What mass must be placed on the cord to keep the pulley from rotating?

AI Thread Summary
To prevent the pulley from rotating, the net torque must equal zero, which involves balancing the forces exerted by the masses on the pulley. The calculations indicate that a mass of 2.5 kg is required to maintain this balance, factoring in the forces from a 5 kg mass on the ramp. However, there is confusion regarding the distances from the center of the pulley, suggesting that the two cables may operate on different radii. The visual representation of the pulley shows distinct inner and outer annuli, which could complicate the torque calculations. Clarification on the pulley design and the correct application of torque principles is needed for accurate results.
I_Try_Math
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Homework Statement
What hanging mass must be placed on the cord to keep the pulley from rotating (see the following figure)? The mass on the frictionless plane is 5.0 kg. The inner radius of the pulley is 20 cm and the outer radius is 30 cm.
Relevant Equations
## \tau = rFsin\theta ##
I suppose to keep the pulley from rotating the net torque has to be zero?
Let ## F_{r} ## be the force that the 5 kg mass on the ramp exerts on the pulley and ## F_{d} ## be the force exerted straight down by the other mass on the pulley.
Let ## r = 0.3 ## m be the outer radius of the pulley.
## \sum \tau_{i} = F_{r}rsin90 - F_{d}rsin90 = 0 ##
## F_{r}r - Mgr = 0 ##
## r5gsin30 - rMg = 0 ##
## \Rightarrow M = 2.5 kg ##

I'm wondering if I solved for ##F_{r} ## incorrectly? Or there's some other mistake? Any hints are appreciated.
 

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It is very unclear, but it looks to me that the two cables are at different distances from the centre of the pulley. This suggests that the two radii given refer to two different wheels side by side on the same pulley. The drawing shows a dark inner annulus and lighter grey outer annulus.
 
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