Classic mechanics problem:two masses with a wedge

[asdff529]

Homework Statement

Given that two masses are connected by a light string over a pulley.They are initial at rest.Suppose all the surfaces are smooth.Find the resultant acceleration of the system

Homework Equations

F=ma
F_pseudo=-ma

The Attempt at a Solution

Suppose I let the resultant a be A,and the acceleration of each mass along the wedge be a
First for mass m
mgsinx+mAcosx-T=ma
N-mgcosx+mAsinx=0
for mass m'
T-m'gsinx'+m'Acosx'=m'a
N'-m'gcosx'-m'Asinx'=0
But then i get stuck because there are 5 unknown but there are only 4 equations
Did I miss anything?

 

[Orodruin]

First off, it is not completely clear what you mean by "resultant", are you implying that it is the tension+gravity? In that case you are essentially trying to take them into account twice. Second, you do not need to consider the force equations orthogonal to the directions of motion as the normal force will settle this to give no direction. You will then have two equations (one from the free body diagram for each block) and two unknowns (the tension in the string and the acceleration). What other unknowns are you thinking of? The angles x and x'? Those are parameters of the problem and your final solution will depend on them. (However, note that there is a simple relationship between them due to the geometry of the problem.)

 

[asdff529]

I think what the resultant acceleration means is that A,which is acceleration of the whole system moving left or right
I still get 3 unknowns,a,A and T,in two equations

 

[Orodruin]

You are saying that the wedge is also allowed to move? This is not clear to me from the problem statement and of course makes the problem significantly harder. In that case you will have to consider several additional things. Would the "resultant acceleration" be the acceleration of the wedge or the acceleration of the centre of mass of the entire system?

 

[HalfLostAndSearching]

Your solution so far looks correct. You have correctly identified the forces on each mass and set up the equations of motion for each mass. However, you have not taken into account the fact that the two masses are connected by a light string over a pulley. This means that the acceleration of one mass will be equal and opposite to the acceleration of the other mass, as they are connected by the string and must move together.

Therefore, you can add the following equation to your system of equations:

a = -a'

This will give you a total of 5 equations and 5 unknowns, allowing you to solve for the acceleration of the system.