Why Did My Second Approach to Finding the Block's Acceleration Fail?

[guyvsdcsniper]

Homework Statement

Find the blocks acceleration on a wedge.

Relevant Equations

F=MA

I am currently solving this problem and approached it two different ways. I have attached a ss of the picture for reference.

On my first attempt, shown on the attached image in pen, I used a component of the normal force and weight as my forces in the y direction. I carried out the work and ended up getting the correct answer according to my book.

I then tried a different approach, as shown by the image with the whiteboard, and used the actual normal force and a component of the force of gravity. When I used this approach I got stuck and did not see anyway of attaining the correct answer.

What is incorrect about the 2nd approach that is preventing me from getting the right answer?

 

[TSny]

In the whiteboard calculation, it looks like you are taking the y-direction to be perpendicular to the ground for the diagram and the constraint equation:

But when you set up the $\sum F_y$ equation, you switch to taking the y-direction to be perpendicular to the plane (i.e., along the normal force $N$):

So, there is an inconsistency in the choice of the y-direction for the whiteboard calculation. $y$ in the constraint equation is not the same as the $y$ in the $\sum F_y$ equation.

 

[guyvsdcsniper]

In the whiteboard calculation, it looks like you are taking the y-direction to be perpendicular to the ground for the diagram and the constraint equation:
View attachment 296975

But when you set up the $\sum F_y$ equation, you switch to taking the y-direction to be perpendicular to the plane (i.e., along the normal force $N$):
View attachment 296976

So, there is an inconsistency in the choice of the y-direction for the whiteboard calculation. $y$ in the constraint equation is not the same as the $y$ in the $\sum F_y$ equation.

Thank you. That makes it very clear.