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 $\alpha _{1}$ 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 $\mu _{k1}\leq \mu _{s1}$ and $\mu _{k2}\leq \mu _{s2}$.

We assume the pulley to be massless so that its moment of inertia can be ignored for the information below. We also assume the string to be massless, so that the tension values at the two ends of the string are equal in magnitude.

Summary of cases starting from rest

These cases will be justified later in the article, based on the force diagram and by solving the resulting equations.

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	$\!a=g(m_{1}\sin \alpha _{1}-m_{2}\sin \alpha _{2}-\mu _{k1}m_{1}\cos \alpha _{1}-\mu _{k2}m_{2}\cos \alpha _{2})/(m_{1}+m_{2})$.
$\!m_{2}\sin \alpha _{2}-m_{1}\sin \alpha _{1}>\mu _{s1}m_{1}\cos \alpha _{1}+\mu _{s2}m_{2}\cos \alpha _{2}$	$m_{2}$ slides downward and $m_{1}$ slides upward, with the same magnitude of acceleration	$\!a=g(m_{2}\sin \alpha _{2}-m_{1}\sin \alpha _{1}-\mu _{k1}m_{1}\cos \alpha _{1}-\mu _{k2}m_{2}\cos \alpha _{2})/(m_{1}+m_{2})$.
$\!|m_{1}\sin \alpha _{1}-m_{2}\sin \alpha _{2}|\leq |\mu _{s1}m_{1}\cos \alpha _{1}+\mu _{s2}m_{2}\cos \alpha _{2}|$	The system remains at rest	$0$

Basic components of force diagram

There are two force diagrams of interest here, namely the force diagrams of the masses $m_{1}$ and $m_{2}$.

There are five candidate forces on each mass. We describe the situation for $m_{1}$ below:

Force (letter)	Nature of force	Condition for existence	Magnitude	Direction
$m_{1}g$	gravitational force	unconditional	$m_{1}g$ where $m_{1}$ is the mass and $g$ is the acceleration due to gravity	vertically downward, hence an angle $\alpha _{1}$ with the normal to the incline and an angle $(\pi /2)-\alpha _{1}$ with the incline
$T$	tension	unconditional	needs to be computed based on solving equations.	pulls the mass up the inclined plane.
$N_{1}$	normal force	unconditional	adjusts so that there is no net acceleration perpendicular to the plane of the incline	outward normal to the incline
$f_{s1}$	static friction	no sliding	adjusts so that there is no net acceleration along the incline. At most equal to $\mu _{s1}N_{1}$.	The direction could be either up or down the incline, depending on whether the remaining forces create a net tendency to pull $m_{1}$ down or pull it up.
$f_{k1}$	kinetic friction	sliding	$\mu _{k1}N_{1}$ where $N_{1}$ is the normal force	opposite direction of motion -- hence down the incline if sliding up, up the incline if sliding down.

A similar description is valid for $m_{2}$:

Force (letter)	Nature of force	Condition for existence	Magnitude	Direction
$m_{2}g$	gravitational force	unconditional	$m_{2}g$ where $m_{2}$ is the mass and $g$ is the acceleration due to gravity	vertically downward, hence an angle $\alpha _{2}$ with the normal to the incline and an angle $(\pi /2)-\alpha _{2}$ with the incline
$T$	tension	unconditional	needs to be computed based on solving equations.	pulls the mass up the inclined plane.
$N_{2}$	normal force	unconditional	adjusts so that there is no net acceleration perpendicular to the plane of the incline	outward normal to the incline
$f_{s2}$	static friction	no sliding	adjusts so that there is no net acceleration along the incline. At most equal to $\mu _{s2}N_{2}$.	The direction could be either up or down the incline, depending on whether the remaining forces create a net tendency to pull $m_{2}$ down or pull it up.
$f_{k2}$	kinetic friction	sliding	$\mu _{k2}N_{2}$ where $N_{2}$ is the normal force	opposite direction of motion -- hence down the incline if sliding up, up the incline if sliding down.

Components perpendicular to the respective inclined planes

For more analysis of this part, see sliding motion along an inclined plane#Component perpendicular to the inclined plane

For $m_{1}$, taking the component perpendicular to the inclined plane $I_{1}$, we get:

$\!N_{1}=m_{1}g\cos \alpha _{1}\qquad (1.1)$

For $m_{2}$, taking the component perpendicular to the inclined plane $I_{2}$, we get:

$\!N_{2}=m_{2}g\cos \alpha _{2}\qquad (1.2)$

Note that this part of the analysis is common to both the sliding and the no sliding cases.

Components along the respective inclined planes assuming no sliding

Note that the no sliding case has two subcases:

• The system has a tendency to slide $m_{1}$ down and $m_{2}$ up, i.e., this is what would happen if there were no static friction. Thus, the static friction $f_{s1}$ acts up the incline $I_{1}$ and the static friction $f_{s2}$ acts down the incline $I_{2}$.
• The system has a tendency to slide $m_{1}$ up and $m_{2}$ down, i.e., this is what would happen if there were no static friction. Thus, the static friction $f_{s1}$ acts down the incline $I_{1}$ and the static friction $f_{s2}$ acts up the incline $I_{2}$.

Let's consider the first case, i.e., $m_{1}$ down and $m_{2}$ up. We get the equation for $m_{1}$:

$m_{1}g\sin \alpha _{1}=T+f_{s1}\qquad (2.1)$

and

$0\leq f_{s1}\leq \mu _{s1}N_{1}\qquad (2.2)$

We get a similar equation for $m_{2}$:

$T=m_{2}g\sin \alpha _{2}+f_{s2}\qquad (2.3)$

and

$0\leq f_{s2}\leq \mu _{s2}N_{2}\qquad (2.4)$

Add (2.1) and (2.3) and rearrange to get:

$m_{1}g\sin \alpha _{1}-m_{2}g\sin \alpha _{2}=f_{s1}+f_{s2}\qquad (2.5)$

Plugging in (2.2) and (2.4) into (2.5), we get:

$0\leq m_{1}g\sin \alpha _{1}-m_{2}g\sin \alpha _{2}\leq \mu _{s1}N_{1}+\mu _{s2}N_{2}\qquad (2.6)$

Plugging in (1.1) and (1.2) into this yields:

$0\leq m_{1}g\sin \alpha _{1}-m_{2}g\sin \alpha _{2}\leq \mu _{s1}m_{1}g\cos \alpha _{1}+\mu _{s2}m_{2}g\cos \alpha _{2}\qquad (2.6)$

Cancel $g$ from all sides to get:

$0\leq m_{1}\sin \alpha _{1}-m_{2}\sin \alpha _{2}\leq \mu _{s1}m_{1}\cos \alpha _{1}+\mu _{s2}m_{2}\cos \alpha _{2}\qquad (2.7)$

This is the necessary and sufficient condition for the system to have a tendency to slide $m_{1}$ down and $m_{2}$ up, but to not in fact slide.

Simialrly, in the other not sliding case (i.e., the system has a tendency to slide $m_{2}$ down, $m_{1}$ up), we get the following necessary and sufficient condition:

$\!0\leq m_{2}\sin \alpha _{2}-m_{1}\sin \alpha _{1}\leq \mu _{s1}m_{1}\cos \alpha _{1}+\mu _{s2}m_{2}\cos \alpha _{2}\qquad (2.8)$

Overall, for the no sliding case, we get the necessary and sufficient condition:

$\!|m_{1}\sin \alpha _{1}-m_{2}\sin \alpha _{2}|\leq \mu _{s1}m_{1}\cos \alpha _{1}+\mu _{s2}m_{2}\cos \alpha _{2}$

Note the following: in all these cases, it is not possible, using these equations, to determine the values of $T$, $f_{s1}$ and $f_{s2}$ individually.

Components along the respective inclined planes assuming sliding

We consider two cases:

• $m_{1}$ is sliding (and accelerating) down and $m_{2}$ is sliding (and accelerating) up, so the force of kinetic friction $f_{k1}$ acts up along $I_{1}$ and the force of kinetic friction acts down along $I_{2}$.
• $m_{1}$ is sliding (and accelerating) up and $m_{2}$ is sliding (and accelerating) down, so the force of kinetic friction $f_{k1}$ acts down along $I_{1}$ and the force of kinetic friction acts up along $I_{2}$.

We first consider the $m_{1}$ down, $m_{2}$ up case. Let $a$ denote the magnitude of acceleration for $m_{1}$. $a$ is also equal to the magnitude of acceleration for $m_{2}$. We get:

$\!m_{1}g\sin \alpha _{1}-f_{k1}-T=m_{1}a\qquad (3.1)$

with

$f_{k1}=\mu _{k1}N_{1}\qquad (3.2)$

and

$T-m_{2}g\sin \alpha _{2}-f_{k2}=m_{2}a\qquad (3.3)$

with

$f_{k2}=\mu _{k2}N_{2}\qquad (3.4)$

Add (3.1) and (3.3), plug in (3.2),(1.1) and (3.4),(1.2) to get:

$m_{1}g\sin \alpha _{1}-m_{2}g\sin \alpha _{2}-\mu _{k1}m_{1}g\cos \alpha _{1}-\mu _{k2}m_{2}g\cos \alpha _{2}=(m_{1}+m_{2})a\qquad (3.5)$

Rearranging, we get:

$a={\frac {m_{1}g\sin \alpha _{1}-m_{2}g\sin \alpha _{2}-\mu _{k1}m_{1}g\cos \alpha _{1}-\mu _{k2}m_{2}g\cos \alpha _{2}}{m_{1}+m_{2}}}\qquad (3.6)$

Since $a>0$ by our sign convention, this case is valid if:

$m_{1}g\sin \alpha _{1}-m_{2}g\sin \alpha _{2}>\mu _{k1}m_{1}g\cos \alpha _{1}+\mu _{k2}m_{2}g\cos \alpha _{2}\qquad (3.7)$

and in particular:

$m_{1}g\sin \alpha _{1}>m_{2}g\sin \alpha _{2}\qquad (3.8)$

The other case ($m_{2}$ down, $m_{1}$ up) occurs if the inequality sign is reversed, and we get, in that case, that:

$a={\frac {m_{2}g\sin \alpha _{2}-m_{1}g\sin \alpha _{1}-\mu _{k1}m_{1}g\cos \alpha _{1}-\mu _{k2}m_{2}g\cos \alpha _{2}}{m_{1}+m_{2}}}\qquad (3.9)$