Change in acceleration with change of angle of applied force

In summary, the block will move forward when the angle of the applied force is increased by 5 degrees.
  • #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
Untitled Diagram.png

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.png

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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  • #2
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.
 
  • #3
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.
 
  • #4
The answer depends on the value of ##\mu##. For ##\mu = 0.5##, your answer is correct.
 
Last edited:
  • Informative
Likes GreenOlive
  • #5
@TSny I did include that under the image. Thanks for the confirmation
 
  • Like
Likes TSny

Related to Change in acceleration with change of angle of applied force

What is change in acceleration with change of angle of applied force?

Change in acceleration with change of angle of applied force refers to the change in the rate of change of an object's velocity when a force is applied at different angles. It describes how the direction of the force affects the acceleration of the object.

How is change in acceleration with change of angle of applied force calculated?

Change in acceleration with change of angle of applied force can be calculated using the formula a = F/m, where a is the acceleration, F is the applied force, and m is the mass of the object. This formula takes into account the angle at which the force is applied.

What factors can affect change in acceleration with change of angle of applied force?

The main factors that can affect change in acceleration with change of angle of applied force are the magnitude and direction of the applied force, the mass of the object, and the angle at which the force is applied. Other factors such as friction and air resistance may also play a role.

How does change in acceleration with change of angle of applied force relate to Newton's laws of motion?

Change in acceleration with change of angle of applied force is closely related to Newton's laws of motion. Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This means that the angle at which the force is applied can affect the acceleration of the object.

What are some real-world examples of change in acceleration with change of angle of applied force?

One example of change in acceleration with change of angle of applied force is a car turning around a curve. The force of the tires on the road changes as the car turns, affecting its acceleration. Another example is a projectile launched at an angle, where the angle of the launch affects the acceleration and trajectory of the projectile.

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