How Fast Does Your Suitcase Travel on a Baggage Carousel?

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To determine the velocity of a suitcase on a baggage carousel inclined at 36 degrees, one must analyze the forces acting on it, including static friction and gravitational forces. The equations of motion involve resolving the components of these forces, particularly the normal force and the frictional force. The maximum static friction must equal the net force acting parallel to the carousel to find the maximum speed. The discussion emphasizes the need to sum forces in both the x and y directions to solve for the centripetal force. Understanding these components and their interactions is crucial for calculating the suitcase's velocity accurately.
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A suitcase on a baggage carousel is spinning around at an unknown velocity. The carousel has an angle of 36 degrees above horizontal. The suitcase is held in place by static friction with a coefficient of 0.760. Find the velocity.

Eqn's
Fn = mg | Fsmax = µsmg | Fc = mV²/r

I get as far as this, and I am stuck as to what to do next.
I need to find Fc

ΣFx = -Fn cos54 + Fs cos36 = mV²/r

ΣFy = Fn sin 54 + Fs sin36 - mg = 0

Where do I go next? My prof tried to explain it to me but it didn't click, I am a step-by-step learner! Thanks all.
 
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I presume you were given the radius of this carousel as well? There is a lot of resolving components summing forces. There should be a normal component from the weight and the centrifugal force and a component parallel to the carousel from bot those forces. To find the max speed the friction must be equal to the parallel forces. The weight and the centrifugal force will act in the opposite direction however.
 

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