حلول التدريبات العملية الديناميكا لجسمين: إيجاد التسارع مع احتكاك

10. جسمان على سطحين مائلين

10.3. إيجاد التسارع مع احتكاك

يتحرك كلا الجسمين على طول السطحين المائلين.

لأن الجسمين موصولين بواسطة خيط مشدود، فإن تسارع كل منهما متساوي المقدار.

لضمان أن يكون لتسارع كل منهما نفس الإشارة، نحدّد المحاور على النحو التالي:

بالنسبة للجسم 1: نقوم بتعيين المحور X لأسفل السطح المائل، والمحور Y متعامد على السطح.

بالنسبة للجسم 2: نحدد المحور X باتجاه السطح المائل لأعلى ، والمحور Y متعامد على السطح.

تعمل ثلاث قوى على الجسم 1: قوة الجاذبية W₁، قوة الاحتكاك f_k1، قوة التوتر T والقوة العمودية N₁.

معطى أن الجسمين يتحركان إلى اليمين، وبالتالي فإن الجسم 1 ينزل نحو أسفل السطح المائل، وبالتالي فإن قوة الاحتكاك في اتجاه نحو أعلى السطح المائل (عكس اتجاه الحركة).

نرسم مخططًا للقوى المؤثرة على الجسم 1:

نكتب معادلة الحركة في اتجاه المحور y، العمودي على السطح المائل، والجسم متزنًا في هذا الاتجاه:

يتحرك الجسم 1 نحو أسفل السطح المائل بتسارع a ، ونكتب معادلة الحركة لهذا الجسم في اتجاه الحركة (في اتجاه المحور x):

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تعمل أربع قوى على الجسم 2: قوة الجاذبية W₂، قوة الاحتكاك f_k2 ، قوة الشد T والقوة العمودية N₂.

يتحرك الجسمان نحو اليمين ، وبالتالي يرتفع الجسم 2 نحو أعلى السطح المائل، وبالتالي فإن قوة الاحتكاك تكون في اتجاه أسفل السطح المائل (عكس اتجاه الحركة).

نرسم مخططًا للقوى المؤثرة على الجسم 2:

نكتب معادلة الحركة في اتجاه المحور y، وهو عمودي على السطح المائل، والجسم متّزنًا في هذا الاتجاه:

يتحرك الجسم 2 نحو أعلى المستوى المائل بالتسارع a ، نكتب معادلة الحركة لهذا الجسم في اتجاه الحركة (في اتجاه المحور x الذي تم اختياره):

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نجمع معادلتا الحركة في الاتجاه الأفقي ونحصل:

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