬ ‫ﻭﺫﻟﻙ‬ ‫ﺒﺎﻟﺯﻨﺒﺭﻙ‬‫ﻠ‬‫ﺍﻟﻜﺘﻠﺔ‬ ‫ﻭﻫﻲ‬ ‫ﻭﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﺠﺴﻡ‬m‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﻭﺜﺎﺒﺕ‬k.
www.trgma.com
) = 2π
‫ﺇﻥ‬ ‫ﻨﺠﺩ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻭﺒﺘﺒﺴﻴﻁ‬
(15.10)
‫ﻴﻌﺭﻑ‬ ‫ﺍﻟﺩﻭﺭﻱ‬ ‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻘﻠﻭﺏ‬‫ﺒﺎﻟﺘﺭﺩﺩ‬frequency
‫ﻓﺘﺭﺓ‬ ‫ﻓﻲ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺒﻬﺎ‬ ‫ﻴﻘﻭﻡ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻻﻫﺘﺯﺍﺯﺍﺕ‬ ‫ﻋﺩﺩ‬ ‫ﻴﻤﺜل‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﻓﺎﻥ‬
(15.11)
‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﻭﺤﺩﺓ‬f‫ﺍﻟﻬﺭﺘﺯ‬ ‫ﺒﺎﺴﻡ‬ ‫ﻭﺘﻌﺭﻑ‬ ‫ﺜﺎﻨﻴﺔ‬ ‫ﻟﻜل‬ ‫ﺩﻭﺭﺓ‬ ‫ﻫﻲ‬
(15.12)
‫ﺍﻟﻤﻌﺎﺩﻻﺕ‬ ‫ﻨﺴﺘﺨﺩﻡ‬ ‫ﺍﻥ‬ ‫ﺍﻟﻤﻤﻜﻥ‬ ‫ﻤﻥ‬9.15‫ﻭ‬
‫ﻟ‬ ‫ﺍﻟﺫﺍﺘﻴﺔ‬ ‫ﺍﻟﺨﺼﺎﺌﺹ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬ ‫ﻭﺫﻟﻙ‬ ‫ﺒﺎﻟﺯﻨﺒﺭﻙ‬

18
k
m
T π
ω
π
2
2
== (15.3)
m
k
T
f
π2
11
== (15.4)
‫ﻋ‬ ‫ﻋﻠﻰ‬ ‫ﻭﻟﻴﺱ‬ ،‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﺜﺎﺒﺕ‬ ‫ﻭﻋﻠﻰ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻜﺘﻠﺔ‬ ‫ﻋﻠﻰ‬ ‫ﻴﻌﺘﻤﺩﺍﻥ‬ ‫ﻭﺍﻟﺘﺭﺩﺩ‬ ‫ﺍﻟﺩﻭﺭﻱ‬ ‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻥ‬ ‫ﻜﻼ‬ ‫ﻴﻜﻭﻥ‬ ‫ﻭﺒﻬﺫﺍ‬،‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﻭﺍﻤل‬
‫ﺯﻨﺒﺭﻙ‬ ‫ﺜﺎﺒﺕ‬ ‫ﺫﻭ‬ ‫ﺃﻱ‬ ‫ﺍﻟﻘﺎﺴﻲ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﺤﺎﻟﺔ‬ ‫ﻓﻲ‬ ‫ﺍﻜﺒﺭ‬ ‫ﻴﻜﻭﻥ‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﻓﺎﻥ‬ ،‫ﻨﺘﻭﻗﻊ‬ ‫ﻭﻜﻤﺎ‬k
‫ﺍﻟﻤﻌﺎﺩﻟﺘﻴﻥ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬ ‫ﻭﺫﻟﻙ‬ ‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻴﺘﺤﺭﻙ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﻭﻋﺠﻠﺘﻪ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺴﺭﻋﺔ‬ ‫ﻨﺠﺩ‬ ‫ﺃﻥ‬ ‫ﺍﻟﻤﻤﻜﻥ‬ ‫ﻤﻥ‬7.15‫ﻭ‬
)sin( ϕωω +−== tA
dt
dx
v
)cos(2
2
2
ϕωω +−== tA
dt
xd
a
www.trgma.com
(15.3)
(15.4)
‫ﻋ‬ ‫ﻋﻠﻰ‬ ‫ﻭﻟﻴﺱ‬ ،‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﺜﺎﺒﺕ‬ ‫ﻭﻋﻠﻰ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻜﺘﻠﺔ‬ ‫ﻋﻠﻰ‬ ‫ﻴﻌﺘﻤﺩﺍﻥ‬ ‫ﻭﺍﻟﺘﺭﺩﺩ‬ ‫ﺍﻟﺩﻭﺭﻱ‬ ‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻥ‬ ‫ﻜﻼ‬ ‫ﻴﻜﻭﻥ‬ ‫ﻭﺒﻬﺫﺍ‬
‫ﺍﻟﺴﻌﺔ‬ ‫ﻫﻲ‬ ‫ﻭﺍﻟﺘﻲ‬A‫ﻭﺍﻟﻁﻭﺭ‬ϕ.‫ﺯﻨﺒﺭﻙ‬ ‫ﺜﺎﺒﺕ‬ ‫ﺫﻭ‬ ‫ﺃﻱ‬ ‫ﺍﻟﻘﺎﺴﻲ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﺤﺎﻟﺔ‬ ‫ﻓﻲ‬ ‫ﺍﻜﺒﺭ‬ ‫ﻴﻜﻭﻥ‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﻓﺎﻥ‬ ،‫ﻨﺘﻭﻗﻊ‬ ‫ﻭﻜﻤﺎ‬
‫ﺍﻟﺠﺴﻡ‬ ‫ﻜﺘﻠﺔ‬ ‫ﺒﺯﻴﺎﺩﺓ‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﻭﻴﻘل‬ ‫ﻜﺒﻴﺭ‬.
‫ﺍﻟﻤﻌﺎﺩﻟﺘﻴﻥ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬ ‫ﻭﺫﻟﻙ‬ ‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺤﺭﻜﺔ‬ ‫ﻴﺘﺤﺭﻙ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﻭﻋﺠﻠﺘﻪ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺴﺭﻋﺔ‬ ‫ﻨﺠﺩ‬ ‫ﺃﻥ‬ ‫ﺍﻟﻤﻤﻜﻥ‬ ‫ﻤﻥ‬
8.15:
(15.15)
(15.16)

19
‫ﺍﻟﻘﻴﻤﺘ‬ ‫ﺒﻴﻥ‬ ‫ﺘﺘﺫﺒﺫﺒﺎﻥ‬ ‫ﺍﻟﺘﻤﺎﻡ‬ ‫ﻭﺠﻴﺏ‬ ‫ﺍﻟﺠﻴﺏ‬ ‫ﺩﺍﻟﺘﻲ‬ ‫ﺒﺴﺒﺏ‬ ‫ﺍﻨﻪ‬ ‫ﻨﺠﺩ‬‫ﺒ‬‫ﻥ‬±1،‫ﻓﺈﻥ‬‫ﺴﻭﻑ‬ ‫ﻟﻠﺴﺭﻋﺔ‬ ‫ﺍﻟﻌﻅﻤﻰ‬ ‫ﺍﻟﻘﻴﻡ‬
‫ﻟﻠﻌﺠﻠﺔ‬ ‫ﺍﻟﻌﻅﻤﻰ‬ ‫ﺍﻟﻘﻴﻡ‬ ‫ﺍﻥ‬ ‫ﺘﻌﻁﻴﻨﺎ‬a‫ﺘﻜﻭﻥ‬±ω2
A.‫ﻓﺎ‬ ‫ﻭﻋﻠﻴﻪ‬‫ﺍﻟﻌﻅﻤﻰ‬ ‫ﺍﻟﻘﻴﻡ‬ ‫ﻥ‬
A
m
k
Av == ωmax (15.17)
A
m
k
Aa == 2
max ω (15.18)
‫ﻟﻠﻁﻭﺭ‬ ‫ﺍﺨﺘﻴﺎﺭﻴﺔ‬ ‫ﻗﻴﻤﺔ‬ ‫ﻋﻨﺩ‬ ‫ﻭﺍﻟﺯﻤﻥ‬ ‫ﺍﻟﻤﻭﻀﻊ‬ ‫ﺒﻴﻥ‬ ‫ﻟﻠﻌﻼﻗﺔ‬ ‫ﻤﺨﻁﻁ‬ ‫ﻴﺒﻴﻥ‬.‫ﺍﻟﺴﺭﻋﺔ‬ ‫ﻋﻼﻗﺔ‬ ‫ﻤﻨﺤﻨﻰ‬‫ﻤﻊ‬‫ﺍﻟﺯﻤﻥ‬
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﻤﻭﻀﺤﺔ‬ ‫ﺍﻟﺯﻤﻥ‬6.15b‫ﻭﺍﻟﺸﻜل‬6.15c.‫ﻁﻭﺭ‬ ‫ﻋﻥ‬ ‫ﻴﺨﺘﻠﻑ‬ ‫ﺍﻟﺴﺭﻋﺔ‬ ‫ﻁﻭﺭ‬ ‫ﺍﻥ‬ ‫ﻨﺠﺩ‬ ‫ﺤﻴﺙ‬
‫ﺘﻜﻭﻥ‬ ‫ﻓﻌﻨﺩﻤﺎ‬x‫ﺼﻔﺭ‬ ‫ﺘﻜﻭﻥ‬ ‫ﺍﻟﺴﺭﻋﺔ‬ ‫ﻓﺎﻥ‬ ،‫ﻗﻴﻤﺔ‬ ‫ﺍﻋﻠﻰ‬ ‫ﺃﻭ‬ ‫ﻗﻴﻤﺔ‬ ‫ﺍﻗﺼﻰ‬ ‫ﻓﻲ‬.‫ﻋﻨﺩﻤﺎ‬ ‫ﻭﺒﺎﻟﻤﺜل‬
‫ﻴﻤﻜﻥ‬ ‫ﻤﺎ‬ ‫ﺍﻗﺼﻰ‬ ‫ﺘﻜﻭﻥ‬ ‫ﺍﻟﺴﺭﻋﺔ‬ ‫ﻓﺎﻥ‬ ‫ﺼﻔﺭ‬ ‫ﺘﺴﺎﻭﻱ‬.‫ﺍﻟﻤﻭﻀﻊ‬ ‫ﻁﻭﺭ‬ ‫ﻋﻥ‬ ‫ﻴﺨﺘﻠﻑ‬ ‫ﺍﻟﻌﺠﻠﺔ‬ ‫ﻁﻭﺭ‬ ‫ﺍﻥ‬ ‫ﻨﻼﺤﻅ‬ ،‫ﺫﻟﻙ‬ ‫ﺍﻟﻰ‬ ‫ﺍﻀﺎﻓﺔ‬
‫ﺘﻜﻭﻥ‬ ‫ﻋﻨﺩﻤﺎ‬ ،‫ﺍﻟﻤﺜﺎل‬ ‫ﺴﺒﻴل‬ ‫ﻋﻠﻰ‬x‫ﺍﻻﺘﺠﺎﻩ‬ ‫ﻓﻲ‬ ‫ﻭﻟﻜﻥ‬ ‫ﻴﻤﻜﻥ‬ ‫ﻤﺎ‬ ‫ﺃﻗﺼﻰ‬ ‫ﺍﻟﻌﺠﻠﺔ‬ ‫ﻓﺎﻥ‬ ،‫ﻗﻴﻤﺔ‬ ‫ﺃﻗﺼﻰ‬
www.trgma.com
‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻤﻥ‬5.15‫ﺍﻟﻘﻴﻤﺘ‬ ‫ﺒﻴﻥ‬ ‫ﺘﺘﺫﺒﺫﺒﺎﻥ‬ ‫ﺍﻟﺘﻤﺎﻡ‬ ‫ﻭﺠﻴﺏ‬ ‫ﺍﻟﺠﻴﺏ‬ ‫ﺩﺍﻟﺘﻲ‬ ‫ﺒﺴﺒﺏ‬ ‫ﺍﻨﻪ‬ ‫ﻨﺠﺩ‬
‫ﺘﻜﻭﻥ‬±ωA.‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻤﻥ‬ ‫ﻭﺒﺎﻟﻤﺜل‬6.15‫ﻟﻠﻌﺠﻠﺔ‬ ‫ﺍﻟﻌﻅﻤﻰ‬ ‫ﺍﻟﻘﻴﻡ‬ ‫ﺍﻥ‬ ‫ﺘﻌﻁﻴﻨﺎ‬
‫ﺘﻜﻭﻥ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﻟﻠﺴﺭﻋﺔ‬
(15.17)
(15.18)
‫ﺍﻟﺸﻜل‬6.15a‫ﻟﻠﻁﻭﺭ‬ ‫ﺍﺨﺘﻴﺎﺭﻴﺔ‬ ‫ﻗﻴﻤﺔ‬ ‫ﻋﻨﺩ‬ ‫ﻭﺍﻟﺯﻤﻥ‬ ‫ﺍﻟﻤﻭﻀﻊ‬ ‫ﺒﻴﻥ‬ ‫ﻟﻠﻌﻼﻗﺔ‬ ‫ﻤﺨﻁﻁ‬ ‫ﻴﺒﻴﻥ‬
‫ﺍﻟﻌﺠﻠﺔ‬ ‫ﻭﻋﻼﻗﺔ‬‫ﻤﻊ‬‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﻤﻭﻀﺤﺔ‬ ‫ﺍﻟﺯﻤﻥ‬
‫ﺒﻤﻘﺩﺍﺭ‬ ‫ﺍﻟﻤﻭﻀﻊ‬90o
‫ﺃﻱ‬π/2.‫ﺘﻜﻭﻥ‬ ‫ﻓﻌﻨﺩﻤﺎ‬
x‫ﻴﻤﻜﻥ‬ ‫ﻤﺎ‬ ‫ﺍﻗﺼﻰ‬ ‫ﺘﻜﻭﻥ‬ ‫ﺍﻟﺴﺭﻋﺔ‬ ‫ﻓﺎﻥ‬ ‫ﺼﻔﺭ‬ ‫ﺘﺴﺎﻭﻱ‬
‫ﺒﻤﻘﺩﺍﺭ‬180o
‫ﺍﻭ‬π.‫ﺘﻜﻭﻥ‬ ‫ﻋﻨﺩﻤﺎ‬ ،‫ﺍﻟﻤﺜﺎل‬ ‫ﺴﺒﻴل‬ ‫ﻋﻠﻰ‬
‫ﺍﻟﻤﻌﺎﻜﺱ‬.

20
www.trgma.com
‫اﻟﺸﻜﻞ‬6.15‫ﻟﻠﺤﺮﻛﺔ‬ ‫ﺑﯿﺎﻧﻲ‬ ‫ﺗﻤﺜﯿﻞ‬ ‫ﯾﻮﺿﺢ‬
‫اﻟﺒﺴﯿﻄﺔ‬ ‫اﻟﺘﻮاﻓﯿﻘﺔ‬) .a(‫اﻟﻤﻮﺿﻊ‬ ‫ﻋﻼﻗﺔ‬
‫ﺑﺎﻟﺰﻣﻦ‬) .b(‫اﻟﺴﺮﻋﺔ‬ ‫ﻋﻼﻗﺔ‬‫اﻟﺰﻣﻦ‬ ‫ﻣﻊ‬.
)c(‫اﻟﺰﻣﻦ‬ ‫ﻣﻊ‬ ‫اﻟﻌﺠﻠﺔ‬ ‫ﻋﻼﻗﺔ‬.‫أي‬ ‫ﻋﻨﺪ‬ ‫ﻻﺣﻆ‬
‫اﻟﺴﺮﻋﺔ‬ ‫ﺗﻜﻮن‬ ‫زﻣﻦ‬90o
‫اﻟﻄﻮر‬ ‫ﻓﻲ‬ ‫ﻣﺨﺘﻠﻔﺔ‬
‫ﻓﻲ‬ ‫ﻣﺨﺘﻠﻔﺔ‬ ‫ﺗﻜﻮن‬ ‫واﻟﻌﺠﻠﺔ‬ ‫اﻟﻤﻮﺿﻊ‬ ‫ﻣﻊ‬
‫ﺑﻤﻘﺪار‬ ‫اﻟﻤﻮﺿﻊ‬ ‫ﻣﻊ‬ ‫اﻟﻄﻮر‬180o
.

21
‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻭﺼﻔﺘﻬﺎ‬ ‫ﻜﻤﺎ‬ ،‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﻟﻠﺤﺭﻜﺔ‬6.15.‫ﻋﻨﺩ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻴﻜﻭﻥ‬ ‫ﻋﻨﺩﻤﺎ‬
a(،‫ﻤﻭﺠﺒﺘﺎﻥ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﺍﻟﺴﺭﻋﺔ‬ ‫ﻤﻥ‬ ‫ﻜﻼ‬)b(،‫ﺴﺎﻟﺒﺘﺎﻥ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﺍﻟﺴﺭﻋﺔ‬ ‫ﻤﻥ‬ ‫ﻜﻼ‬
،‫ﺼﻔﺭ‬ ‫ﺘﺴﺎﻭﻱ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﻤﻭﺠﺒﺔ‬ ‫ﺍﻟﺴﺭﻋﺔ‬)d(،‫ﺼﻔﺭ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﺴﺎﻟﺒﺔ‬ ‫ﺍﻟﺴﺭﻋﺔ‬)e(،‫ﺴﺎﻟﺒﺔ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﻤﻭﺠﺒﺔ‬ ‫ﺍﻟﺴﺭﻋﺔ‬)f(
‫ﺍﻨﻪ‬ ‫ﻋﻠﻰ‬ ‫ﻭﻤﺴﺠل‬ ‫ﻤﻘﺎﺱ‬ ‫ﻟﻠﺘﺫﺒﺫﺏ‬ ‫ﺍﻟﺩﻭﺭﻱ‬ ‫ﻟﺯﻤﻥ‬T.‫ﺍﻟﺠﺴﻡ‬ ‫ﺍﺴﺘﺒﺩﺍل‬ ‫ﺘﻡ‬ ‫ﻓﺎﺫﺍ‬m
‫ﺍﻟﺠﺴﻡ‬ ‫ﻴﺘﺫﺒﺫﺏ‬ ‫ﻋﻨﺩﻤﺎ‬2m‫ﻴﻜﻭﻥ‬ ‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺍﻟﺩﻭﺭﻱ‬ ‫ﺍﻟﺯﻤﻥ‬ ‫ﻓﺎﻥ‬)a(2T،)b(T2،)c(T)d(
www.trgma.com
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﺍﻟﺒﻴﺎﻨﻲ‬ ‫ﺍﻟﺘﻤﺜﻴل‬ ‫ﺍﻓﺘﺭﺽ‬4.15‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻭﺼﻔﺘﻬﺎ‬ ‫ﻜﻤﺎ‬ ،‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﻟﻠﺤﺭﻜﺔ‬
‫ﺍﻟﻨﻘﻁﺔ‬A‫ﻓﺎ‬ ‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﻤﻭﻀﺢ‬ ‫ﻫﻭ‬ ‫ﻜﻤﺎ‬‫ﻥ‬)a
)c(،‫ﺼﻔﺭ‬ ‫ﺘﺴﺎﻭﻱ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﻤﻭﺠﺒﺔ‬ ‫ﺍﻟﺴﺭﻋﺔ‬
‫ﻤﻭﺠﺒﺔ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﺴﺎﻟﺒﺔ‬ ‫ﺍﻟﺴﺭﻋﺔ‬.
‫ﻜﺘﻠﺘﻪ‬ ‫ﺠﺴﻡ‬m‫ﻴﺘﺫﺒﺫﺏ‬ ‫ﺯﻨﺒﺭﻙ‬ ‫ﻓﻲ‬ ‫ﻤﻌﻠﻕ‬.‫ﺍ‬‫ﺍﻨﻪ‬ ‫ﻋﻠﻰ‬ ‫ﻭﻤﺴﺠل‬ ‫ﻤﻘﺎﺱ‬ ‫ﻟﻠﺘﺫﺒﺫﺏ‬ ‫ﺍﻟﺩﻭﺭﻱ‬ ‫ﻟﺯﻤﻥ‬
‫ﻜﺘﻠﺘﻪ‬ ‫ﺍﺨﺭ‬ ‫ﺒﺠﺴﻡ‬2m.‫ﺍﻟﺠﺴﻡ‬ ‫ﻴﺘﺫﺒﺫﺏ‬ ‫ﻋﻨﺩﻤﺎ‬
T/ 2،)e(T/2.

22
‫ﻋﺎﻤﺔ‬ ‫ﺒﺼﻔﺔ‬ ‫ﻟﻠﺠﺴﻡ‬ ‫ﺴﻴﻁﺔ‬،‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺜﻭﺍﺒﺕ‬ ‫ﻨﺤﺴﺏ‬ ‫ﻜﻴﻑ‬ ‫ﺍﻻﻥ‬ ‫ﻟﻨﺭﻯ‬.‫ﻓ‬‫ﺎﻟﺘﺭﺩﺩ‬
‫ﻭﺍﻟﺜﻭﺍﺒﺕ‬A‫ﻭ‬ϕ‫ﺍﻟﺯﻤﻥ‬ ‫ﻋﻨﺩ‬ ‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺍﻻﺒﺘﺩﺍﺌﻴﺔ‬ ‫ﺍﻟﺤﺎﻟﺔ‬ ‫ﺘﺤﺩﺩ‬t=0.
‫ﻤﺴﺎﻓﺔ‬ ‫ﺍﻻﺘﺯﺍﻥ‬ ‫ﻤﻭﻀﻊ‬ ‫ﻤﻥ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺴﺤﺏ‬ ‫ﻋﻨﺩ‬ ‫ﺒﺩﺃﺕ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺇﻥ‬ ‫ﺍﻓﺘﺭﺽ‬A‫ﺘﺭﻙ‬ ‫ﺜﻡ‬ ‫ﻭﻤﻥ‬‫ﺍﻟﺯﻤﻥ‬ ‫ﻋﻨﺩ‬ ‫ﺍﻟﺴﻜﻭﻥ‬ ‫ﻤﻥ‬t=0،
‫ﻋﻨﺪ‬ ‫اﻟﺴﻜﻮن‬ ‫ﻣﻦ‬ ‫اﻟﺤﺮﻛﺔ‬ ‫ﺗﺒﺪأ‬ ‫ﺑﺰﻧﺒﺮك‬ ‫ﻣﺘﺼﻠﺔ‬ ‫ﻛﺘﻠﺔ‬x=A‫و‬t=0‫ﺗﻜﻮن‬ ‫اﻟﺤﺎﻟﺔ‬ ‫ھﺬه‬ ‫وﻓﻲ‬ ،ϕ=0‫ﻓﺎن‬ ‫ﻋﻠﯿﮫ‬ ‫و‬
x=Acosωt
www.trgma.com
‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬6.15‫ﺍﻟﺒ‬ ‫ﺍﻟﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺘﺼﻑ‬‫ﻋﺎﻤﺔ‬ ‫ﺒﺼﻔﺔ‬ ‫ﻟﻠﺠﺴﻡ‬ ‫ﺴﻴﻁﺔ‬
‫ﺍﻟﺯﺍﻭﻱ‬ω‫ﻴ‬‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻤﻥ‬ ‫ﺤﺴﺏ‬9.15.‫ﻭﺍﻟﺜﻭﺍﺒﺕ‬
‫ﻤﺴﺎﻓﺔ‬ ‫ﺍﻻﺘﺯﺍﻥ‬ ‫ﻤﻭﻀﻊ‬ ‫ﻤﻥ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺴﺤﺏ‬ ‫ﻋﻨﺩ‬ ‫ﺒﺩﺃﺕ‬ ‫ﺍﻟﺤﺭﻜﺔ‬ ‫ﺇﻥ‬ ‫ﺍﻓﺘﺭﺽ‬
‫ﻓﻲ‬ ‫ﻜﻤﺎ‬‫ﺍﻟﺸﻜل‬7.15.
‫اﻟﺸﻜﻞ‬7.15‫ﻋﻨﺪ‬ ‫اﻟﺴﻜﻮن‬ ‫ﻣﻦ‬ ‫اﻟﺤﺮﻛﺔ‬ ‫ﺗﺒﺪأ‬ ‫ﺑﺰﻧﺒﺮك‬ ‫ﻣﺘﺼﻠﺔ‬ ‫ﻛﺘﻠﺔ‬

23
‫ﺑﺸﺮوط‬ ‫ﺑﺴﯿﻄﺔ‬ ‫ﺗﻮاﻓﻘﯿﺔ‬ ‫ﺣﺮﻛﺔ‬ ‫ﯾﺘﺤﺮك‬ ‫ﻟﺠﺴﻢ‬ ‫اﻟﺰﻣﻦ‬ ‫ﻓﻲ‬ ‫ﻛﺪاﻟﺔ‬ ‫واﻟﻌﺠﻠﺔ‬ ،‫واﻟﺴﺮﻋﺔ‬ ،‫اﻟﻤﻮﺿﻊ‬
v() .b(‫ﯾﻮﺿﺢ‬‫ﯾﺘﺤﺮك‬ ‫ﻟﺠﺴﻢ‬ ‫اﻟﺰﻣﻦ‬ ‫ﻓﻲ‬ ‫ﻛﺪاﻟﺔ‬ ‫واﻟﻌﺠﻠﺔ‬ ،‫واﻟﺴﺮﻋﺔ‬ ،‫اﻟﻤﻮﺿﻊ‬
‫ھﻲ‬ ‫اﺑﺘﺪاﺋﯿﺔ‬ ‫ﺑﺸﺮوط‬ ‫ﺑﺴﯿﻄﺔ‬ ‫ﺗﻮاﻓﻘﯿﺔ‬ ‫ﺣﺮﻛﺔ‬t=0‫و‬x(0) =A‫و‬v(0)=vi.
www.trgma.com
‫اﻟﺸﻜﻞ‬8.15)a(‫ﯾﻮﺿﺢ‬‫ﺑﺸﺮوط‬ ‫ﺑﺴﯿﻄﺔ‬ ‫ﺗﻮاﻓﻘﯿﺔ‬ ‫ﺣﺮﻛﺔ‬ ‫ﯾﺘﺤﺮك‬ ‫ﻟﺠﺴﻢ‬ ‫اﻟﺰﻣﻦ‬ ‫ﻓﻲ‬ ‫ﻛﺪاﻟﺔ‬ ‫واﻟﻌﺠﻠﺔ‬ ،‫واﻟﺴﺮﻋﺔ‬ ،‫اﻟﻤﻮﺿﻊ‬
‫ھﻲ‬ ‫اﺑﺘﺪاﺋﯿﺔ‬t=0‫و‬x(0) =A‫و‬v(0)=0
‫ھﻲ‬ ‫اﺑﺘﺪاﺋﯿﺔ‬ ‫ﺑﺸﺮوط‬ ‫ﺑﺴﯿﻄﺔ‬ ‫ﺗﻮاﻓﻘﯿﺔ‬ ‫ﺣﺮﻛﺔ‬

24
v(t))‫ﺍﻟﻤﻌﺎﺩﻟﺘﻴﻥ‬6.15‫ﻭ‬5.15(‫ﺘ‬‫ﺤﻘﻘ‬‫ﺎﻥ‬‫ﻭﻫﻲ‬ ‫ﺍﻻﺒﺘﺩﺍﺌﻴﺔ‬ ‫ﺍﻟﺸﺭﻭﻁ‬x(0) = A
0sin)0(
cos)0(
=−=
==
φω
φ
Av
AAx
‫ﺍﻻﺒﺘﺩﺍﺌﻲ‬ ‫ﺍﻟﻁﻭﺭ‬ ‫ﺍﺨﺘﺭﻨﺎ‬ ‫ﺇﺫﺍ‬ ‫ﺘﺘﺤﻘﻕ‬ ‫ﺍﻟﺸﺭﻭﻁ‬ ‫ﻫﺫﻩ‬initial phaseϕ=0‫ﻭ‬ ،x=A cosωt‫ﻜﺤل‬.‫ﻴﺠﺏ‬ ‫ﺍﻟﺤل‬ ‫ﻤﻥ‬ ‫ﻭﻟﻠﺘﺤﻘﻕ‬
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﻤﻤﺜﻠﺔ‬ ‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻊ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﻭﺍﻟﺴﺭﻋﺔ‬ ‫ﺍﻟﻤﻭﻀﻊ‬ ‫ﻋﻼﻗﺔ‬8.15a‫ﺍﻟﺨﺎﺼﺔ‬ ‫ﺍﻟﺤﺎﻟﺔ‬ ‫ﻟﻬﺫﻩ‬.‫ﺘﺼل‬‫ﺃﻗﺼﻰ‬ ‫ﺇﻟﻰ‬ ‫ﺍﻟﻌﺠﻠﺔ‬
‫ﻭﻫﻲ‬ ‫ﻟﻪ‬ ‫ﻗﻴﻤﺔ‬ ‫ﺃﻗﺼﻰ‬ ‫ﻓﻲ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻭﻀﻊ‬ ‫ﻴﻜﻭﻥ‬ ‫ﻋﻨﺩﻤﺎ‬±A.‫ﺃﻗﺼﻰ‬ ‫ﻟﻬﺎ‬ ‫ﺍﻟﺴﺭﻋﺔ‬ ‫ﺇﻥ‬ ‫ﺫﻟﻙ‬ ‫ﺇﻟﻰ‬ ‫ﺃﻀﻑ‬
‫ﻨﺩﺭﺴﻬﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﺤﺎﻟﺔ‬ ‫ﻤﻊ‬ ‫ﻤﺘﻔﻘﺎ‬ ‫ﻴﻜﻭﻥ‬ ‫ﺍﻟﺤل‬ ‫ﻓﺎﻥ‬ ‫ﻭﻋﻠﻴﻪ‬.
‫ﺍﻟﺯﻤﻥ‬ ‫ﻭﺍﺨﺘﺭﻨﺎ‬ ،‫ﺍﻟﺠﺴﻡ‬ ‫ﻴﺘﺫﺒﺫﺏ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﻭﻫﻲ‬ ‫ﺃﺨﺭﻯ‬ ‫ﺍﺤﺘﻤﺎﻟﻴﺔ‬ ‫ﻟﻨﻔﺘﺭﺽ‬t=0‫ﻭﻫﻭ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﺍﺘﺯﺍﻥ‬ ‫ﺒﻨﻘﻁﺔ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻴﻤﺭ‬ ‫ﻋﻨﺩﻤﺎ‬
9.15.‫ﻟـ‬ ‫ﺍﻟﺤﺎﻟﺔ‬ ‫ﻫﺫﻩ‬ ‫ﻴﺤﻘﻕ‬ ‫ﺍﻟﺫﻱ‬ ‫ﺍﻟﺤل‬ ‫ﻴﻜﻭﻥ‬ ‫ﺃﻥ‬ ‫ﻴﺠﺏ‬ ‫ﺍﻟﺤﺎﻟﺔ‬ ‫ﻫﺫﻩ‬ ‫ﻓﻲ‬x(t)‫ﻭ‬
x(0)=0‫ﻭ‬v(0)=vi:
ivAv
Ax
=−=
==
φω
φ
sin)0(
0cos)0(
www.trgma.com
‫ﻤﻥ‬ ‫ﻟﻜﻼ‬ ‫ﺤﻼ‬ ‫ﻨﺠﺩ‬ ‫ﺃﻥ‬ ‫ﻴﺘﻁﻠﺏ‬ ‫ﻭﻫﺫﺍ‬x(t)‫ﻭ‬v(t)
‫ﻭ‬v(0)=0:
‫ﺍﻻﺒﺘﺩﺍﺌﻲ‬ ‫ﺍﻟﻁﻭﺭ‬ ‫ﺍﺨﺘﺭﻨﺎ‬ ‫ﺇﺫﺍ‬ ‫ﺘﺘﺤﻘﻕ‬ ‫ﺍﻟﺸﺭﻭﻁ‬ ‫ﻫﺫﻩ‬
‫ﻴﻜﻭﻥ‬ ‫ﺃﻥ‬x(0)=A‫ﻻﻥ‬cos0=1.
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﻤﻤﺜﻠﺔ‬ ‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻊ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﻭﺍﻟﺴﺭﻋﺔ‬ ‫ﺍﻟﻤﻭﻀﻊ‬ ‫ﻋﻼﻗﺔ‬
‫ﻭﻫﻲ‬ ‫ﻟﻬﺎ‬ ‫ﻗﻴﻤﺔ‬±ω2
A‫ﻭﻫﻲ‬ ‫ﻟﻪ‬ ‫ﻗﻴﻤﺔ‬ ‫ﺃﻗﺼﻰ‬ ‫ﻓﻲ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻭﻀﻊ‬ ‫ﻴﻜﻭﻥ‬ ‫ﻋﻨﺩﻤﺎ‬
‫ﻭﻫﻲ‬ ‫ﻗﻴﻤﺔ‬±ωA‫ﻋﻨﺩ‬ ،x=0.‫ﻨﺩﺭﺴﻬﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﺤﺎﻟﺔ‬ ‫ﻤﻊ‬ ‫ﻤﺘﻔﻘﺎ‬ ‫ﻴﻜﻭﻥ‬ ‫ﺍﻟﺤل‬ ‫ﻓﺎﻥ‬ ‫ﻭﻋﻠﻴﻪ‬
‫ﺍﻟﺯﻤﻥ‬ ‫ﻭﺍﺨﺘﺭﻨﺎ‬ ،‫ﺍﻟﺠﺴﻡ‬ ‫ﻴﺘﺫﺒﺫﺏ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﻭﻫﻲ‬ ‫ﺃﺨﺭﻯ‬ ‫ﺍﺤﺘﻤﺎﻟﻴﺔ‬ ‫ﻟﻨﻔﺘﺭﺽ‬
‫ﻓﻲ‬‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﻜﻤﺎ‬ ‫ﺍﻟﻴﻤﻴﻥ‬ ‫ﺇﻟﻰ‬ ‫ﺤﺭﻜﺘﻪ‬9.15
v(t)‫ﺇﻥ‬ ‫ﻭﻫﻲ‬ ‫ﺍﻻﺒﺘﺩﺍﺌﻴﺔ‬ ‫ﺍﻟﺸﺭﻭﻁ‬ ‫ﻴﺤﻘﻕ‬x(0)=0

25
‫ﻟـ‬ ‫ﺍﻟﺨﻴﺎﺭﺍﻥ‬ ‫ﻭﻟﻬﺫﻴﻥ‬ϕ،‫ﻭﻤﻥ‬‫ﺍﻟﺜﺎﻨﻲ‬ ‫ﺍﻟﺸﺭﻁ‬‫ﺍ‬ ‫ﻋﻠﻰ‬ ‫ﻨﺤﺼل‬‫ﻟﺴﻌﺔ‬A=±vi/ω،‫ﻭ‬‫ﻻﻥ‬
‫ﺘﻜﻭﻥ‬ ‫ﻭﺍﻥ‬ ‫ﻻﺒﺩ‬ ‫ﻭﺍﻟﺴﻌﺔ‬‫ﻤﻭﺠﺒﺔ‬‫ﺃﻴﻀﺎ‬،‫ﻋﻠﻰ‬ ‫ﻨﺤﺼل‬ ‫ﻓﺈﻨﻨﺎ‬ϕ = -π/2.‫ﻴﻌﻁﻰ‬ ‫ﺍﻟﺤل‬ ‫ﻓﺎﻥ‬ ‫ﻭﻟﻬﺫﺍ‬






−=
2
cos
π
ω
ω
t
v
x i
‫ﻋﻨﺩ‬ ‫ﺍﻟﺤﺎﻟﺔ‬ ‫ﻟﻬﺫﻩ‬ ‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻊ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﻭﺍﻟﺴﺭﻋﺔ‬ ‫ﺍﻟﻤﻭﻀﻊ‬ ‫ﻋﻼﻗﺔ‬t=0‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﻤﻭﻀﺤﺔ‬8.15b.‫ﺇ‬ ‫ﻻﺤﻅ‬‫ﻫﺫﻩ‬ ‫ﻥ‬
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﺍﻟﻤﻨﺤﻨﻴﺎﺕ‬ ‫ﻨﻔﺴﻬﺎ‬ ‫ﻫﻲ‬ ‫ﺍﻟﻤﻨﺤﻨﻴﺎﺕ‬8.15b‫ﺩﻭﺭﺓ‬ ‫ﺒﺭﺒﻊ‬ ‫ﺍﻟﻴﻤﻴﻥ‬ ‫ﺇﻟﻰ‬ ‫ﻤﺯﺍﺤﺔ‬ ‫ﻭﻟﻜﻨﻬﺎ‬ ،.‫ﺭﻴﺎﻀﻴﺎ‬ ‫ﺫﻟﻙ‬ ‫ﻋﻥ‬ ‫ﻭﻴﻌﺒﺭ‬
ϕ=-π‫ﻤﻥ‬ ‫ﺩﻭﺭﺓ‬ ‫ﺭﺒﻊ‬ ‫ﻴﻤﺜل‬ ‫ﻭﻫﺫﺍ‬ ،2π.
‫و‬ ،‫ﺑﺴﯿﻄﺔ‬ ‫ﺗﻮاﻓﻘﯿﺔ‬ ‫ﺣﺮﻛﺔ‬ ‫ﺗﺘﺤﺮك‬ ‫ﺑﺰﻧﺒﺮك‬ ‫ﻣﺘﺼﻠﺔ‬ ‫ﻛﺘﻠﺔ‬t=0‫اﻟﺘﻲ‬ ‫اﻟﻠﺤﮭﺔ‬ ‫ﻓﻲ‬ ‫ﺗﻌﺮف‬‫ﻧﻘﻄﺔ‬ ‫اﻟﻜﺘﻠﺔ‬ ‫ﻓﯿﮫ‬ ‫ﺗﻤﺮ‬
‫اﻻﺗﺰان‬x=0‫ﺑﺴﺮﻋﺔ‬ ‫ﻟﻠﯿﻤﯿﻦ‬ ‫وﺗﺘﺤﺮك‬vi.
www.trgma.com
‫ﺇﻥ‬ ‫ﻴﺨﺒﺭﻨﺎ‬ ‫ﺍﻷﻭل‬ ‫ﺍﻟﺸﺭﻁ‬ϕ=±π/2.‫ﻟـ‬ ‫ﺍﻟﺨﻴﺎﺭﺍﻥ‬ ‫ﻭﻟﻬﺫﻴﻥ‬
‫ﻤﻭﺠﺒﺔ‬ ‫ﺍﻻﺒﺘﺩﺍﺌﻴﺔ‬ ‫ﺍﻟﺴﺭﻋﺔ‬‫ﺘﻜﻭﻥ‬ ‫ﻭﺍﻥ‬ ‫ﻻﺒﺩ‬ ‫ﻭﺍﻟﺴﻌﺔ‬
‫ﻤﻨﺤﻨﻴﺎﺕ‬‫ﻋﻨﺩ‬ ‫ﺍﻟﺤﺎﻟﺔ‬ ‫ﻟﻬﺫﻩ‬ ‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻊ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﻭﺍﻟﺴﺭﻋﺔ‬ ‫ﺍﻟﻤﻭﻀﻊ‬ ‫ﻋﻼﻗﺔ‬
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﺍﻟﻤﻨﺤﻨﻴﺎﺕ‬ ‫ﻨﻔﺴﻬﺎ‬ ‫ﻫﻲ‬ ‫ﺍﻟﻤﻨﺤﻨﻴﺎﺕ‬
‫ﺍﻟﻁﻭﺭ‬ ‫ﺜﺎﺒﺕ‬ ‫ﺒﻭﺍﺴﻁﺔ‬phase constantπ/2
‫اﻟﺸﻜﻞ‬9.15‫و‬ ،‫ﺑﺴﯿﻄﺔ‬ ‫ﺗﻮاﻓﻘﯿﺔ‬ ‫ﺣﺮﻛﺔ‬ ‫ﺗﺘﺤﺮك‬ ‫ﺑﺰﻧﺒﺮك‬ ‫ﻣﺘﺼﻠﺔ‬ ‫ﻛﺘﻠﺔ‬
‫اﻻﺗﺰان‬

26
‫ﺍﻟﺘﺎﻟﻴﺔ‬ ‫ﻟﻠﻤﻌﺎﺩﻟﺔ‬ ‫ﻁﺒﻘﺎ‬ ‫ﻟﻠﺯﻤﻥ‬ ‫ﺒﺎﻟﻨﺴﺒﺔ‬ ‫ﻴﺘﻐﻴﺭ‬ ‫ﻤﻭﻗﻌﻪ‬:
= cos)00.4( mx
) .A(‫ﺤ‬‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺍﻟﺩﻭﺭﻱ‬ ‫ﻭﺍﻟﺯﻤﻥ‬ ،‫ﻭﺍﻟﺘﺭﺩﺩ‬ ،‫ﺍﻟﺴﻌﺔ‬ ‫ﻤﻘﺩﺍﺭ‬ ‫ﺩﺩ‬) .B(‫ﺍﺤﺴﺏ‬
‫ﺍﻟﺠﺯﺀ‬ ‫ﺤل‬ ‫ﻤﻥ‬ ‫ﻋﻠﻴﻬﺎ‬ ‫ﺴﺘﺤﺼل‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻨﺘﺎﺌﺞ‬ ‫ﻤﻥ‬)B(‫ﻭﻋﺠﻠﺘﻪ‬ ‫ﻭﺴﺭﻋﺘﻪ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻭﻀﻊ‬ ‫ﺒﺤﺴﺎﺏ‬ ‫ﻗﻡ‬
E(‫ﺒﻴﻥ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺇﺯﺍﺤﺔ‬ ‫ﺍﻭﺠﺩ‬t=0‫ﻭ‬t=1.
x(t) = A cos (
‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﻓﺎﻥ‬ ‫ﻭﻋﻠﻴﻪ‬f=ω/2π=π/2π= 0.5Hz‫ﺍﻟﺩﻭﺭﻱ‬ ‫ﻭﺍﻟﺯﻤﻥ‬T=1/f=2sec
‫ﻟﻠﺯﻤﻥ‬ ‫ﺒﺎﻟﻨﺴﺒﺔ‬t.‫ﻋ‬ ‫ﻨﺤﺼل‬‫ﻠﻰ‬
sin)/00.4( sm
dt
dx
v −==
−= /00.4( π smv
www.trgma.com
‫ﻤﺜﺎل‬1.15‫ﻤﺘﺫﺒﺫﺏ‬ ‫ﺠﺴﻡ‬
‫ﻤﺤﻭﺭ‬ ‫ﻋﻠﻰ‬ ‫ﺒﺴﻴﻁﺔ‬ ‫ﺘﻭﺍﻓﻘﻴﺔ‬ ‫ﺒﺤﺭﻜﺔ‬ ‫ﻴﺘﺫﺒﺫﺏ‬ ‫ﺠﺴﻡ‬x.‫ﺍﻟﺘﺎﻟﻴﺔ‬ ‫ﻟﻠﻤﻌﺎﺩﻟﺔ‬ ‫ﻁﺒﻘﺎ‬ ‫ﻟﻠﺯﻤﻥ‬ ‫ﺒﺎﻟﻨﺴﺒﺔ‬ ‫ﻴﺘﻐﻴﺭ‬ ‫ﻤﻭﻗﻌﻪ‬






−
4
cos
π
πt
‫ﺤﻴﺙ‬t‫ﺒ‬ ‫ﻤﻘﺩﺭﺓ‬ ‫ﺍﻟﻘﻭﺴﻴﻥ‬ ‫ﺒﻴﻥ‬ ‫ﻭﺍﻟﺯﺍﻭﻴﺔ‬ ‫ﺒﺎﻟﺜﻭﺍﻨﻲ‬ ‫ﺍﻟﺯﻤﻥ‬‫ﻭﺤﺩﺓ‬rad) .
‫ﺯﻤﻥ‬ ‫ﺃﻱ‬ ‫ﻋﻨﺩ‬ ‫ﻟﻠﺠﺴﻡ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﺍﻟﺴﺭﻋﺔ‬t) .C(‫ﺍﻟﺠﺯﺀ‬ ‫ﺤل‬ ‫ﻤﻥ‬ ‫ﻋﻠﻴﻬﺎ‬ ‫ﺴﺘﺤﺼل‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻨﺘﺎﺌﺞ‬ ‫ﻤﻥ‬
‫ﺍﻟﺯﻤﻥ‬ ‫ﻋﻨﺩ‬t=1s) .D(‫ﻟﻠﺠﺴﻡ‬ ‫ﻋﺠﻠﺔ‬ ‫ﻭﺃﻋﻅﻡ‬ ‫ﺴﺭﻋﺔ‬ ‫ﺃﻋﻅﻡ‬ ‫ﻗﻴﻤﺔ‬ ‫ﺤﺩﺩ‬) .E
‫ﺍﻟﺤل‬:‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻤﻊ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻫﺫﻩ‬ ‫ﺒﻤﻘﺎﺭﻨﺔ‬15.6
x(t) = A cos (ωt+ϕ)
)A(‫ﺃﻥ‬ ‫ﻨﺠﺩ‬A=4.00m‫ﺍﻟﺯﺍﻭﻱ‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﻭ‬ω=π rad/s‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﻓﺎﻥ‬ ‫ﻭﻋﻠﻴﻪ‬
)B(‫ﺍﻟﺴﺭﻋﺔ‬ ‫ﻹﻴﺠﺎﺩ‬v‫ﻟـ‬ ‫ﺍﻟﺘﻔﺎﻀل‬ ‫ﻋﻤﻠﻴﺔ‬ ‫ﺒﺈﺠﺭﺍﺀ‬ ‫ﻨﻘﻭﻡ‬x‫ﻟﻠﺯﻤﻥ‬ ‫ﺒﺎﻟﻨﺴﺒﺔ‬
)(
4
sin t
dt
d
t π
π
π 





+






+
4
sin)
π
πts

27
‫ﺍﻟﺘﺎﻟﻴﺔ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻋﻠﻰ‬ ‫ﻓﻨﺤﺼل‬ ‫ﻟﻠﺯﻤﻥ‬ ‫ﺒﺎﻟﻨﺴﺒﺔ‬:
)/00.4( sm
dt
dv
a π−==
−= /00.4( 22
π sma
‫ﺍﻟﻤﻁﻠﻭﺒﺔ‬ ‫ﺍﻟﻤﻌﻠﻭﻤﺎﺕ‬ ‫ﻋﻠﻰ‬ ‫ﻨﺤﺼل‬ ‫ﻋﻥ‬ ‫ﻨﺴﺘﻁﻴﻊ‬‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﻭﺍﻟﺴﺭﻋﺔ‬ ‫ﺍﻟﻤﻭﻗﻊ‬ ‫ﻤﻥ‬ ‫ﻟﻜل‬.
mmx cos)00.4(
4
cos)00.4( =





+−=
π
π
smv .4(
4
5
sin)/00.4( −=





−=
π
π
22
.4(
4
5
cos)/00.4( sma −=





−=
π
π
‫ﺍﻟﺠﺯﺀ‬ ‫ﻓﻲ‬ ‫ﺒﺤﺴﺎﺒﻬﺎ‬ ‫ﻗﻤﻨﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﻫﻲ‬ ‫ﻟﻠﺠﺴﻡ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﺍﻟﺴﺭﻋﺔ‬ ‫ﻓﺎﻥ‬ ‫ﻋﺎﻤﺔ‬ ‫ﺒﺼﻔﺔ‬)B(‫ﻭﺃﻋﻅﻡ‬‫ﺩﺍﻟﺔ‬ ‫ﻤﻥ‬ ‫ﻜل‬ ‫ﺘﻜﻭﻥ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﻴﻜﻭﻥ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﻟﻠﺴﺭﻋﺔ‬ ‫ﻗﻴﻤﺔ‬
‫ﺍﻟﻘﻴﻤﺘﻴﻥ‬ ‫ﺒﻴﻥ‬ ‫ﺘﺘﻐﻴﺭ‬±4.00πm/s‫ﻭﺍﻟﻌﺠﻠﺔ‬ ،a‫ﺍﻟﻘﻴﻤﺘﻴﻥ‬ ‫ﺒﻴﻥ‬ ‫ﺘﺘﻐﻴﺭ‬±4.00π2
m/s2
.‫ﻭﻟﻬﺫﺍ‬
vmax = 4.00 πm/s = 12.6m/s
www.trgma.com
‫ﻟﻠﺴﺭﻋﺔ‬ ‫ﺍﻟﺘﻔﺎﻀل‬ ‫ﻋﻤﻠﻴﺔ‬ ‫ﺒﺈﺠﺭﺍﺀ‬ ‫ﻨﻘﻭﻡ‬ ‫ﺍﻟﻌﺠﻠﺔ‬ ‫ﻋﻠﻰ‬ ‫ﻭﻟﻠﺤﺼﻭل‬v‫ﺍﻟﺘﺎﻟﻴﺔ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻋﻠﻰ‬ ‫ﻓﻨﺤﺼل‬ ‫ﻟﻠﺯﻤﻥ‬ ‫ﺒﺎﻟﻨﺴﺒﺔ‬
)(
4
cos) t
dt
d
t π
π
π 





+






+
4
cos)2 π
πt
)C(‫ﺍﻟﺯﻤﻥ‬ ‫ﻋﻥ‬ ‫ﺒﺎﻟﺘﻌﻭﻴﺽ‬t=1s‫ﺍﻟﻤﻁﻠﻭﺒﺔ‬ ‫ﺍﻟﻤﻌﻠﻭﻤﺎﺕ‬ ‫ﻋﻠﻰ‬ ‫ﻨﺤﺼل‬ ‫ﻋﻥ‬ ‫ﻨﺴﺘﻁﻴﻊ‬
mm 83.2)707.0)(00.4(
4
5
cos −=−=




 π
smsm /89.8)707.0)(/00. =−π
222
/9.27)707.0)(/00. smsm =−π
)D(‫ﺍﻟﺠﺯﺀ‬ ‫ﻓﻲ‬ ‫ﺒﺤﺴﺎﺒﻬﺎ‬ ‫ﻗﻤﻨﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﻫﻲ‬ ‫ﻟﻠﺠﺴﻡ‬ ‫ﻭﺍﻟﻌﺠﻠﺔ‬ ‫ﺍﻟﺴﺭﻋﺔ‬ ‫ﻓﺎﻥ‬ ‫ﻋﺎﻤﺔ‬ ‫ﺒﺼﻔﺔ‬
‫ﺍﻟﻭﺤﺩ‬ ‫ﺘﺴﺎﻭﻱ‬ ‫ﺍﻟﺘﻤﺎﻡ‬ ‫ﻭﺠﻴﺏ‬ ‫ﺍﻟﺠﻴﺏ‬‫ﺓ‬.‫ﺍﻟﺴﺭﻋﺔ‬ ‫ﻓﺈﻥ‬ ‫ﻭﻟﻬﺫﺍ‬v‫ﺍﻟﻘﻴﻤﺘﻴﻥ‬ ‫ﺒﻴﻥ‬ ‫ﺘﺘﻐﻴﺭ‬
‫ﻓﺎﻥ‬
m/s = 12.6m/s

28
amax = 4.00π2
m/s
amax=ω2
‫ﺍﻥ‬ ‫ﺤﻴﺙ‬A=4.00m‫ﻭ‬ω=π rad/s
mxi 0cos)00.4( 


=
-2.83m‫ﺒﻴﻥ‬ ‫ﺍﻹﺯﺍﺤﺔ‬ ‫ﻓﺎﻥ‬ ‫ﻭﻟﻬﺫﺍ‬t=0‫ﻭ‬t=1‫ﺍﻟﺘﺎﻟﻲ‬ ‫ﺍﻟﻨﺤﻭ‬ ‫ﻋﻠﻰ‬ ‫ﻴﻜﻭﻥ‬:
mxxx if 83.2−=−=∆
‫ﺯﻨﺒﺭﻙ‬ ‫ﻜل‬ ‫ﺜﺎﺒﺕ‬20,000N/m.‫ﻜﺘﻠﺘﻬﻤﺎ‬ ‫ﺭﺠﻠﻴﻥ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﺼﻌﺩ‬ ‫ﺇﺫﺍ‬160kg‫ﺘﺭﺩﺩ‬ ‫ﺍﻭﺠﺩ‬ ،
‫ﻟ‬ ‫ﻨﺘﻴﺠﺔ‬ ‫ﻓﺎﻨﻪ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﻨﺭﻜﺏ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﺍﻨﻨﺎ‬ ‫ﺤﻴﺙ‬ ‫ﺍﻟﻌﻤﻠﻴﺔ‬ ‫ﺨﺒﺭﺘﻨﺎ‬ ‫ﻨﺴﺘﺭﺠﻊ‬ ‫ﺩﻋﻨﺎ‬ ‫ﺍﻟﻤﺜﺎل‬ ‫ﻫﺫﺍ‬ ‫ﺤل‬ ‫ﻤﻥ‬ ‫ﻟﻨﺘﻤﻜﻥ‬‫ﻘﻭﺓ‬‫ﻭﺯﻥ‬‫ﺍﻟﺠﺴﻡ‬‫ﺒﻘﺩ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﺘﻨﺨﻔﺽ‬‫ﺭ‬
‫ﻋﺩﺓ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﻟﺘﺘﺫﺒﺫﺏ‬ ‫ﻟﻸﺴﻔل‬ ‫ﺒﺎﻟﻘﻭﺓ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﻤﻘﺩﻤﺔ‬ ‫ﻋﻠﻰ‬ ‫ﺒﺎﻟﻀﻐﻁ‬ ‫ﺘﻘﻭﻡ‬ ‫ﺍﻥ‬ ‫ﻤﺭﺓ‬ ‫ﺤﺎﻭﻟﺕ‬ ‫ﻗﺩ‬ ‫ﻭﺍﻨﻙ‬ ‫ﺸﻙ‬ ‫ﻻ‬ ‫ﺍﻨﻙ‬ ‫ﻜﻤﺎ‬
www.trgma.com
m/s2
= 39.5m/s2
‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﺒﺎﺴﺘﺨﺩﺍﻡ‬ ‫ﺍﻟﻨﺘﺎﺌﺞ‬ ‫ﻨﻔﺱ‬ ‫ﻋﻠﻰ‬ ‫ﻨﺤﺼل‬ ‫ﺃﻥ‬ ‫ﻭﻴﻤﻜﻥ‬vmax=ωA‫ﻭ‬2
A
)E(‫ﺍﻟﺯﻤﻥ‬ ‫ﻋﻨﺩ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻭﻀﻊ‬t=0‫ﻫﻭ‬
m
t
83.20 =


+
π
‫ﺍﻟﺠﺯﺀ‬ ‫ﺤل‬ ‫ﻤﻥ‬ ‫ﺇﻨﻨﺎ‬ ‫ﻭﺤﻴﺙ‬)C(‫ﺍﻟﺯﻤﻥ‬ ‫ﻋﻨﺩ‬ ‫ﺍﻟﻤﻭﻀﻊ‬ ‫ﺇﻥ‬ ‫ﻭﺠﺩﻨﺎ‬ ،t=1‫ﻫﻭ‬2.83m
mmm 66.583.2 −=−
‫ﻤﺜﺎل‬2.15
‫ﻜﺘﻠﺘﻬﺎ‬ ‫ﺴﻴﺎﺭﺓ‬1300kg‫ﻤﺜﺒﺘﺔ‬‫ﺯﻨﺒﺭﻜﺎ‬ ‫ﺃﺭﺒﻌﺔ‬ ‫ﻋﻠﻰ‬‫ﺕ‬.‫ﺯﻨﺒﺭﻙ‬ ‫ﻜل‬ ‫ﺜﺎﺒﺕ‬
‫ﺍﻟﻁﺭﻴﻕ‬ ‫ﻋﻠﻰ‬ ‫ﻀﻴﻕ‬ ‫ﺃﺨﺩﻭﺩ‬ ‫ﻋﻠﻰ‬ ‫ﺘﻤﺭ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﻟﻠﺴﻴﺎﺭﺓ‬ ‫ﺍﻻﻫﺘﺯﺍﺯ‬.
‫ﺍﻟﺤل‬:‫ﻟ‬ ‫ﻨﺘﻴﺠﺔ‬ ‫ﻓﺎﻨﻪ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﻨﺭﻜﺏ‬ ‫ﻋﻨﺩﻤﺎ‬ ‫ﺍﻨﻨﺎ‬ ‫ﺤﻴﺙ‬ ‫ﺍﻟﻌﻤﻠﻴﺔ‬ ‫ﺨﺒﺭﺘﻨﺎ‬ ‫ﻨﺴﺘﺭﺠﻊ‬ ‫ﺩﻋﻨﺎ‬ ‫ﺍﻟﻤﺜﺎل‬ ‫ﻫﺫﺍ‬ ‫ﺤل‬ ‫ﻤﻥ‬ ‫ﻟﻨﺘﻤﻜﻥ‬
‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﻻﻨﻀﻐﺎﻁ‬ ‫ﻨﺘﻴﺠﺔ‬ ‫ﺒﺴﻴﻁ‬.‫ﻋﺩﺓ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﻟﺘﺘﺫﺒﺫﺏ‬ ‫ﻟﻸﺴﻔل‬ ‫ﺒﺎﻟﻘﻭﺓ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﻤﻘﺩﻤﺔ‬ ‫ﻋﻠﻰ‬ ‫ﺒﺎﻟﻀﻐﻁ‬ ‫ﺘﻘﻭﻡ‬ ‫ﺍﻥ‬ ‫ﻤﺭﺓ‬ ‫ﺤﺎﻭﻟﺕ‬ ‫ﻗﺩ‬ ‫ﻭﺍﻨﻙ‬ ‫ﺸﻙ‬ ‫ﻻ‬ ‫ﺍﻨﻙ‬ ‫ﻜﻤﺎ‬

29
‫ﺭﻜﻭﺏ‬ ‫ﺴﺒﺒﻬﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻹﺯﺍﺤﺔ‬ ‫ﺍﻥ‬ ‫ﻭﺤﻴﺙ‬ ،‫ﻤﻜﺎﻓﺊ‬ ‫ﺯﻨﺒﺭﻙ‬ ‫ﻤﻊ‬ ‫ﺒﺎﻟﺘﻌﺎﻤل‬ ‫ﻨﻘﻭﻡ‬ ‫ﺴﻭﻑ‬ ‫ﻟﺫﻟﻙ‬ ‫ﺯﻨﺒﺭﻜﺎﺕ‬
kxFtotal ∑ −= )(
‫ﺍﻟﻤﺠﻤﻭﻉ‬ ‫ﺃﻗﻭﺍﺱ‬ ‫ﻤﻥ‬ ‫ﺇﺨﺭﺍﺠﻬﺎ‬ ‫ﺘﻡ‬ ‫ﻟﺫﻟﻙ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﻓﻲ‬ ‫ﺯﻨﺒﺭﻜﺎﺕ‬ ‫ﻟﻸﺭﺒﻊ‬ ‫ﻤﺘﺴﺎﻭﻴﺔ‬ ‫ﺇﺯﺍﺤﺔ‬ ‫ﺘﻤﺜل‬.‫ﻴﻌﻁﻰ‬ ‫ﺍﻟﻤﻜﺎﻓﺊ‬ ‫ﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﺜﺎﺒﺕ‬ ‫ﺍﻥ‬ ‫ﻨﻼﺤﻅ‬ ‫ﻭﻫﻨﺎ‬
kkeff 000,204×== ∑
15.14،
m
k
f
eff 80
2
1
2
1
==
ππ
m‫ﺍﻟﺴﺎﺒﻘﺔ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻓﻲ‬.
‫ﻟﺘﺘﺫﺒﺫﺏ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﻤﻘﺩﻤﺔ‬ ‫ﻋﻠﻰ‬ ‫ﺒﺎﻟﻀﻐﻁ‬ ‫ﺍﺤﺩﻫﻤﺎ‬ ‫ﻭﻗﺎﻡ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﻤﻥ‬ ‫ﺨﺭﺠﺎ‬ ‫ﺭﺠﻠﻴﻥ‬ ‫ﺍﻥ‬ ‫ﺍﻓﺘﺭﺽ‬ ‫ﻟﻭ؟‬ ‫ﻤﺎﺫﺍ‬.‫ﻓﻬل‬‫ﺴﺒﻕ؟‬ ‫ﻤﺎ‬ ‫ﻋﻥ‬ ‫ﻴﺨﺘﻠﻑ‬ ‫ﺍﻻﻫﺘﺯﺍﺯﺍﺕ‬ ‫ﺘﺭﺩﺩ‬
‫ﺃﻋﻠﻰ‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﻴﻜﻭﻥ‬ ‫ﺍﻥ‬ ‫ﻨﺘﻭﻗﻊ‬ ‫ﻭﻟﻬﺫﺍ‬ ‫ﺍﻗل‬ ‫ﺴﺘﻜﻭﻥ‬ ‫ﺍﻟﻜﺘﻠﺔ‬ ‫ﻫﻨﺎ‬ ‫ﻭﻟﻜﻥ‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﺒﺤﺴﺎﺏ‬ ‫ﺴﻨﻘﻭﻡ‬ ‫ﺍﻟﺘﺴﺎﺅل‬ ‫ﻫﺫﺍ‬ ‫ﻋﻠﻰ‬ ‫ﻟﻺﺠﺎﺒﺔ‬.
m
k
f
eff 80
2
1
2
1
==
ππ
www.trgma.com
‫ﻤﺭﺍﺕ‬.‫ﺒﻭﺍﺴﻁﺔ‬ ‫ﻤﺤﻤﻭﻟﺔ‬ ‫ﺘﻜﻭﻥ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﺍﻥ‬ ‫ﻭﺤﻴﺙ‬4‫ﺭﻜﻭﺏ‬ ‫ﺴﺒﺒﻬﺎ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻹﺯﺍﺤﺔ‬ ‫ﺍﻥ‬ ‫ﻭﺤﻴﺙ‬ ،‫ﻤﻜﺎﻓﺊ‬ ‫ﺯﻨﺒﺭﻙ‬ ‫ﻤﻊ‬ ‫ﺒﺎﻟﺘﻌﺎﻤل‬ ‫ﻨﻘﻭﻡ‬ ‫ﺴﻭﻑ‬ ‫ﻟﺫﻟﻙ‬ ‫ﺯﻨﺒﺭﻜﺎﺕ‬
‫ﺍﻟﺭﺠﻠ‬‫ﺇﺯﺍﺤﺔ‬ ‫ﻫﻲ‬ ‫ﻟﻠﺴﻴﺎﺭﺓ‬ ‫ﻴﻥ‬x‫ﺍﻟﺘﺎﻟﻴﺔ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻤﻥ‬ ‫ﻨﺠﺩﻫﺎ‬ ‫ﺍﻥ‬ ‫ﻴﻤﻜﻥ‬:
( )xk∑−=)
‫ﺍﻥ‬ ‫ﺤﻴﺙ‬x‫ﺍﻟﻤﺠﻤﻭﻉ‬ ‫ﺃﻗﻭﺍﺱ‬ ‫ﻤﻥ‬ ‫ﺇﺨﺭﺍﺠﻬﺎ‬ ‫ﺘﻡ‬ ‫ﻟﺫﻟﻙ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﻓﻲ‬ ‫ﺯﻨﺒﺭﻜﺎﺕ‬ ‫ﻟﻸﺭﺒﻊ‬ ‫ﻤﺘﺴﺎﻭﻴﺔ‬ ‫ﺇﺯﺍﺤﺔ‬ ‫ﺘﻤﺜل‬
‫ﺒﺎﻟﻌﻼﻗﺔ‬
mNmN /000,80/ =
‫ﺘﺭﺩﺩ‬ ‫ﻓﺎﻥ‬ ‫ﻭﻋﻠﻴﻪ‬‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻤﻥ‬ ‫ﻴﺤﺴﺏ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﻓﻲ‬ ‫ﻴﺤﺩﺙ‬ ‫ﺍﻟﺫﻱ‬ ‫ﺍﻻﻫﺘﺯﺍﺯ‬15.14
Hz
kg
mN
18.1
1460
/000,80
=
‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﻜﺘﻠﺔ‬ ‫ﺍﺴﺘﺨﺩﻤﻨﺎ‬ ‫ﺍﻨﻨﺎ‬ ‫ﻫﻨﺎ‬ ‫ﻻﺤﻅ‬+‫ﻋﻥ‬ ‫ﺍﻟﺘﻌﻭﻴﺽ‬ ‫ﻋﻨﺩ‬ ‫ﺍﻟﺭﺠﻠﻴﻥ‬ ‫ﻜﺘﻠﺔ‬m
‫ﻟﺘﺘﺫﺒﺫﺏ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﻤﻘﺩﻤﺔ‬ ‫ﻋﻠﻰ‬ ‫ﺒﺎﻟﻀﻐﻁ‬ ‫ﺍﺤﺩﻫﻤﺎ‬ ‫ﻭﻗﺎﻡ‬ ‫ﺍﻟﺴﻴﺎﺭﺓ‬ ‫ﻤﻥ‬ ‫ﺨﺭﺠﺎ‬ ‫ﺭﺠﻠﻴﻥ‬ ‫ﺍﻥ‬ ‫ﺍﻓﺘﺭﺽ‬ ‫ﻟﻭ؟‬ ‫ﻤﺎﺫﺍ‬
‫ﺃﻋﻠﻰ‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﻴﻜﻭﻥ‬ ‫ﺍﻥ‬ ‫ﻨﺘﻭﻗﻊ‬ ‫ﻭﻟﻬﺫﺍ‬ ‫ﺍﻗل‬ ‫ﺴﺘﻜﻭﻥ‬ ‫ﺍﻟﻜﺘﻠﺔ‬ ‫ﻫﻨﺎ‬ ‫ﻭﻟﻜﻥ‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﺒﺤﺴﺎﺏ‬ ‫ﺴﻨﻘﻭﻡ‬ ‫ﺍﻟﺘﺴﺎﺅل‬ ‫ﻫﺫﺍ‬ ‫ﻋﻠﻰ‬ ‫ﻟﻺﺠﺎﺒﺔ‬
Hz
kg
mN
25.1
1300
/000,80
=

30
‫ﺍﻻﺤﺘ‬ ‫ﻋﺩﻴﻡ‬ ‫ﺴﻁﺢ‬ ‫ﻋﻠﻰ‬ ‫ﺃﻓﻘﻴﺎ‬ ‫ﻴﺘﺫﺒﺫﺏ‬‫ﻜﺎﻙ‬.‫ﻤﻘﺩﺍﺭﻫﺎ‬ ‫ﻤﺴﺎﻓﺔ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﺴﺤﺏ‬ ‫ﺘﻡ‬ ‫ﻓﺈﺫﺍ‬5.00cm‫ﻤﻥ‬
،‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺍﻟﺩﻭﺭﻱ‬ ‫ﺍﻟﺯﻤﻥ‬ ‫ﺍﻭﺠﺩ‬)B(،‫ﻟﻠﺠﺴﻡ‬ ‫ﺴﺭﻋﺔ‬ ‫ﺃﻗﺼﻰ‬)C(،‫ﻟﻠﺠﺴﻡ‬ ‫ﻋﺠﻠﺔ‬ ‫ﺃﻗﺼﻰ‬
‫ﺍﻟﺘﺎﻟﻲ‬ ‫ﺍﻟﻨﺤﻭ‬ ‫ﻋﻠﻰ‬ ‫ﻭﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﻟﻠﺠﺴﻡ‬ ‫ﺍﻟﺯﺍﻭﻱ‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﻨﺤﺴﺏ‬ ‫ﺍﻥ‬ ‫ﻴﻤﻜﻥ‬:
10200
00.5
2
1 N
m
k
×
==
π
ω
00.5
22
==
rad
T
π
ω
π
vmax = ωA = (5.00 rad/s)(5.00x10
amax = ω2
A = (5.00 rad/s)2
(5.00x10
‫ﺤﻴﺙ‬ ‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺍﻻﺒﺘﺩﺍﺌﻴﺔ‬ ‫ﺍﻟﺸﺭﻭﻁ‬ ‫ﺍﺴﺘﺨﺩﺍﻡ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬x=A‫ﻭ‬t=0‫ﻋﻠﻰ‬ ‫ﻨﺤﺼل‬
www.trgma.com
‫ﻤﺜﺎل‬3.15‫ﺒﺯﻨﺒﺭﻙ‬ ‫ﻤﺘﺼل‬ ‫ﺠﺴﻡ‬
‫ﻜﺘﻠﺘﻪ‬ ‫ﺠﺴﻡ‬200g‫ﺒﺯﻨﺒﺭﻙ‬ ‫ﻤﺘﺼل‬)k=5.00N/m(‫ﺍﻻﺤﺘ‬ ‫ﻋﺩﻴﻡ‬ ‫ﺴﻁﺢ‬ ‫ﻋﻠﻰ‬ ‫ﺃﻓﻘﻴﺎ‬ ‫ﻴﺘﺫﺒﺫﺏ‬
‫ﺍﻟﺸﻜل‬ ‫ﻓﻲ‬ ‫ﻜﻤﺎ‬ ‫ﺍﻟﺴﻜﻭﻥ‬ ‫ﻤﻥ‬ ‫ﺘﺭﻙ‬ ‫ﺜﻡ‬ ‫ﺍﻻﺘﺯﺍﻥ‬ ‫ﻨﻘﻁﺔ‬7.15)A(،‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺍﻟﺩﻭﺭﻱ‬ ‫ﺍﻟﺯﻤﻥ‬ ‫ﺍﻭﺠﺩ‬
)D(‫ﺍﻟﺯﻤﻥ‬ ‫ﻤﻊ‬ ‫ﺒﺩﺍﻟﺔ‬ ‫ﻭﻋﺠﻠﺘﻪ‬ ‫ﻭﺴﺭﻋﺘﻪ‬ ‫ﺍﻟﺠﺴﻡ‬ ‫ﻤﻭﻀﻊ‬ ‫ﻋﻥ‬ ‫ﻋﺒﺭ‬.
‫ﺍﻟﺤل‬:)A(‫ﺍﻟ‬ ‫ﻤﻊ‬ ‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﻫﺫﻩ‬ ‫ﺒﻤﻘﺎﺭﻨﺔ‬‫ﻤﻌﺎﺩﻟﺔ‬9.15‫ﻭ‬10.15‫ﺍﻟﺘﺎﻟﻲ‬ ‫ﺍﻟﻨﺤﻭ‬ ‫ﻋﻠﻰ‬ ‫ﻭﺍﻟﺯﻨﺒﺭﻙ‬ ‫ﻟﻠﺠﺴﻡ‬ ‫ﺍﻟﺯﺍﻭﻱ‬ ‫ﺍﻟﺘﺭﺩﺩ‬ ‫ﻨﺤﺴﺏ‬ ‫ﺍﻥ‬ ‫ﻴﻤﻜﻥ‬
sec/00.5
10
/
3
rad
kg
mN
=−
‫ﻴﻜﻭﻥ‬ ‫ﺍﻟﺩﻭﺭﻱ‬ ‫ﻭﺍﻟﺯﻤﻥ‬
sec26.1
sec/
=
rad
)B(‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﺒﺎﺴﺘﺨﺩﺍﻡ‬17.15‫ﻋﻠﻰ‬ ‫ﻨﺤﺼل‬
A = (5.00 rad/s)(5.00x10-2
m)=0.250 m/s
)C(‫ﺍﻟﻤﻌﺎﺩﻟﺔ‬ ‫ﺒﺎﺴﺘﺨﺩﺍﻡ‬18.15‫ﻋﻠﻰ‬ ‫ﻨﺤﺼل‬
(5.00x10-2
m)=1.25 m/s2
)D(‫ﺍﻟﻁﻭﺭ‬ ‫ﺜﺎﺒﺕ‬ ‫ﺒﺤﺴﺎﺏ‬ ‫ﺴﻨﻘﻭﻡ‬phase constant‫ﺤﻴﺙ‬ ‫ﻟﻠﺤﺭﻜﺔ‬ ‫ﺍﻻﺒﺘﺩﺍﺌﻴﺔ‬ ‫ﺍﻟﺸﺭﻭﻁ‬ ‫ﺍﺴﺘﺨﺩﺍﻡ‬ ‫ﺨﻼل‬ ‫ﻤﻥ‬

Ch15 oscillatory motion

Related slideshows

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

Recommended for you

More Related Content

What's hot

What's hot (20)

Viewers also liked

Viewers also liked (20)

Recently uploaded

Recently uploaded (10)

Ch15 oscillatory motion