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

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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, thus making it a [[involves::double inclined plane]]. A [[involves::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, thus making it a [[involves::double inclined plane]]. A [[involves::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>.
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We assume the pulley to be massless so that its moment of inertia can be ignored for the information below.
  
 
==Summary of cases starting from rest==
 
==Summary of cases starting from rest==

Revision as of 00:07, 13 August 2011

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, thus making it a double inclined plane. 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}.

We assume the pulley to be massless so that its moment of inertia can be ignored for the information below.

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_1 + m_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 + m_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