I am copying and pasting this question I asked in the physics stack exchange. The reason I am doing so is because the question was automatically closed for being too "homework-like" and not asking about concepts. To be clear, my purpose with asking this question is to disambiguate the concepts of decomposing forces into their components, i.e., decomposing vectors into components using trigonometry. I feel it is appropriate to ask here as my quandary is inherently mathematical, it just so happens that it shows up in a physical context more often than not. I am really bad at this stuff and I'm hoping for an answer that can help me to see why it was that was unable to come up with the correct trigonometric multiplicative factor. I am hoping to gain insight into perhaps a more systematic approach to these kinds of problems. I believe my work below is sufficient to get the point across, so I'll leave the addendum here. Thanks!
Problem
I am working on an exercise that asks us to determine the initial vertical and horizonal acceleration components for a massless pole that is leaning against a wall with angle $\theta$ with respect to the horizontal. The pole is connected to a pivoting block of mass$M$ at each end. I understand the process to complete the solution but upon getting stuck at a certain point I decided to refer to the solution to check to see that my equations of motion were correct. I found that one of them wasn't - the solution referred to the relevant component of force of the lower block as being $F_p\sin\theta$ where I determined that component of force to be $F_p\cos\theta$. I believe the solution is correct as the component of force referenced by the solutions manual does allow me to proceed. I just don't know how they got that component as $F_p\sin\theta$ and not $F_p\cos\theta$. I will provide some images and some relevant work below and I would appreciate some guidance is seeing where I went wrong in my thinking. Thank you!
Images
The $3$ images here from left to bottom left clockwise are:
- The force diagram from the solutions (mine was the same)
- My thinking regarding the horizontal component of $F_p$ on the lower block
- The setup
My attempt
For this problem I determined the relevant Equations of motion to be
- From Upper block vertical $2$nd law $$F_p\sin\theta - W = M\ddot{y}$$
- From lower Block horizontal $2$nd law: $$F_p\cos\theta = M\ddot{x}$$
- Constraint equation: $$\ddot{x} = -\frac{y}{x}\ddot{y} = -\tan\theta \ddot{y}$$
However, the solutions manual says equation number $2$ (lower block horizontal) should in fact be $$F_p\sin\theta = M\ddot{x}$$ However I'm having trouble seeing why the horizontal component of the $F_p$ force is multiplied by $\sin$ and not $\cos$. My diagram above shows my thinking - just simple SOHCAHTOA trig. Could it be the case that the fact that the $F_p$ force is pointed downwards (and hence in the 4th quadrant of an imaginary unit circle that has the triangle inscribed) that leads to this result? If so, any help to be able to identify and see this going forward would be greatly appreciated.