What is the Relationship Between Mu and Theta in the Inclined Plane Problem?

In summary, the conversation discusses the concept of forces, specifically in the context of an object on a ramp. The forces involved are gravity, normal force, friction, and pull-force. The sum of y-components yields the normal force, and the sum of x-components yields the equation F = mg(cos(theta) + mu*sin(theta)). The conversation also mentions taking the derivative with respect to theta to find a minimum value, and how the first derivative is zero at mu = cot(theta). There is a disagreement about whether this corresponds to a minimum or maximum value.
  • #1
tricky_tick
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  • #2
i chose d

marlon
 
  • #3
tricky_tick said:
can you please explain how you arrived at that conclusion?

You have four forces. Suppose the x-acis is along the ramp and the force F for pulling the object is aligned along this x-axis. Y-axis perpendicular to x-axis.

gravity
[tex]-mg\sin(\theta)*e_{x} - mg\cos(\theta)*e_{y}[/tex]

normal force
[tex] N * e_{y} [/tex]

friction
[tex]-{\mu}N*e_{x} [/tex]

pull-force
[tex]F*e_{x} [/tex]

The sum of y-componets yield :

[tex] N = mg\cos(\theta) [/tex]

The sum of x-componets yield :

[tex] -{\mu}N -mg\sin(\theta) + F = 0 [/tex]

So we have that [tex] F = {\mu}mg\cos(\theta) + mg\sin(\theta) [/tex]

or [tex] F = mg(\mu\cos(\theta) + \sin(\theta)) [/tex]

let's take the derivative with respect to the angle theta and set this equal to 0. Thus we get :

[tex] 0 = mg(-{\mu}\sin(\theta) + \cos(\theta)) [/tex]
or
[tex] 0 =-{\mu}\sin(\theta) + \cos(\theta) [/tex]

or
[tex] \mu = \frac {\cos(\theta)}{\sin(\theta)} = \cot(\theta)[/tex]
Thus [tex]\tan(\theta) = \frac {1}{\mu}[/tex]

marlon.
 
  • #4
Answer d yields a maximum value of Work. We are looking for a minimum value.
 
  • #5
tricky_tick said:
Answer d yields a maximum value of Work. We are looking for a minimum value.

please prove your statement...

marlon
 
  • #6
marlon said:
please prove your statement...

marlon

Besides i disagree because the first derivative is zero in [tex]\mu = \cot(\theta)[/tex]

Yet if [tex]\mu > \cot(\theta)[/tex] then thefirst derivative is negative
Yet if [tex]\mu < \cot(\theta)[/tex] then the first derivative is positive

So the mu corrsponds to a minimal value here. You are mixing extrema with maxima. An extremum is a general name for both local max and min values of a function
marlon
 

What is an inclined plane?

An inclined plane is a simple machine that is a flat surface with one end higher than the other. It is commonly used to reduce the amount of force needed to move an object up or down, by spreading the work over a longer distance.

How do you calculate the mechanical advantage of an inclined plane?

The mechanical advantage of an inclined plane is the length of the ramp divided by its height. In other words, it is the ratio of the length of the incline to its height.

What is the relationship between the angle of inclination and the effort force required?

The angle of inclination is directly related to the effort force required. As the angle of inclination increases, the effort force required to move an object up the incline decreases. This is because a steeper incline provides a longer distance over which the work is distributed.

What is the difference between a frictionless inclined plane and a real inclined plane?

A frictionless inclined plane is a theoretical concept where there is no friction between the object and the surface of the incline. In reality, there is always some amount of friction present, which makes the effort force required to move an object up the incline slightly higher than the calculated value using the mechanical advantage formula.

How is the concept of an inclined plane used in everyday life?

Inclined planes are used in many everyday objects and activities. Some examples include ramps for wheelchairs and strollers, escalators, slides, and even roads on hilly terrain. They are also used in construction, such as in the construction of roads, stairs, and roofs.

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