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

[Potatochip911]

Homework Statement

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?

Homework Equations

$\vec{\tau}=\vec{\mu}\times \vec{B}$
$\vec{\tau}=\vec{r}\times \vec{F}$

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:

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.

 

[haruspex]

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.

Why do you think it does not make sense?
Please show your working with regard to the torque from gravity.

 

[Potatochip911]

Why do you think it does not make sense?
Please show your working with regard to the torque from gravity.

I messed up the first cross product, it should be $\vec{u}\times\vec{B}=\hat{j}$, and for torque from gravity:

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 $\tau_{net}=0$

 

[haruspex]

I just can't see how setting the net torque equal to zero in the y plane will hold the cylinder in place.

There isn't a y plane, there's a y axis. If it were to roll, which axis would it rotate about?

 

[Potatochip911]

There isn't a y plane, there's a y axis. If it were to roll, which axis would it rotate about?

I suppose it would roll about the y axis

 

[haruspex]

I suppose it would roll about the y axis

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