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

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(Summary of cases starting from rest)
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! Case !! What happens qualitatively !! Magnitude of accelerations
 
! Case !! What happens qualitatively !! Magnitude of accelerations
 
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| <math>\! m_1\sin \alpha_1 - m_2\sin \alpha_2 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2</math> || <math>m_1</math> slides downward and <math>m_2</math> slides upward, with the same magnitude of acceleration || <math>\! 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]</math>.
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| <math>\! m_1\sin \alpha_1 - m_2\sin \alpha_2 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2</math> || <math>m_1</math> slides downward and <math>m_2</math> slides upward, with the same magnitude of acceleration || <math>\! 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)</math>.
 
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| <math>\! m_2\sin \alpha_2 - m_1\sin \alpha_1 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2</math> || <math>m_2</math> slides downward and <math>m_1</math> slides upward, with the same magnitude of acceleration || <math>\! 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]</math>.
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| <math>\! m_2\sin \alpha_2 - m_1\sin \alpha_1 > \mu_{s1}m_1\cos \alpha_1 + \mu_{s2}m_2\cos \alpha_2</math> || <math>m_2</math> slides downward and <math>m_1</math> slides upward, with the same magnitude of acceleration || <math>\! 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)</math>.
 
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| <math>\! |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|</math> || The system remains at rest || <math>0</math>
 
| <math>\! |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|</math> || The system remains at rest || <math>0</math>
 
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Revision as of 18:13, 18 September 2010

This article discusses a scenario/arrangement whose statics/dynamics/kinematics can be understood using the ideas of classical mechanics.
View other mechanics scenarios
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_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