# How do I find Theta critical

1. Oct 8, 2007

### 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.

2. Oct 8, 2007

### 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:

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

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