Change in acceleration with change of angle of applied force

[GreenOlive]

Homework Statement

How will the acceleration of the block change if the angle of the applied force is increase by 5 degrees? Write Increase, Decrease or Stay the same.

Relevant Equations

Newton 2, Fnet = ma
friction = N * μ

Let $μ_k$ = 0.5
$F_a$ = 10 Newtons
$\theta$ is the angel of the Applied force.
How will the acceleration of the block change if the angle of the applied force is increase by $5^o$? Write Increase, Decrease or Stay the same.

Recently we were discussing a question similar to this in class, I believe the acceleration should INCREASE however the memorandum for the question said it would DECREASE. I would like to just confirm my maths is correct (This is not the first time I have seen such a question).

This is my general solution.Let right be positive.

$$F_{net} = ma$$
$$F_{net} = F_{a,x} - f$$
$$ma = F_{a,x} - (N \times μ_k)$$
$$ma = F_a \times cos\theta - ((F_g-F_{a,y}) \times μ_k) $$
$$ma = F_a \times cos\theta - ((F_g-(F_a\times sin\theta)) \times μ_k) $$

As $\theta$ is the only variable changing let's isolate the expressions it is part of.

$$ma = F_a \times cos\theta + F_a\times sin\theta \times μ_k - F_g \times μ_k $$
$$ma = F_a (cos\theta + sin\theta \times μ_k) - F_g \times μ_k $$

lets graph $(cos\theta + sin\theta \times μ_k)$

Desmos graph

Looking at the graph we can see $(cos\theta + sin\theta \times μ_k)$increases from 0 until its maximum at 28.955 degrees from which it starts to decrease.
And as $$a = \frac{F_a (cos\theta + sin\theta \times μ_k) - F_g \times μ_k}m $$

Therefore by increasing the angle of the applied force by 5 degrees the acceleration will INCREASE.
By looking closer at the graph you could see that $\theta$ would need to be greater than 48.76 degrees for the acceleration to Descrease(or less than 5).

Who is Incorrect, the Memo or I?

 

[Halc]

Your graph, if proportional to acceleration, still shows positive acceleration when θ is > 90 degrees, which seems to imply that the block will move forward when you're applying negative force to it. That intuitively suggests something amiss.
It being positive for a while with a negative θ is OK. A force pulling a bit down can still move the block until the added friction eventually prevents any motion. So the left side of the graph seems ok.

 

[GreenOlive]

Can we create a limit: -90 < $\theta$ < 90. Angels not in this range would imply a negative applied force and then positive friction(if right is still positive) so you would have to change the equation.

 

[TSny]

The answer depends on the value of $\mu$. For $\mu = 0.5$, your answer is correct.

 

[GreenOlive]

@TSny I did include that under the image. Thanks for the confirmation