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

1. Dec 9, 2015

### Potatochip911

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?

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:

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. Dec 10, 2015

### haruspex

Why do you think it does not make sense?

3. Dec 10, 2015

### Potatochip911

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 $\tau_{net}=0$

4. Dec 10, 2015

### haruspex

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

5. Dec 10, 2015

### Potatochip911

I suppose it would roll about the y axis

6. Dec 10, 2015

### haruspex

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