# Pulley with a rope

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1. Feb 4, 2017

### DaniV

1. The problem statement, all variables and given/known data
a circular homogenic pulley (disk) with radius R and mass of M hanging on a axis that passes through his center,
the pulley rotating without friction.
we wrap the pulley with a rope with a total mass of mr and length of L and in the other side of the rope we ataching body with a mass of m* , m* is starting to go down while rotating the disk
find the angular velocity of the pullay as function of the length of the suspanded rope -x
2. Relevant equations
Ipulley=MR^2/2 - inertia torque of the pulley (in the center of mass)
Irope=mr*R^2 - inertia torque of the rope (at the beginning when wrapped)
E=0.5Iω^2 -energy to rotate the disk pulley
U=m*gh-potential energy of the mass m* at the beginning (h is not given)
3. The attempt at a solution
ive tried to do equations of preserving energy between the start point when we have only potential of the mass m* and to equalize it to the point when we have a total hanging mass of m**= m* + mr(x/L) in a height of (h-x) -potential energy plus kinetic energy of 0.5m**v^2 when v equal to ωR. this energy also going for the rotation of the pulley E=0.5Iω^2. I suceeded by finding the height of h with those equations: h=x+m*L/mr-ω^2R^2/2g-m*ω^2R^2L/2mrg.
but it doesnt helps me finding the relation between x and ω...
I cant find more equation, couldnt think of an equation that link also the inertia of the "ring" of the wrapped rope....

2. Feb 4, 2017

### haruspex

If h is not given, why introduce it? Can't you take it as zero?
By the way, I think your energy equation is not quite right.

3. Feb 4, 2017

### DaniV

how could I express potential energy without h? or the potential energy isnt relevantic at all?

4. Feb 4, 2017

### haruspex

maybe I misunderstood what your h is. I took it to be the initial length of the hanging rope. If it means the height of the pulley above the ground, its contribution to PE is constant. Just take the reference ground as the height of the centre of the pulley.