# Motion of cylinder

1. Jun 29, 2016

### Hamal_Arietis

1. The problem statement, all variables and given/known data
At the height h, from the top of inclined plane forming an angle α with the horizontal, a disk slips down which has radius R, thickness d, mass m and resistivity ρ. On inclined plane, an area S is very small square that has a magnetic field B. The magnetic field B is directed horizontally, intensity is constant, at distance R from surface of inclined plane (figure)
Find the velocity of center of mass at time when the disk leaves the magnetic area. Suppose that the disk rolls without slide
2. Relevant equations
Newton's law: F=ma $$\vec{E}=\vec{v}×\vec{B}$$
Current density $$\vec{j}=\frac{I(\vec{l}\vec{n})}{S}\vec{n}$$ with \vec{l} is vector unit alongs electric current
Differential form of Ohm's law: $$\vec{E}=ρ\vec{j}$$
3. The attempt at a solution
Force is appeared by magnetic field, along x-axis and y-axis are relational. Althought I can write Newton's law but I cant find the velocity

Last edited: Jun 29, 2016
2. Jun 29, 2016

### Amrator

This looks like an awesome problem.

Since the ball rolls without slipping, what can you say about its total kinetic energy before it goes through the magnetic field, as it goes through the magnetic field, and after it goes through the magnetic field?

Last edited: Jun 29, 2016
3. Jun 29, 2016

### Hamal_Arietis

I thinked that using kinetic energy theorem. ΔW=ΣA
But I cant find the work of the x-axis is create by Fx (electromagnetism force). Because while disk rolls, the area S moves too. And size of velocity of area S isnt constant. It is relational with the velocity of disk. So equations differential is very difficult.

4. Jun 29, 2016

### Delta²

It seems hard to model the interaction between the magnetic field and the rolling cylinder in terms of forces.

Instead use conservation of energy: I would first use the equations you present in OP to determine the current density $J=\frac{E}{\rho}$ and the current $I=JS=\frac{Bv}{\rho}S$ that is induced, then calculate properly the power dissipated as $P=I^2R$ where $R=\rho\frac{\sqrt{S}}{d\sqrt{S}}$ and assume that this power comes from reducing the kinetic energy (translational+rotational) of the cylinder.

5. Jun 29, 2016

### Hamal_Arietis

We must find the loss enery by heat. $Q=I^2Rt$ , not P. So we must find t, it seems very hard. Your I isnt correct. $I=\frac{B\sqrt{S}d}{\rho}v$. And this v is velocity of area S in laboratory reference frame. and it changes when the time changes. Thanks your good idea. After I use force to find work. There are the same result. And it cant find the work :(

Last edited: Jun 30, 2016
6. Jun 30, 2016

### Delta²

Write down the expression for the total mechanical energy $E=E(h(t),v(t))$ involving the term $mgh(t)$ and the translational and rotational kinetic energy terms that depend on $v(t)$ and the radius of the cylinder R (we assume the cylinder keeps rolling without slipping through the whole phase).

Then we ll have the differential equation $I^2R=-\frac{dE}{dt}$.

Last edited: Jun 30, 2016
7. Jun 30, 2016

### Hamal_Arietis

the I have v(t) but that is a velocity of area S. $\vec{v}=-\vec{\omega}X\vec{r}+ \vec{v'}$ Moever dE= f(h(t);v '(t)) . dE/dt => will appear a(t) . The differential equation seem hard

8. Jun 30, 2016

### Delta²

Ah yes I didn't notice that the velocity of the area S will vary also from $\omega(t)R$ (as the area S enters the cylinder) to 0 (as the area passes through the center of the disk) and then to $-\omega(t)R$ when the area leaves the cylinder.

Either we 'll do an approximation and assume that it remains "constant" $\omega(t)R$ or otherwise the differential equation will be very very hard.

9. Jun 30, 2016

### haruspex

This is not an area I know anything about, so I may be completely off here. When a conducting plate passes across a magnetic field for some distance does the total work done depend on the speed of the plate? If not, you just need to compute the path length.
I tried searching for equations on this, but could only find an equation for alternating fields.

10. Jun 30, 2016

### Delta²

Magnetic field cannot do work on matter, and there are some threads in physics forums about this. However magnetic field creates an electric field which does work on matter and the electric field generated in this case depends on the magnetic field and on the velocity of the surface S $\vec{v}$ , it is $\vec{E}=\vec{v}\times \vec{B}$.

This problem involves a few approximations
1) We neglect the displacement current term $dE/dt$ which will create an additional magnetic field according to Maxwell-Ampere Law. But ok this is the most usual approximation in these kind of problems

2) We neglect the self induction term. The current density $\vec{J}=\sigma\vec{E}$ that will be generated will create additional magnetic field again according to Maxwell-Ampere Law.

3) In the velocity of the surface S, $v$, it is $v=\vec{\omega(t)}\times\vec{R(t)}+\vec{v'(t)}$ where $v'(t)$ the translational velocity of the cylinder which is $\omega(t)R$ since the cylinder is rolling without slipping. Seems we must neglect the term $\vec{\omega(t)}\times\vec{R(t)}$ otherwise the differential equation will have additional complexity cause because $R(t)=R-\int v'(t)dt$. This R(t) varies from +R to -R thus has an average value of 0, maybe is a good idea to neglect it..

11. Jun 30, 2016

### Hamal_Arietis

today, that is what I can do

Is forum in Australia or America ?

Last edited by a moderator: May 8, 2017
12. Jun 30, 2016

### Hamal_Arietis

E is just a virtual field, not real. I argree with you that magnetic field cannot do work on matter. And work is done by visual E equals the loss enery by heat

13. Jun 30, 2016

### Delta²

Ok I see you preferred to work with forces, I ll check it and reply tomorrow , I got to go now.

14. Jun 30, 2016

### Nidum

Is this problem based on Faraday's disk ?

15. Jun 30, 2016

### haruspex

Ok, so the retarding force is some constant k times the normal velocity. That's what I needed to know.
If x is the displacement of the disc down the slope, with x=-r representing the point at which the leading edge reaches the field, I get
$I\ddot x=A-k\dot x(x^2+r^2)$, for some constants A and I.
We can integrate that directly, or convert to the time-independent form using $\dot v= v dv/dx$, but neither produces anything I can see how to solve.

Last edited by a moderator: May 8, 2017
16. Jun 30, 2016

### Delta²

I sense a problem with your working

1) If the electric field is virtual, then the current density (which for sure is real) is solely due to the Lorentz force $q(v\times B)$. BUT then the Lorentz force can NOT be responsible for two things simultaneously: for generating the current density, and for generating the laplace force $BI\sqrt{S}$ on the current density.

2) If the electric field is real, then where does it come from since there is no term $\frac{\partial B}{\partial t}$. I suppose it comes from separation of charges , with a mechanism similar that happens in a conductor that has velocity v inside a magnetic field B and has real voltage $E=Bvl$ in its two edges. I believe the top and left side of the square area S will accumulate positive charges, while the bottom and right side of the area S will accumulate negative charges. There will be current due to that voltage difference that will span across the rest portion of the cylinder (the area of the circle disk minus the area S). But I believe the current will be quite different from what you calculate in your workings.

17. Jul 1, 2016

### Delta²

I have to ask this, is this problem found in a book, or is it a problem you made yourself Hamal_Arietis?

Cause it just seems hard to model current density (at least the way I view it, with the area S acting as EMF source and the current density in the whole cylinder) hard to model the differential equation, and without exact solution for the differential equation...

18. Jul 1, 2016

### Hamal_Arietis

It is the exam for gifted student in my country. And the solutions in all exam has been keeped secret since 2010. Yesterday I tried to find $-\frac{dE}{dt}$ but that appeared a(t) and I cant contrinous.
I think we need to stop this problem.

19. Jul 1, 2016

### Delta²

Ok , when you learn the solution please post here, I am almost sure what we missing is some sort of approximation/simplification that simplifies the problem and unlocks the solution.

20. Jul 2, 2016

### haruspex

Do you agree with Hamal's expression for the retarding force in post #11: F=B2dSv/ρ, where F acts oppositely to the relative velocity v?
If so, you can obtain the differential equation in my post #15. The constants k, A, I are all known. It may be necessary to make an approximation to solve it, though.