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Find i such that ##t_{net}=0## for a loop/(Cross prod. help)

  1. Dec 9, 2015 #1
    1. The problem statement, all variables and given/known data
    A wood cylinder of mass ##m## and length ##L## with ##N## turns of wire wrapped around it longitudinally, so that the plane of the wire coil contains the long central axis of the cylinder. The cylinder is released on a plane inclined at an angle ##\theta## to the horizontal, with the plane of the coil parallel to the incline plane. If there is a vertical uniform magnetic field ##B##, what is the least current ##i## through the coil that keeps the cylinder from rolling down the plane?

    woodcylinder.jpg

    2. Relevant equations
    ##\vec{\tau}=\vec{\mu}\times \vec{B}##
    ##\vec{\tau}=\vec{r}\times \vec{F}##
    3. The attempt at a solution

    Since the cylinder isn't rolling down the plane we have that ##\tau_{net}=0##, the torque from coil on the cylinder is ##\tau_{coil}=\vec{\mu}\times\vec{B}=NiAB\sin\theta## and the torque from the cylinder's weight is ##\tau_{w}=\vec{r}\times\vec{F}=rmg\sin\theta##, now setting these equal to each other solves the problem but I'm confused about the direction of these forces from the right hand rule/cross product. I'll draw a picture to show you: ngqdcz.png

    Which makes no sense and I also can't get the torque from the cylinders weight to go in the correct direction, if someone could show/tell me what's wrong with my vector diagram that would be great.
     
  2. jcsd
  3. Dec 10, 2015 #2

    haruspex

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    Why do you think it does not make sense?
    Please show your working with regard to the torque from gravity.
     
  4. Dec 10, 2015 #3
    I messed up the first cross product, it should be ##\vec{u}\times\vec{B}=\hat{j}##, and for torque from gravity: 2hwmnpl.png
    I will explain why this doesn't make sense to me. The question is asking what the smallest value of current is that will stop the cylinder from rolling down the plane. I just can't see how setting the net torque equal to zero in the y plane will hold the cylinder in place.

    I would think that for the cylinder we would need ##F_{netx}=0## and ##F_{nety}=0## not ##\tau_{net}=0##
     
  5. Dec 10, 2015 #4

    haruspex

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    There isn't a y plane, there's a y axis. If it were to roll, which axis would it rotate about?
     
  6. Dec 10, 2015 #5
    I suppose it would roll about the y axis
     
  7. Dec 10, 2015 #6

    haruspex

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    Right. So if the torques about the y axis balance, it won't roll.
     
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