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 $I_{1}$ and $I_{2}$ making angles $α_{1}$ and $α_{2}$ 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 $μ_{s 1}$ and $μ_{k 1}$ respectively. The coefficients of static and kinetic friction between $m_{2}$ and $I_{2}$ are $μ_{s 2}$ and $μ_{k 2}$ respectively. Assume that $μ_{k 1} \leq μ_{s 1}$ and $μ_{k 2} \leq μ_{s 2}$ .

Summary of cases starting from rest

Case	What happens qualitatively	Magnitude of accelerations
$m_{1} \sin α_{1} - m_{2} \sin α_{2} > μ_{s 1} m_{1} \cos α_{1} + μ_{s 2} m_{2} \cos α_{2}$	$m_{1}$ slides downward and $m_{2}$ slides upward, with the same magnitude of acceleration	$a = g (m_{1} \sin α_{1} - m_{2} \sin α_{2} - μ_{k 1} m_{1} \sin α_{1} - μ_{k 2} m_{2} \sin α_{2}) / (m_{1} + m_{2})$ .
$m_{2} \sin α_{2} - m_{1} \sin α_{1} > μ_{s 1} m_{1} \cos α_{1} + μ_{s 2} m_{2} \cos α_{2}$	$m_{2}$ slides downward and $m_{1}$ slides upward, with the same magnitude of acceleration	$a = g (m_{2} \sin α_{2} - m_{1} \sin α_{1} - μ_{k 1} m_{1} \sin α_{1} - μ_{k 2} m_{2} \sin α_{2}) / (m_{1} + m_{2})$ .
$\| m_{1} \sin α_{1} - m_{2} \sin α_{2} \| \leq \| μ_{s 1} m_{1} \cos α_{1} + μ_{s 2} m_{2} \cos α_{2} \|$	The system remains at rest	$0$