Mechanics problem in a non-inertial system

[RoloJosh16]

Homework Statement

This is problem $26$ from Jaan kalda mechanics booklet of study guides for ipho.

A wedge has been made out of a very light and slippery material. Its upper surface consists of two slopes making an angle $α$ with the horizontal and inclined towards one another. The block is situated on a horizontal plane; a ball with mass m lies at the bottom of the hole on its upper surface. Another ball with mass M is placed higher than the first ball and the system is released. On what condition will the small ball with mass m start slipping upwards along the slope? Friction can be neglected.

Relevant Equations

Non inertial mechanics

Well, the problem suggests to analyse it in a system attached to the wedge. If mass m ball is slipping upwards then it would have an acceleration along the wedge pointing upwards (in the suggested system). The forces acting on that ball would be the weight, a normal and a force due to the acceleration of the system. If the acceleration of the wedge is $a=g$ tan$α$ then the mass m ball is at rest on that system, if the acceleration is bigger it slips upwards. But I do not know how to relate that with the other ball. That is all I have. Thanks in advance.

 

[Lnewqban]

How is that the wedge has an acceleration?

 

[RoloJosh16]

The wedge moves freely on the surface. The normal forces acting on the wedge accelerates it.

 

[TSny]

The normal forces acting on the wedge accelerates it.

Yes. Can you work out expressions for the normal forces?

 

[RoloJosh16]

If the acceleration of the wedge is $a> g$ tan $α$ (that is the condition for the mass m to slip upwards I think) then $N_m= ma$ sen $α + mg$ cos $α$ and $N_M = Mg$ cos $α - Ma$ sen $α$.

 

[TSny]

If the acceleration of the wedge is $a> g$ tan $α$ (that is the condition for the mass m to slip upwards I think) then $N_m= ma$ sen $α + mg$ cos $α$ and $N_M = Mg$ cos $α - Ma$ sen $α$.

That looks correct to me. What next?

 

[RoloJosh16]

I´ d like to relate those forces to the acceleration of the wedge, but I do not know its mass.

My statement of the minimum acceleration is correct?

 

[TSny]

I think you can interpret "very light material" to mean that you can take the wedge to be massless.

 

[RoloJosh16]

Ahhh thank you I already found the answer with that. When thinking about this problem I thought that there had to happen some time since the ball is placed in top of the wedge until the wedge acquires the needed acceleration but would it not be strange that acquires it immediately?

 

[TSny]

Ahhh thank you I already found the answer with that.

Good.

When thinking about this problem I thought that there had to happen some time since the ball is placed in top of the wedge until the wedge acquires the needed acceleration but would it not be strange that acquires it immediately?

I think it's OK that the acceleration is "immediate". If you place a block on a fixed, frictionless incline, the block immediately acquires its acceleration down the slope.

 

[RoloJosh16]

Though in this case it is like a combined acceleration. The acceleration of the masses depend on the acceleration of the wedge and both would accelerate immediately to the needed value.

 

[TSny]

Though in this case it is like a combined acceleration. The acceleration of the masses depend on the acceleration of the wedge and both would accelerate immediately to the needed value.

Yes. That sounds right to me.

 

[RoloJosh16]

Thanks, you have helped me a lot :)

 

[TSny]

Thanks, you have helped me a lot :)

You're welcome. I didn't do much. You did all the work!