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

In summary, to keep the pulley from rotating, the mass placed on the cord must create a torque that balances the torque produced by any other forces acting on the pulley. This involves calculating the gravitational force acting on the mass and ensuring it is sufficient to counteract the forces causing the pulley to move. The exact mass required can be determined using the principles of equilibrium and the relationship between torque, force, and distance from the pivot point.
  • #1
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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FAQ: What mass must be placed on the cord to keep the pulley from rotating?

What is the role of mass in keeping the pulley from rotating?

The mass placed on the cord creates a gravitational force that exerts tension in the cord. This tension must be balanced against any opposing forces, such as friction or the weight of the pulley system itself, to prevent rotation.

How do I calculate the mass needed to prevent the pulley from rotating?

To calculate the mass, you can use the formula for torque. The mass (m) must be such that the torque produced by the mass (m * g * r) is equal to the opposing torque due to the pulley system. Here, g is the acceleration due to gravity, and r is the radius of the pulley.

What factors affect the mass required to keep the pulley stationary?

Several factors affect the required mass, including the radius of the pulley, the friction in the pulley system, the weight of the pulley itself, and any additional loads connected to the pulley. All these factors must be considered to ensure the correct mass is applied.

Can the pulley still rotate if the mass is not sufficient?

Yes, if the mass is insufficient to counteract the forces acting on the pulley, it will begin to rotate. The rotation will continue until a balance of forces is achieved or until the mass is increased to the required level.

How does the angle of the cord affect the required mass?

The angle of the cord can change the effective tension in the cord. If the cord is at an angle, the vertical component of the tension must be considered, which may require a different mass to achieve the same effect of preventing rotation compared to a vertical setup.

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