What Is Theta Critical for an Object on an Inclined Plane?

[pyrojelli]

We have Uk and/or Us and angle of inclination (theta)


Are these eq-ns relevant? sin(theta critical)=Us(tan(theta critical))=Us
and sin(theta static)/cos(theta static)=Uk(tan(theta static)=Uk



How do I even interpret these eq-ns? Exam is tomorrow and I need to know how to find the angle of inclination that allows an object to start sliding (theta critical) and the angle of inclination so that the object will slide w/o accelaration. This is for an object on an inclined plane w/ friction.

 

[learningphysics]

I'm confused by the equations you wrote... can you write them exactly as they are?

The moment when sliding occurs is when the static frictional force becomes $$\mu_s*F_n$$.

Take the equation perpendicular to the plane... $$F_n - mgcos(\theta) = 0$$, so $$F_n = mgcos(\theta)$$

The equation parallel to the plane is: $$mgsin(\theta) - f = 0$$

so this is while the block is not sliding...

ie: $$f = mgsin(\theta)$$ (1)

so this equation is always true while the block is not sliding... you will notice that as theta becomes larger (the incline becomes steeper)... f becomes larger... this is all while the block is still not sliding... but there is a limit to how long this can go on... the limit occurs when f becomes $$\mu_s*F_n = \mu_s*mgcos(\theta)$$.

so to find the angle at which this limit occurs substitute $$f = \mu_s*mgcos(\theta)$$ into (1)

so you get:

$$\mu_s*mgcos(\theta) = mgsin(\theta)$$

$$\mu_s = tan(\theta)$$

 

[m2003]

To find theta critical, you can use the equation sin(theta critical)=Us(tan(theta critical))=Us, where Us is the coefficient of static friction. This equation represents the point at which the object just begins to slide down the inclined plane. To find the angle of inclination where the object will slide without acceleration, you can use the equation sin(theta static)/cos(theta static)=Uk(tan(theta static)=Uk, where Uk is the coefficient of kinetic friction. This equation represents the point at which the object will slide down the inclined plane at a constant velocity. These equations are relevant because they relate the angle of inclination to the coefficients of friction, which are important factors in determining the motion of an object on an inclined plane with friction.

To interpret these equations, you can think of them as representing the balance between the force of gravity pulling the object down the inclined plane and the frictional force trying to keep the object from sliding. As the angle of inclination increases, the force of gravity pulling the object down the plane also increases, until it reaches a point where it overcomes the frictional force and the object begins to slide. The critical angle represents this point of balance, and any angle beyond it will cause the object to slide.

For the second equation, the coefficient of kinetic friction is used because once the object is in motion, the force of friction changes and becomes lower than the force of static friction. This allows the object to slide down the inclined plane at a constant velocity without any acceleration. The angle at which this occurs is represented by the equation and can be found by solving for theta static.

In summary, to find theta critical and theta static, you can use the equations provided and plug in the given values for the coefficients of friction. It is important to understand the concept behind these equations and how they relate to the motion of an object on an inclined plane with friction. Good luck on your exam!