A block on a wedge which itself sliding

[center o bass]

Homework Statement

A small block of mass m is sliding along the inclined slope of a wedge of mass M.
The block slides without friction on the wedge and the wedge slides without friction on the floor. The slope on the wedge has an inclination $$\alpha$$ with the floor.

Homework Equations

Newtons etc... gravity, normal force.

The Attempt at a Solution

I reason as follows: Gravity pulls the upper box down along the wedge with a force $$mg \sin \alpha$$
but it also has a component normal to the wegde being $$mg \cos \alpha$$.
Now this has to equal the normal force from the wedge on the box, so $$N =mg cos \alpha$$, but this is in an action-reaction par with the weight of the box acting on the wedge so the wedge feels a force in the negative direction along the plane of the floor of
$$mg \sin \alpha \cos \alpha$$ indicating that it would get an acelleration of $$a_M = \frac{mg}{M} \sin \alpha \cos \alpha$$ while the little box would get an acceleration of $$a_m = g\cos\alpha$$.

This is what I would naivly do, but after some time i realize that the box on top is in an
accelerated frame so I can't blindly apply Newtons laws here and therefore the reasoning fails. How would I then go about analyzing this problem?

 

[gneill]

Perhaps try looking at it from the center of momentum frame? If the block and wedge begin from rest, then the center of mass should stay put in the x direction, and the momenta of the block and wedge should be equal and opposite (summing to zero).

Then perhaps you can finesse their relative motions with conservation of energy to come up with something.

 

[center o bass]

That's a good idea and I will certainly try it. The only thing is that this problem was introduced in a book in an earlier chapter before encountering any conservation of anything. The only thing that has been introduced is Newtons laws, constrained motion etc. so I suspect there is a way to do this just using Newtons laws in a straight forward way...

But then again, they do break down in an accelerated frame...

 

[SammyS]

In their textbook: Elements of Newtonian Mechanics, 3rd Ed., J. M. Knudsen and P. G. Hjorth show 2 solutions to this problem.

In one solution, they use conservation laws: for both momentum and energy.

Their other solution uses Newton's equations of motion. They do this using a non-inertial reference frame fixed to the wedge, and use a fictitious force to compensate for using a non-inertial reference frame.

You can find this textbook on Google books. The solution begins on page 209.

 

[wsender]

Sammy, that page isn't available on Google Books. It looks like it might stop around page 74 unfortunately. Any other leads on where I might be able to find it?