Difference between revisions of "Pulley system on a double inclined plane"

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[[File:Pulleysystemondoubleinclinedplane.png|thumb|400px|right]]
  
 
This article is about the following scenario. A fixed triangular wedge has two inclines <math>I_1</math> and <math>I_2</math> making angles <math>\alpha_1</math> and <math>\alpha_2</math> with the horizontal. A pulley is affixed to the top vertex of the triangle. A string through the pulley has attached at its two ends blocks of masses <math>m_1</math> and <math>m_2</math>, resting on the two inclines <math>I_1</math> and <math>I_2</math> respectively. The string is inextensible. The coefficients of static and kinetic friction between <math>m_1</math> and <math>I_1</math> are <math>\mu_{s1}</math> and <math>\mu_{k1}</math> respectively. The coefficients of static and kinetic friction between <math>m_2</math> and <math>I_2</math> are <math>\mu_{s2}</math> and <math>\mu_{k2}</math> respectively. Assume that <math>\mu_{k1} \le \mu_{s1}</math> and <math>\mu_{k2} \le \mu_{s2}</math>.
 
This article is about the following scenario. A fixed triangular wedge has two inclines <math>I_1</math> and <math>I_2</math> making angles <math>\alpha_1</math> and <math>\alpha_2</math> with the horizontal. A pulley is affixed to the top vertex of the triangle. A string through the pulley has attached at its two ends blocks of masses <math>m_1</math> and <math>m_2</math>, resting on the two inclines <math>I_1</math> and <math>I_2</math> respectively. The string is inextensible. The coefficients of static and kinetic friction between <math>m_1</math> and <math>I_1</math> are <math>\mu_{s1}</math> and <math>\mu_{k1}</math> respectively. The coefficients of static and kinetic friction between <math>m_2</math> and <math>I_2</math> are <math>\mu_{s2}</math> and <math>\mu_{k2}</math> respectively. Assume that <math>\mu_{k1} \le \mu_{s1}</math> and <math>\mu_{k2} \le \mu_{s2}</math>.

Revision as of 01:12, 17 September 2010

This article discusses a scenario/arrangement whose statics/dynamics/kinematics can be understood using the ideas of classical mechanics.
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Pulleysystemondoubleinclinedplane.png

This article is about the following scenario. A fixed triangular wedge has two inclines I_1 and I_2 making angles \alpha_1 and \alpha_2 with the horizontal. A pulley is affixed to the top vertex of the triangle. A string through the pulley has attached at its two ends blocks of masses m_1 and m_2, resting on the two inclines I_1 and I_2 respectively. The string is inextensible. The coefficients of static and kinetic friction between m_1 and I_1 are \mu_{s1} and \mu_{k1} respectively. The coefficients of static and kinetic friction between m_2 and I_2 are \mu_{s2} and \mu_{k2} respectively. Assume that \mu_{k1} \le \mu_{s1} and \mu_{k2} \le \mu_{s2}.

Summary of cases starting from rest

Case What happens qualitatively Magnitude of accelerations
\! m_1\sin \alpha_1 - m_2\sin \alpha_2 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2 m_1 slides downward and m_2 slides upward, with the same magnitude of acceleration \! a = g[m_1 \sin \alpha_1 - m_2 \sin \alpha_2 - \mu_{k1}m_1\sin \alpha_1 - \mu_{k2}m_2 \sin\alpha_2].
\! m_2\sin \alpha_2 - m_1\sin \alpha_1 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2 m_2 slides downward and m_1 slides upward, with the same magnitude of acceleration \! a = g[m_2 \sin \alpha_2 - m_1 \sin \alpha_1 - \mu_{k1}m_1\sin \alpha_1 - \mu_{k2}m_2 \sin\alpha_2].
\! |m_1 \sin \alpha_1 - m_2 \sin \alpha_2| \le |\mu_{s1}m_1 \cos \alpha_1 + \mu_{s2}m_2 \cos \alpha_2| The system remains at rest 0