Flat Belt and Pullies with different Coefficients

[Spencer25]

Homework Statement

Please Help...I have searched for hours but cannot find an example of this question. There is a good chance it will be on an upcoming exam. Two different sized pullies with a flat belt. Tension given for both sides of the belt (I think but maybe only one side)...the angle of wrap is given, but the pullies do not have the same coefficients of friction (the smaller pulley having the higher coefficient). Which pulley will slip first? Every question I have found shows the pullies having the same coefficient so I guess the formula T1 = T2 x e^(mu x beta) does not apply if only one of the tensions is given. Does it matter which is the driver and which is the driven? I don't know where to start and any information would be greatly appreciated. I have a picture to attach if that is possible.

Relevant Equations

T1 = T2 x e^(mu x beta)

Source https://www.physicsforums.com/forums/introductory-physics-homework-help.153/post-thread

I would like to attached a picture here...

 

[Spencer25]

 

[BvU]

Hello,

For your own benefit, you want to learn to separate a problem statement from further stages such as collecting relevant equations and specifying which is which, units, given/unknown, etc etc.

Your picture identifies the right pulley as the driving one, but in the text (which is not -- should not be -- part of the problem statement), you wonder about the distinction.

The relevant capstan equation you bring in correctly, is valid for each and every pulley individually. Does that help you get started ?

Notes: the PF rules and guidelines deserve your attention: we are only allowed to help if you post an attempt at solution (and "I don't know where to start is not enough" )
In addition: no need to refer to the "post thread"source with a link.

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[Spencer25]

Yes, I am sorry about not following the proper format but i wasn't sure that this was even a valid quiestion. I don't think I am very clear at conveying what I am getting at. I understand:

"If both pulleys are not of the same diameter, then the smaller pulley is the one that determines the maximum torque and power that can be transmitted. Not only does the smaller pulley have a shorter length of contact, but it also has a smaller angle of contact than the larger pulley, so it will always be the first to slip. In cases where the pulleys are not the same diameter, the angle of contact will depend also upon the centre distance between the shafts. The greater the centre distance, the greater the angle of contact. For this reason, centre distances should not be below the recommended minimum value (sum of the pulley pitch diameters) unless there are special circumstances.

but what if the smaller pulley has a much high coefficient of friction? How does that translate to slippage...i would assume we would need to consider the area in contact or the moment using net force? Would it be maximum monment x the coeffcient...so still use Mu=Friction Force/REaction Normal...and reaction normal is just net tension...then just compare and the one with the lower frction force would slip first?

 

[BvU]

Why don't you do some calculations and show your work ?
All these 'what ifs' don't get it done !


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[Lnewqban]

You also need to consider how much torque each pulley is overcoming; no friction is needed if no input (motor) or output (resistive) torque is present.
With the diameter of the pulley, the tangential force changes for the same transferred torque.
Smaller diameters of pulleys create higher centrifugal effect and more bending stress for the belt.