Two blocks on a third block which is split into two ramps

[burian]

Homework Statement

Two cubes of masses $m_1$ and $m_2$ be on two frictionless slopes of block A which rests on a horizontal table. The cubes are connected by string which passes over a pulley as shown in the figure. To what horizontal acceleration f should the whole system ( that is blocks and cubes) be subjected so that the cubes do not slide down the planes. What is the tension of the string in this situation?

Relevant Equations

[tex] N_{m_1} \sin \alpha - T \cos \alpha = m_1 f \\
T \sin \alpha + N_{m_1} \cos \alpha = m_1 g [\tex]

So, for this question first I did a free body diagram in the perpendicular x-y axes, and, I got some equations with the normals, but the answer is independent of normal forces. So, I'm not sure how to eliminate the two normals. Further I find it quite weird that big block-A doesn't come into the picture, why is this? I have attached pics of my work

Free body analysis + Final equation I got boxed in blue+ reference picture

Thanks for reading!

 

[Lnewqban]

Homework Statement:: Two cubes of masses $m_1$ and $m_2$ be on two frictionless slopes of block A which rests on a horizontal table.

... Further I find it quite weird that big block-A doesn't come into the picture, why is this?...

There is a net force making those two small blocks naturaly slide down that contour.
Simultaneously accelerating those masses (plus big block A at same rate) horizontally should generate an additional net force that cancells the other one.
Big block A has nothing to do with all that, except going for the ride and providing that sliding countour for our smaler blocks.

 

[haruspex]

I do not understand your FBD. What is that arrow pointing up the slope? The letter written next to it looks like a distorted lower case y.

 

[burian]

I do not understand your FBD. What is that arrow pointing up the slope? The letter written next to it looks like a distorted lower case y.

Oh that's my tension

 

[burian]

There is a net force making those two small blocks naturaly slide down that contour.
Simultaneously accelerating those masses (plus big block A at same rate) horizontally should generate an additional net force that cancells the other one.
Big block A has nothing to do with all that, except going for the ride and providing that sliding countour for our smaler blocks.

See this is the part I don't get, why does simultaneously accelerating generate a net force which cancels the force on small blocks?

 

[haruspex]

See this is the part I don't get, why does simultaneously accelerating generate a net force which cancels the force on small blocks?

There are two ways of expressing this.
We can use the reference frame of the large block, which means we have a virtual inertial force on each small block going the other way.
Or we can say that in order to produce accelerations that are equal and horizontal on the small blocks we need an acceleration of the large block so as to increase the two normal forces.

 

[Lnewqban]

See this is the part I don't get, why does simultaneously accelerating generate a net force which cancels the force on small blocks?

@haruspex has explained it better than I could.

Gravity → acceleration of small blocks → $\vec F_1=(m_1+m_2)~\vec a_1$

Then, we have the question:
To what horizontal acceleration f should the whole system (that is blocks and cubes) be subjected so that the cubes do not slide down the planes?

New horizontal acceleration → Induces new additional force (hopefully of same magnitude and opposite direction, so it cancels the first one) → $\vec F_2=(m_1+m_2)~\vec a_2$

 

[burian]

There are two ways of expressing this.
We can use the reference frame of the large block, which means we have a virtual inertial force on each small block going the other way.
Or we can say that in order to produce accelerations that are equal and horizontal on the small blocks we need an acceleration of the large block so as to increase the two normal forces.

What is a 'virtual inertial force'? And, how exactly does acceleration increase the normal force?

 

[burian]

@haruspex has explained it better than I could.

Gravity → acceleration of small blocks → $\vec F_1=(m_1+m_2)~\vec a_1$

Then, we have the question:
To what horizontal acceleration f should the whole system (that is blocks and cubes) be subjected so that the cubes do not slide down the planes?

New horizontal acceleration → Induces new additional force (hopefully of same magnitude and opposite direction, so it cancels the first one) → $\vec F_2=(m_1+m_2)~\vec a_2$

For this part, how do I finish the problem with equations I have ..or are they wrong? I get an equation relating the pseudo force stuff and normal force which big block applies on the two little blocks