An air-conditioning unit is fastened to a roof that slopes at an angle of 35° above the horizontal...

Question

The book doesn't provide an answer key for this question, so I want to know if the process I followed is ok.

1) I set a coordinate system and draw a line parallel to the horizontal, that intersects the origin of it.

2) From alternate internal angles and vertical angles, it follows

3) So

4) Now, $A_{y}$ the y component of $\overrightarrow{F}$ is $-425N$ given how the coordinates are defined, so the condition for the problem is that $F_{y}≥-425N$

5) Now

$F_{y}≥-425N$

and

$F_{y}=F\sin(235°)$

so

$F\sin(235°)≥-425N$

Now, since $\sin(235°)<0$

$F≤\frac{-425N}{\sin(235°)}$

and

$F≤519N$

Is this reasoning correct?

Thanks in advance.

Answer 1

It looks right, but the fact that the question gives the conversion to pounds suggests that it might want the answer in pounds. Also, with this sort of question, you only need to deal with the magnitude of the force, you don't need the direction. The angle between gravity and your y axis is 35 degrees, so the weight projected onto y is the weight times cos(35).

Answer 2

Here's an easier way to visualize the angle computation. The green line (representing the ground) is 35° south from negative x axis. It forms a right triangle with the force vector. So the angle of the force vector is 90° - 35° = 55° south of negative x axis. Add 180° for the rotation and you get 235° from the positive x axis.