 #1
GreenOlive
 3
 1
 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.
$$F_{net} = ma$$
$$F_{net} = F_{a,x}  f$$
$$ma = F_{a,x}  (N \times μ_k)$$
$$ma = F_a \times cos\theta  ((F_gF_{a,y}) \times μ_k) $$
$$ma = F_a \times cos\theta  ((F_g(F_a\times sin\theta)) \times μ_k) $$
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?
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