# Finding the acceleration of a mass/pulley system without knowing the direction of the displacement

I have the following question:

I noticed that if I solve for the acceleration assuming that the acceleration is occuring to the left, I get the equation:

$\displaystyle a = \frac{m_3g-\mu_km_1gcos\theta-m_1gsin\theta}{m_1+m_2+m_3}$.

If I assume that the acceleration is to the right as in the diagram, I get the equation:

$\displaystyle a= \frac{m_1gsin\theta-\mu_km_1gcos\theta-m_3g}{m_1+m_2+m_3}$

Each one produces a different answer as I have to specify whether friction aids or opposed the tension in the rope. My question is, is there a way to know beforehand what direction the acceleration is in so that you can make the necessary assumption about the direction of friction?

• In question it is given m1 is moving down. This necessaries a to be right. – Anubhav Goel Aug 4 '16 at 13:21

your two equations are correct when the velocity is going to the left, and right respectively. If the velocity is zero then the acceleration could be anywhere in between.

This means if the accelerations have opposite signs the blocks will decelerate to a stop and then stay stopped. If they have the same sign then the friction won't hold the system in place. If the acceleration is the opposite as the velocity it will first accelerate quickly to zero velocity and then continue accelerating at the lower magnitude acceleration.

Which direction the friction will act is is dependent on the relative velosity not the acceleration.

• Oh and in the problem it states that the blocks are sliding to the right, so the second equation is correct. – Rick Jan 22 '16 at 16:39

There are three forces which are acting on the system of three masses and causing them to accelerate:

$m_3g$ always acts so as to pull the $m_2$ to the left.
$m_1g \sin \theta$ always acts so as to pull the $m_2$ to the right.
The direction of the frictional force $\mu_k m_1 g \cos \theta$ will depend on the direction of motion.

If $m_3g-m_1gsin\theta > \mu_km_1gcos\theta$ then there will be a net force to the left and the acceleration will be to the left, and if the inequality is the other way around then the acceleration will be to the right.

In the first sentence it is mentioned that mass m1 is moving down the incline. So it is a given :-)

There is a way to find the direction of friction in a given problem. Assume that there is no friction in the given problem. U get to know which side the mass m1 moves on the plane and now the friction Since it opposes relative motion between objects it must act in the direction opposite to the acceleration of the block. Hence the direction is found.

You are assuming that $a$ is positive in the direction of motion. One of the two solutions will be negative because the forces are inconsistent, the correct answer will be the positive one.