defis 1 : physique - le double plan incliné (dynamique)

deux blocs, $m_1 = 3,0\text{ kg}$ et $m_2 = 5,0\text{ kg}$, sont reliés par une corde idéale passant par une poulie sans friction.
- $m_1$ est placé sur un plan incliné à $30^\circ$ (vers la gauche).
- $m_2$ est placé sur un plan incliné à $45^\circ$ (vers la droite).

le coefficient de friction cinétique est le même pour les deux plans : $\mu_k = 0,15$. le système se déplace vers la droite ($m_2$ descend). (prendre $g = 9,8\text{ m/s}^2$)

question :
1) dessinez les diagrammes de corps franc (free body diagram) pour chaque bloc.
2) écrivez les équations de mouvement pour $m_1$ et $m_2$.
3) calculez laccélération (a) du système (arrondie à 2 décimales).

Question

defis 1 : physique - le double plan incliné (dynamique)

deux blocs, $m_1 = 3,0\text{ kg}$ et $m_2 = 5,0\text{ kg}$, sont reliés par une corde idéale passant par une poulie sans friction.
- $m_1$ est placé sur un plan incliné à $30^\circ$ (vers la gauche).
- $m_2$ est placé sur un plan incliné à $45^\circ$ (vers la droite).

le coefficient de friction cinétique est le même pour les deux plans : $\mu_k = 0,15$. le système se déplace vers la droite ($m_2$ descend). (prendre $g = 9,8\text{ m/s}^2$)

question :
1) dessinez les diagrammes de corps franc (free body diagram) pour chaque bloc.
2) écrivez les équations de mouvement pour $m_1$ et $m_2$.
3) calculez laccélération (a) du système (arrondie à 2 décimales).

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [], "new_concepts": [ "Newton's Second Law", "Friction on Inclined Plane", "Two-Body Systems" ], "current_concepts": [ "Newton's Second Law", "Friction on Inclined Plane", "Two-Body Systems" ] } ## Identify given parameters and coordinate systems $$ \begin{aligned} &m_1 = 3.0\text{ kg},\quad \theta_1 = 30^\circ\text{ (inclined to the left, moving up the incline)}\ &m_2 = 5.0\text{ kg},\quad \theta_2 = 45^\circ\text{ (inclined to the right, moving down the incline)}\ &\mu_k = 0.15,\quad g = 9.8\text{ m/s}^2 \end{aligned} $$ ## Formulate equations of motion for each block $$ \begin{aligned} &\text{For } m_1 \text{ (moving up the incline):}\ &N_1 = m_1 g \cos\theta_1\ &T - m_1 g \sin\theta_1 - \mu_k N_1 = m_1 a\ &T - m_1 g \sin\theta_1 - \mu_k m_1 g \cos\theta_1 = m_1 a\ \ &\text{For } m_2 \text{ (moving down the incline):}\ &N_2 = m_2 g \cos\theta_2\ &m_2 g \sin\theta_2 - T - \mu_k N_2 = m_2 a\ &m_2 g \sin\theta_2 - T - \mu_k m_2 g \cos\theta_2 = m_2 a \end{aligned} $$ ## Solve for the system acceleration $$ \begin{aligned} &a = \frac{m_2 g (\sin\theta_2 - \mu_k \cos\theta_2) - m_1 g (\sin\theta_1 + \mu_k \cos\theta_1)}{m_1 + m_2}\ &a = \frac{5.0 \cdot 9.8 (\sin 45^\circ - 0.15 \cos 45^\circ) - 3.0 \cdot 9.8 (\sin 30^\circ + 0.15 \cos 30^\circ)}{3.0 + 5.0}\ &a = \frac{49.0 (0.7071 - 0.1061) - 29.4 (0.5000 + 0.1299)}{8.0}\ &a = \frac{29.45 - 18.52}{8.0} \approx 1.37\text{ m/s}^2 \end{aligned} $$ ## Question 1 * **Free Body Diagram for $m_1$** (moving up the incline to the right): * Tension force $\vec{T}$ pointing up the incline (parallel to the surface, towards the peak). * Gravity force $\vec{F}_{g1} = m_1\vec{g}$ pointing vertically downwards. * Normal force $\vec{N}_1$ pointing perpendicular to and away from the incline surface. * Kinetic friction force $\vec{f}_{k1}$ pointing down the incline (opposite to the direction of motion). * **Free Body Diagram for $m_2$** (moving down the incline to the right): * Tension force $\vec{T}$ pointing up the incline (parallel to the surface, towards the peak). * Gravity force $\vec{F}_{g2} = m_2\vec{g}$ pointing vertically downwards. * Normal force $\vec{N}_2$ pointing perpendicular to and away from the incline surface. * Kinetic friction force $\vec{f}_{k2}$ pointing up the incline (opposite to the direction of motion). ## Question 2 * **For $m_1$**: $$T - m_1 g \sin(30^\circ) - \mu_k m_1 g \cos(30^\circ) = m_1 a$$ * **For $m_2$**: $$m_2 g \sin(45^\circ) - T - \mu_k m_2 g \cos(45^\circ) = m_2 a$$ ## Question 3 $$a \approx 1.37\text{ m/s}^2$$ { "elements": [ { "type": "polygon", "params": [[-5, -3], [0, 1], [5, -3]], "properties": { "fillColor": "#583C87", "fillOpacity": 0.1, "strokeColor": "#583C87", "strokeWidth": 2 } }, { "type": "line", "params": [[-6, -3], [6, -3]], "properties": { "strokeColor": "black", "strokeWidth": 1 } }, { "type": "point", "params": [[-2.5, -1]], "properties": { "name": "m1", "color": "#8C55F2", "size": 6 } }, { "type": "point", "params": [[2.5, 1]], "properties": { "name": "m2", "color": "#F2557F", "size": 8 } }, { "type": "point", "params": [[0, 1]], "properties": { "name": "Pulley", "color": "black", "size": 3 } }, { "type": "segment", "params": [[-2.5, -1], [0, 1]], "properties": { "strokeColor": "grey", "strokeWidth": 2 } }, { "type": "segment", "params": [[2.5, 1], [0, 1]], "properties": { "strokeColor": "grey", "strokeWidth": 2 } }, { "type": "text", "params": [[-4, -2.5], "30°"] }, { "type": "text", "params": [[3.5, -2.5], "45°"] } ] } { "subject": "Natural Science", "question_type": "Multi-part", "knowledge_point": [ "Natural Science", "Physics", "Dynamics" ] }

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Explanation:

Identify given parameters and coordinate systems

Formulate equations of motion for each block

Solve for the system acceleration

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