حلول التدريبات العملية 3- مسارات رأسية

1. 1.1-

نعبر عن سرعة الجسم في النقطة C كدالة للارتفاع h_A باستخدام مبادئ الكينماتيكا والديناميكا: 

نرسم مخططًا للقوى المؤثرة على الجسم في حركته أسفل المسار، ونقوم بتحليل قوة الجاذبية الى مركباتها.

نختار المحور X بحيث يكون موجهًا في الاتجاه أسفل السكة، والمحور Y يكون موجهًا في اتجاه القوة العادية.

نكتب معادلات الحركة:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#931;F«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»sin«/mi»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨box¨»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»sin«/mi»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«/menclose»«/mstyle»«/math»                                                    «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#931;F«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Y«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»0«/mn»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Y«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨box¨»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»cos«/mi»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»N«/mi»«/menclose»«/mstyle»«/math»

_{نعبر عن السرعة في النقطة C بمساعدة الكينماتيكا، نستخدم تعبير مربع السرعات:} 

_{«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msup»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»v«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msup»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»v«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»0«/mn»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»x«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msup»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»v«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»c«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»a«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msup»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»0«/mn»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»sin«/mi»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»x«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»v«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»c«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»sin«/mi»«mrow»«mo stretchy=¨true¨ mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo stretchy=¨true¨ mathvariant=¨bold¨»)«/mo»«/mrow»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨»§#8710;«/mo»«mi mathvariant=¨bold¨»x«/mi»«/msqrt»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«/mstyle»«/math»}

لوصف سرعة الجسم في النقطة C كدالة للارتفاع hA ، يتم استخدام الهندسة 

 نعبّر عن الازاحة  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»x«/mi»«/math» بدلالة الارتفاع   h_{A ، بواسطة النسبة المثلثية sin:}

_{«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»sin«/mi»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«mo mathvariant=¨bold¨»=«/mo»«/mrow»«mfrac mathcolor=¨#0000FF¨»«msub»«mi mathvariant=¨bold¨»h«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mrow»«mo mathvariant=¨bold¨»§#8710;«/mo»«mi mathvariant=¨bold¨»x«/mi»«/mrow»«/mfrac»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«msub»«mi mathvariant=¨bold¨»h«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mrow»«mi mathvariant=¨bold¨»sin«/mi»«mrow»«mo stretchy=¨true¨ mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo stretchy=¨true¨ mathvariant=¨bold¨»)«/mo»«/mrow»«/mrow»«/mfrac»«mspace linebreak=¨newline¨/»«/mstyle»«/math»}

_{نُعوّض تعبير الازاحة في تعبير السرعة في النقطة  C:}

_{«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»v«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»c«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»sin«/mi»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«mo mathvariant=¨bold¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨»§#8710;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»x«/mi»«/msqrt»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»v«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»c«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»sin«/mi»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«mo mathvariant=¨bold¨»§#183;«/mo»«mfrac mathcolor=¨#FF0000¨»«msub»«mi mathvariant=¨bold¨»h«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mrow»«mi mathvariant=¨bold¨»sin«/mi»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«/mrow»«/mfrac»«/msqrt»«/mstyle»«/math»}

_{نختزل دالة sin ونحصل على تعبير لسرعة الجسم في النقطة C بدلالة الارتفاع الذي تم تحرير الجسم منه  hA:}

_{«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»V«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»c«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«menclose notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨»sin«/mi»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«/menclose»«mo mathvariant=¨bold¨»§#183;«/mo»«mfrac»«msub»«mi mathvariant=¨bold¨»h«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«menclose notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨»sin«/mi»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«/menclose»«/mfrac»«/msqrt»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨»V«/mi»«mi mathvariant=¨bold¨»c«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«msqrt»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»h«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«/msqrt»«/menclose»«/mstyle»«/math»}

2. 1.2-

أثناء حركة الجسم من النقطة A إلى النقطة C فقط قوة الجاذبية هي التي تبذل شغل، ويتم حفظ الطاقة الميكانيكية.

نكتب معادلة حفظ الطاقة ونعبر منها عن سرعة الجسم في النقطة C كدالة للارتفاع hA: 

نقارن الطاقة الميكانيكية في النقطة A بالطاقة الميكانيكية في النقطة C: 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»v«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mspace linebreak=¨newline¨/»«/math»

يتحرك الجسم من حالة السكون، وبالتالي فإن الطاقة الحركية في النقطة A تساوي صفرًا.

يتم اختيار المستوى النسبي على مستوى الأرض، بحيث تكون الطاقة الوضعية في النقطة C مساوية للصفر.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»h«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mspace linebreak=¨newline¨/»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨»V«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«msqrt»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»h«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«/msqrt»«/menclose»«/mstyle»«/math»

3. 1.3-

نرسم مخططًا للقوى المؤثرة على الجسم عندما يمر بالنقطة  B:

نكتب معادلات الحركة:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#931;F«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»sin«/mi»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨box¨»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»sin«/mi»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«/menclose»«/mstyle»«/math»                                                    «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#931;F«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Y«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»0«/mn»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Y«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨box¨»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»cos«/mi»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«mo mathvariant=¨bold¨»=«/mo»«msub»«mi mathvariant=¨bold¨»N«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«/menclose»«/mstyle»«/math»

من معادلة الحركة باتجاه المحور Y نعبر عن القوة العمودية:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨24px¨»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨»N«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»cos«/mi»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«/menclose»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«/mstyle»«/math»

4. 1.4-

نرسم مخططًا للقوى المؤثرة على الجسم في النقطة D

نكتب معادلة الحركة في اتجاه عمودي على السكة، في هذا الاتجاه يكون الجسم في حالة استمرارية. 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#931;F«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»y«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»0«/mn»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨»N«/mi»«mi mathvariant=¨bold¨»D«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«/menclose»«mspace linebreak=¨newline¨/»«/mstyle»«/math»

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»D«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»0«/mn»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨»N«/mi»«mi mathvariant=¨bold¨»D«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«/menclose»«mspace linebreak=¨newline¨/»«/math»

5. 2.1-

أثناء حركة الجسم من النقطة A إلى النقطة C، يتحرك الجسم بتسارع متغير. A  

نظرًا لأن تسارع الجسم يتغير، فلا يمكن استخدام الكينماتيكا للتعبير عن سرعة الجسم في النقطة C. 

نظرًا لأن الجاذبية فقط هي التي تبذل شغلًا، يتحقق حفظ الطاقة الميكانيكية. 

نكتب معادلة حفظ الطاقة ونعبر منها عن سرعة الجسم في النقطة C كدالة للارتفاع h_A:

نقارن الطاقة الميكانيكية في النقطة A بالطاقة الميكانيكية في النقطة C: 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»v«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mspace linebreak=¨newline¨/»«/mstyle»«/math»

يتحرك الجسم من حالة السكون ، وبالتالي فإن الطاقة الحركية في النقطة A تساوي صفرًا.

يتم اختيار المستوى النسبي على مستوى الأرض، بحيث تكون الطاقة الوضعية في النقطة C مساوية للصفر.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»h«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mspace linebreak=¨newline¨/»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨»V«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«msqrt»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»h«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«/msqrt»«/menclose»«/mstyle»«/math»

6. 2.2-

نرسم مخططًا للقوى المؤثرة على الجسم عندما يكون في النقطة B: 

نختار هيئة محاور: المحور X في الاتجاه المماسي والمحور Y في الاتجاه نحو مركز الدوران. 

نكتب معادلات الحركة:

 في الاتجاه المركزي (اتجاه المحور Y) وفي الاتجاه المماسي :

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#931;F«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»X«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»X«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»X«/mi»«/msub»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»sin«/mi»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»X«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨box¨»«msub»«mi mathvariant=¨bold¨»a«/mi»«mi mathvariant=¨bold¨»X«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»sin«/mi»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«/menclose»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«/mstyle»«/math»                                                    «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#931;F«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#931;F«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»B«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Y«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨box¨»«msub»«mi mathvariant=¨bold¨»N«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»cos«/mi»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#946;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mfrac»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«/menclose»«/mstyle»«/math»

نعبر عن القوة العمودية من معادلة الحركة نحو المركز (الرادياليت): 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»N«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨ mathcolor=¨#0000FF¨»B«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»cos«/mi»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨ mathsize=¨20px¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨ mathsize=¨20px¨»§#946;«/mi»«mo mathvariant=¨bold¨ mathsize=¨20px¨ stretchy=¨true¨»)«/mo»«/mrow»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«msub»«mi mathvariant=¨bold¨ mathsize=¨20px¨»v«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨»B«/mi»«/msub»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/msup»«mi mathvariant=¨bold¨ mathsize=¨20px¨»R«/mi»«/mfrac»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨ mathsize=¨20px¨»N«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨»B«/mi»«/msub»«mo mathvariant=¨bold¨ mathsize=¨20px¨»=«/mo»«mi mathvariant=¨bold¨ mathsize=¨20px¨»m«/mi»«mo mathvariant=¨bold¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathsize=¨20px¨»cos«/mi»«mrow»«mo stretchy=¨true¨ mathvariant=¨bold¨ mathsize=¨20px¨»(«/mo»«mi mathvariant=¨bold¨ mathsize=¨20px¨»§#946;«/mi»«mo stretchy=¨true¨ mathvariant=¨bold¨ mathsize=¨20px¨»)«/mo»«/mrow»«mo mathvariant=¨bold¨ mathsize=¨20px¨»+«/mo»«mi mathvariant=¨bold¨ mathsize=¨20px¨»m«/mi»«mo mathvariant=¨bold¨ mathsize=¨20px¨»§#183;«/mo»«mfrac»«msup»«msub»«mi mathvariant=¨bold¨ mathsize=¨20px¨»v«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨»B«/mi»«/msub»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/msup»«mi mathvariant=¨bold¨ mathsize=¨20px¨»R«/mi»«/mfrac»«/menclose»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«/math»

7. 2.3-

نرسم مخططًا للقوى المؤثرة على الجسم عندما يكون في النقطة  B.

نختار هيئة محاور: المحور X في الاتجاه المماسي والمحور Y في الاتجاه نحو مركز الدوران. 

نكتب معادلات الحركة:

 في الاتجاه المركزي (اتجاه المحور Y) وفي الاتجاه المماسي :

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#931;F«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»X«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»X«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»X«/mi»«/msub»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»sin«/mi»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»X«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨box¨»«msub»«mi mathvariant=¨bold¨»a«/mi»«mi mathvariant=¨bold¨»X«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»sin«/mi»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#945;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«/menclose»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«/mstyle»«/math»                                                    «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#931;F«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#931;F«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»B«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»W«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»Y«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨box¨»«msub»«mi mathvariant=¨bold¨»N«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»cos«/mi»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#946;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mfrac»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«/menclose»«/mstyle»«/math»

نعبر من حفظ الطاقة عن السرعة في النقطة  B:

                                                     «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»B«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»B«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»B«/mi»«/msub»«/mstyle»«/math»

الارتفاع h_A يساوي نصف قطر الدوران R. يتحرك الجسم من حالة السكون وسرعته في النقطة A تساوي صفرًا. 

 نعبر عن السرعة في النقطة B من معادلة حفظ الطاقة: 

                                                «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«menclose mathcolor=¨#0000FF¨ notation=¨updiagonalstrike¨»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»R«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»B«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»B«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»V«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»-«/mo»«msub»«mi mathvariant=¨bold¨»h«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨box¨»«msub»«mi mathvariant=¨bold¨»V«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«msqrt»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mrow»«mo stretchy=¨true¨ mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»-«/mo»«msub»«mi mathvariant=¨bold¨»h«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mo stretchy=¨true¨ mathvariant=¨bold¨»)«/mo»«/mrow»«/msqrt»«/menclose»«/mstyle»«/math»

للوصول إلى التعبير المطلوب، نعبر عن الارتفاع h_B هندسيًا اعتمادًا على الزاوية β ونصف قطر الدوران R. 

مجموع الارتفاعات X و h_B يساوي نصف قطر الدوران R.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»X«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»B«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«/mstyle»«/math»

حسب نسبة cos يتحقق:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»cos«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»(«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#946;«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»)«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»X«/mi»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«/mstyle»«/math»

نعبر عن الارتفاع h_B من هاتين المعادلتين

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نعوّض الارتفاع h_B في تعبير السرعة  v_B  

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»V«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»B«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»-«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»B«/mi»«/msub»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«/msqrt»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»V«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»B«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mrow»«mo stretchy=¨true¨ mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»(«/mo»«mi mathcolor=¨#FF0000¨ mathvariant=¨bold¨»R«/mi»«mo mathcolor=¨#FF0000¨ mathvariant=¨bold¨»-«/mo»«mi mathcolor=¨#FF0000¨ mathvariant=¨bold¨»R«/mi»«mo mathcolor=¨#FF0000¨ mathvariant=¨bold¨»§#183;«/mo»«mi mathcolor=¨#FF0000¨ mathvariant=¨bold¨»cos«/mi»«mo mathcolor=¨#FF0000¨ mathvariant=¨bold¨»(«/mo»«mi mathcolor=¨#FF0000¨ mathvariant=¨bold¨»§#946;«/mi»«mo mathcolor=¨#FF0000¨ mathvariant=¨bold¨»)«/mo»«mo stretchy=¨true¨ mathvariant=¨bold¨»)«/mo»«/mrow»«/msqrt»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»V«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»B«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mrow»«mo stretchy=¨true¨ mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»+«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»cos«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»§#946;«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo stretchy=¨true¨ mathvariant=¨bold¨»)«/mo»«/mrow»«/msqrt»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»V«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»B«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mrow»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»cos«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»§#946;«/mi»«mo mathvariant=¨bold¨»)«/mo»«/mrow»«/msqrt»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«/mstyle»«/math»

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»V«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»B«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»-«/mo»«msub»«mi mathvariant=¨bold¨»h«/mi»«mi mathvariant=¨bold¨»B«/mi»«/msub»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«/msqrt»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»-«/mo»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»cos«/mi»«mo mathvariant=¨bold¨»(«/mo»«mi mathvariant=¨bold¨»§#946;«/mi»«mo mathvariant=¨bold¨»)«/mo»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«/msqrt»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»R«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»cos«/mi»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«mi mathvariant=¨bold¨»§#946;«/mi»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«/msqrt»«/mstyle»«/math»

نعبر عن القوة العمودية في النقطة B من معادلة الحركة الدائرية ، ونعوّض تعبير السرعة V_B: 

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8. 3.1-

أثناء حركة الجسم من النقطة A إلى النقطة C ، فقط قوة الجاذبية هي التي تبذل شغلًا، ويتحقق حفظ الطاقة الميكانيكية.

نجد سرعة الجسم في النقطة C باستخدام معادلة حفظ الطاقة: 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»v«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«/mstyle»«/math»

يتحرك الجسم من حالة السكون، وبالتالي فإن الطاقة الحركية في النقطة A تساوي صفرًا. ارتفاع النقطة A يساوي R1. ارتفاع النقطة C يساوي R2.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mfrac»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msub»«mspace linebreak=¨newline¨/»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»-«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mrow mathcolor=¨#0000FF¨»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»-«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«msqrt»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mrow»«mo stretchy=¨true¨ mathvariant=¨bold¨»(«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»-«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo stretchy=¨true¨ mathvariant=¨bold¨»)«/mo»«/mrow»«/msqrt»«/menclose»«mspace linebreak=¨newline¨/»«/mstyle»«/math»

9. 3.2-

نرسم مخطط للقوى المؤثرة على الجسم عندما يمر بالنقطة C. 

نكتب معادلة الحركة في اتجاه نحو مركز المسار الدائري:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#931;F«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»r«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»a«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»r«/mi»«/msub»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#931;F«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»r«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨»N«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mfrac»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«/menclose»«/mstyle»«/math»

10. 3.3-

في البند السابق، وجدنا تعبيرًا للقوة العمودية في النقطة C كدالة لسرعة الجسم في النقطة C.

نجد من هذا التعبير السرعة عندما تكون السرعة العمودية في النقطة C تساوي صفرًا.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«mspace linebreak=¨newline¨/»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»0«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«mspace linebreak=¨newline¨/»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mspace linebreak=¨newline¨/»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«msqrt»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/msqrt»«/menclose»«mspace linebreak=¨newline¨/»«/mstyle»«/math»

11. 3.4-

وجدنا في البند 3.1 أن تعبير سرعة الجسم في النقطة C هو:  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»v«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»-«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«/msqrt»«/mstyle»«/math»

وجدنا في البند 3.3 أن تعبير سرعة الجسم في النقطة C التي تكون فيها القوة العمودية في النقطة C يساوي صفرًا: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«msup»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»V«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#953;«/mi»«/msup»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»c«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/msqrt»«/mstyle»«/math»

عندما يتحرر الجسم من حالة السكون وتكون القوة العمودية في النقطة C مساويًا للصفر، فإن كلا التعبيرين سوف يتحققان. 

لإيجاد النسبة بين نصفي القطر، نقارن بين تعبيري السرعة: 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msqrt mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/msqrt»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mrow»«mo mathvariant=¨bold¨ stretchy=¨true¨»(«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»-«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo mathvariant=¨bold¨ stretchy=¨true¨»)«/mo»«/mrow»«/msqrt»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨»g«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨»g«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mrow mathcolor=¨#0000FF¨»«mo stretchy=¨true¨ mathvariant=¨bold¨»(«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«mo mathvariant=¨bold¨»-«/mo»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«mo stretchy=¨true¨ mathvariant=¨bold¨»)«/mo»«/mrow»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»3«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»R«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»1«/mn»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«mfrac»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»1«/mn»«/msub»«msub»«mi mathvariant=¨bold¨»R«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msub»«/mfrac»«mo mathvariant=¨bold¨»=«/mo»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨».«/mo»«mn mathvariant=¨bold¨»5«/mn»«/menclose»«/mstyle»«/math»

12. 4.1-

نرسم مخطط للقوى المؤثرة على الجسم عندما يمر بالنقطة C. 

نكتب معادلة الحركة الدائرية عندما يمر الجسم بالنقطة  C .

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#931;F«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»r«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨box¨»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»+«/mo»«msub»«mi mathvariant=¨bold¨»N«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mfrac»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«/menclose»«/mstyle»«/math»

نعبّر عن القوة العمودية التي تشغّلها السكة على الجسم في النقطة C من معادلة الحركة الدائرية: 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨»N«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mfrac»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«/menclose»«/mstyle»«/math»

13. 4.2-

في البند السابق وجدنا تعبيرًا للقوة العمودية في النقطة C كدالة لسرعة الجسم في النقطة C. 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«/mstyle»«/math»

أثناء حركة الجسم على المسار الأملس، لم تبذل شغلل على الجسم سوى قوة الجاذبية. لذلك يتم حفظ الطاقة الميكانيكية. 

 نعبّر عن السرعة V_C كدالة للسرعة V_A, بواسطة معادلة حفظ الطاقة.

                                                                                         «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«/mstyle»«/math»

نختار مستوى الانتساب في ارتفاع النقطة A ، بحيث تكون الطاقة الوضعية في النقطة A مساوية لصفر.

ارتفاع النقطة C يساوي ضعف نصف قطر الدوران. 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨ mathsize=¨20px¨»1«/mn»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»v«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨»A«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»+«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»m«/mi»«mo mathvariant=¨bold¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathsize=¨20px¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathsize=¨20px¨»h«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨»A«/mi»«/msub»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨ mathsize=¨20px¨»1«/mn»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»v«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨ mathsize=¨20px¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨ mathsize=¨20px¨»C«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨ mathsize=¨20px¨»1«/mn»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»v«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨»A«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨ mathsize=¨20px¨»1«/mn»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»v«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#FF0000¨ mathsize=¨20px¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#FF0000¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨ mathsize=¨20px¨»R«/mi»«/math»

نختزل الكتلة ونعبر عن سرعة V_C:

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mathvariant=¨bold¨ mathsize=¨20px¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»V«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»+«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»R«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»V«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨»A«/mi»«/msub»«mn mathcolor=¨#0000FF¨ mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»=«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»v«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨»C«/mi»«/msub»«mn mathcolor=¨#0000FF¨ mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»+«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»R«/mi»«mspace linebreak=¨newline¨/»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mo mathsize=¨20px¨»§#160;«/mo»«mspace linebreak=¨newline¨/»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»V«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»=«/mo»«msup»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»v«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»A«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»-«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»4«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»R«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»V«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«msup»«msub»«mi mathvariant=¨bold¨ mathsize=¨20px¨»v«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨»A«/mi»«/msub»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathsize=¨20px¨»-«/mo»«mn mathvariant=¨bold¨ mathsize=¨20px¨»4«/mn»«mo mathvariant=¨bold¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathsize=¨20px¨»R«/mi»«/msqrt»«mspace linebreak=¨newline¨/»«/math»

نعوّض السرعة Vc في تعبير القوة العمودية في النقطة C: 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»v«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»C«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«mstyle mathvariant=¨bold¨»«mo stretchy=¨true¨»(«/mo»«msqrt mathcolor=¨#FF0000¨»«msup»«msub»«mi»v«/mi»«mi»A«/mi»«/msub»«mn»2«/mn»«/msup»«mo»-«/mo»«mn»4«/mn»«mo»§#183;«/mo»«mi»g«/mi»«mo»§#183;«/mo»«mi»R«/mi»«/msqrt»«mo stretchy=¨true¨»)«/mo»«/mstyle»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»4«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mn mathvariant=¨bold¨»4«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»R«/mi»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mn mathvariant=¨bold¨»4«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«menclose notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨»R«/mi»«/menclose»«/mrow»«menclose notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨»R«/mi»«/menclose»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»4«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨»N«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mfrac»«msup»«msub»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨»-«/mo»«mn mathvariant=¨bold¨»5«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»g«/mi»«/menclose»«/mstyle»«/math»

14. 4.3-

السرعة الدنيا التي يكمل بها الجسم دورة كاملة هي السرعة التي عندها القوة العمودية في النقطة C تساوي صفرًا. 

نعوّض «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»N«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»0«/mn»«/math» في التعبير الذي حصلنا عليه في التعبير 4.1 ، ونعبر عن السرعة بـ  V_C 

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨»V«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»N«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨»V«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mrow»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»0«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«msup»«mrow»«mi mathvariant=¨bold¨»m«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»V«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«/mrow»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«msup»«mrow»«menclose notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨»V«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«/mrow»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mi mathvariant=¨bold¨»R«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨updiagonalstrike¨»«mi mathvariant=¨bold¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨»V«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mo mathvariant=¨bold¨»=«/mo»«msqrt»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»R«/mi»«/msqrt»«/menclose»«/mstyle»«/math»

15. 4.4-

الحد الأدنى للارتفاع الذي يجب أن يتحرر منه الجسم من حالة السكون هو الارتفاع الذي ستكون فيه سرعة الجسم في النقطة C مساوية  «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»V«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»R«/mi»«/msqrt»«/math»

نعبّر عن هذا الارتفاع من معادلة حفظ الطاقة: 

                                              «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»v«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«/mstyle»«/math»

يبدأ الجسم في التحرك من حالة السكون ، فليس لديه طاقة حركية في النقطة  A.

ارتفاع النقطة C  يساوي ضعف نصف قطر الدوران.  

وبناءً عليه نعبر عن ارتفاع تحرير الجسم اللازم لإكمال حركته.

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«menclose mathcolor=¨#0000FF¨ notation=¨updiagonalstrike¨»«mfrac»«mn mathvariant=¨bold¨ mathsize=¨20px¨»1«/mn»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathsize=¨20px¨»m«/mi»«mo mathvariant=¨bold¨ mathsize=¨20px¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨ mathsize=¨20px¨»v«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨»A«/mi»«/msub»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/msup»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»+«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨ mathsize=¨20px¨»1«/mn»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«msup»«msqrt mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathsize=¨20px¨»R«/mi»«/msqrt»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»+«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»m«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»R«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»g«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»h«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨ mathsize=¨20px¨»1«/mn»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»g«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»R«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»+«/mo»«menclose mathcolor=¨#0000FF¨ notation=¨downdiagonalstrike¨»«mi mathvariant=¨bold¨ mathsize=¨20px¨»g«/mi»«/menclose»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»R«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»h«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨ mathsize=¨20px¨»1«/mn»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»R«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»+«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»2«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨ mathsize=¨20px¨»R«/mi»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«menclose mathcolor=¨#0000FF¨ notation=¨circle¨»«msub»«mi mathvariant=¨bold¨ mathsize=¨20px¨»h«/mi»«mi mathvariant=¨bold¨ mathsize=¨20px¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathsize=¨20px¨»=«/mo»«mn mathvariant=¨bold¨ mathsize=¨20px¨»2«/mn»«mo mathvariant=¨bold¨ mathsize=¨20px¨».«/mo»«mn mathvariant=¨bold¨ mathsize=¨20px¨»5«/mn»«mo mathvariant=¨bold¨ mathsize=¨20px¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathsize=¨20px¨»R«/mi»«/menclose»«mspace linebreak=¨newline¨/»«/math»

16. 4.5-

نجد الارتفاع h_A الذي يجب أن يقذف الجسم منه بسرعة V₀ بحيث تكون سرعة الجسم في النقطة C  مساوية «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msub»«mi mathvariant=¨bold-italic¨ mathcolor=¨#0000FF¨»V«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msqrt mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»g«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»R«/mi»«/msqrt»«/math»

يتم التعبير عن هذا الارتفاع من معادلة حفظ الطاقة:

                                              «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨20px¨»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mspace linebreak=¨newline¨/»«mspace linebreak=¨newline¨/»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»A«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»E«/mi»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»k«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»U«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«mspace linebreak=¨newline¨/»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»v«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»A«/mi»«/msub»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mn mathvariant=¨bold¨»1«/mn»«mn mathvariant=¨bold¨»2«/mn»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msup»«msub mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨»v«/mi»«mi mathvariant=¨bold¨»C«/mi»«/msub»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»m«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»g«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«msub»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»h«/mi»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»C«/mi»«/msub»«/mstyle»«/math»

سرعة الجسم بالنقطة A هي V₀.

ارتفاع النقطة C يساوي ضعف نصف قطر الدوران. 

وبناءً عليه سوف نعبر عن ارتفاع تحرير الجسم اللازم لإكمال حركته. 

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نضرب المعادلة في 2 ونعبر عن h_A: 

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