# The acceleration of a ramp and and mass on the ramp

• Luca 123
In summary, the problem involves a ramp with mass M and a mass m resting on it, with no friction between them or between the ramp and the floor. The acceleration of the ramp is given by mgsinxcosx/M+(msinx)^2, where x is the angle of the ramp. For the acceleration of mass m, the equation is [(M+m)gsinx]/M+(msinx)^2 and takes into account the fact that both the ramp and the block are accelerating. To solve the problem, one needs to consider the distances the objects move and the forces acting on them. The method involves setting certain forces equal to each other and considering the conservation of energy.

## Homework Statement

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 don't 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?[/B]

## Homework Equations

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

## The Attempt at a Solution

I tried but failed[/B]

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.