How Fast Does Your Suitcase Travel on a Baggage Carousel?

[rcrx]

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.

 

[Kurdt]

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.

 

[baba89]

To find the velocity, we need to use the equation ΣFx = mV²/r, where ΣFx represents the sum of all the forces acting in the horizontal direction (x-axis). In this case, we have two forces acting in the horizontal direction: Fn cos54 and Fs cos36.

First, we need to find the value of Fn, which represents the normal force acting on the suitcase. We can use the equation Fn = mg, where m is the mass of the suitcase and g is the acceleration due to gravity (9.8 m/s²). Since we do not know the mass of the suitcase, we can assign a variable, let's say m.

Next, we need to find the value of Fs, which represents the maximum static friction force. We can use the equation Fsmax = µsmg, where µs is the coefficient of static friction and mg is the weight of the suitcase. Again, we do not know the weight of the suitcase, so we can assign a variable, let's say w.

Now, we can substitute these values into the equation ΣFx = mV²/r:

mV²/r = -mg cos54 + µsmg cos36

Next, we can simplify this equation by factoring out the common term m:

m(V²/r) = m(-g cos54 + µsg cos36)

Now, we can cancel out the m's on both sides of the equation, leaving us with:

V²/r = -g cos54 + µsg cos36

Finally, we can rearrange this equation to solve for V:

V = √[(r(-g cos54 + µsg cos36)]

Now we have our final equation for finding the velocity of the suitcase on the baggage carousel. We just need to plug in the values for the given variables (r, g, µs), and solve for V. Keep in mind that r represents the radius of the carousel, which you may need to measure or look up.

I hope this helps and makes sense to you! Let me know if you have any further questions.