The acceleration of a ramp and and mass on the ramp

1. Feb 4, 2015

Luca 123

1. The problem statement, all variables and given/known data
A ramp with mass M rests on a frictionless floor, and another mass m rests on the ramp itself. There is no friction between the ramp and the mass. Find the acceleration of ramp and mass m. The ans are given but I dont understand them. For the acceleration of the ramp, why is there (msinx)^2 in the mass component. For the acceleration of mass m, how does mass M factor into the eqn? Can someone please show me how to get the ans?

2. Relevant equations
Ans given are
Acceleration of ramp=mgsinxcosx/M+(msinx)2
Acceleration of mass m=[(M+m)gsinx]/M+(msinx)^2

3. The attempt at a solution
I tried but failed

2. Feb 4, 2015

DEvens

Ah! The infamous Flying Wedge! The first time I saw this one was on a prize exam I took when I was in high school.

https://uwaterloo.ca/sir-isaac-newton-exam/

You have some things to think about. If the ramp moves to the right by distance D, then how far to the left must the mass m move? Remember that everything is frictionless, and remember what you have to conserve. When the mass moves this distance, call it d, then how far vertically down has it moved? Remember that the ramp is moving, so the horizontal location on the ramp has changed by more than d. When you have those then start thinking about free body diagrams, and work out all the forces involved.

For completeness you should think about such things as: Can the ramp ever pop "out from under" the mass? That is, can the mass ever lose contact with the ramp?

For your ego: In grade 12 when I first took this exam, I did not manage to solve this question. Of course, it was the 15th question on the exam, and the exam was 2 hours.

3. Feb 4, 2015

Luca 123

I will try it using the method. But before I posted this, I attempted the qns but it is not correct. Would you mind telling me what is wrong with my method?
For acceleration of the ramp, I used Ma(x)=(Normal force due to mass m)sinx to find a(x).This seems correct but I cannot get (msinx)^2
Thanks!

4. Feb 4, 2015

DEvens

As I said, you need to resolve all of the forces. There has to be a net force on each object because there is an acceleration. But you will be able to set certain forces equal to certain other forces. And you will be able to set the sum of certain forces equal to the sum of other forces. That's why I mentioned working out the relative values of d and D, and so the relative sizes of the horizontal acceleration of the block and the ramp.

5. Feb 5, 2015

Luca 123

Thanks, I solved the problem:)