# Homework Help: Block in a rotating cylinder

1. May 21, 2017

### fedecolo

1. The problem statement, all variables and given/known data
A cylinder with radius R spins around its axis with an angular speed ω. On its inner surface there lies a small block; the coefficient of friction between the block and the inner sur- face of the cylinder is μ. Find the values of ω for which the block does not slip (stays still with respect to the cylinder). Consider the cases where (a) the axis of the cylinder is hori- zontal; (b) the axis is inclined by angle α with respect to the horizon.

2. Relevant equations

3. The attempt at a solution
I tried to figure out which are the forces applied to the block in the non inertial frame of the cylinder. There is $m \vec{g}$, the centrifugal force $\vec{F_{c}} = m \omega^2 \vec{R}$, the normal reaction $\vec{N}$ and the frictional force $\vec{F_{s}}= \vec{N} \mu$. If the block is at rest the sum $m \vec{g} + \vec{F_{c}} + \vec{N} + \vec{F_{s}} = 0$.
Considering only case (a) of the problem, I don't know how to work with this equation in a three dimension, because my first solution where that $\omega = \sqrt{ \frac{\mu g}{R}}$ but I think it's correct in a two-dimensional geometry. Any help or reference to see?

2. May 21, 2017

### Isaac0427

What is the centrifugal force? Is it a force on its own?

3. May 21, 2017

### fedecolo

No, it's the centripetal force in the reference frame of the rotating cylinder.

4. May 21, 2017

### haruspex

That looks right. (I don't see that using the noninertial frame helps here. You get the same equation immediately.)

The 3D version is going to hurt the head.
Consider when the cylinder has rotated θ from the position where the mass is at its highest.
What are the components of the normal unit vector?

By the way, it is not true that $\vec F_s=\vec N\mu_s$. You only know that on the point of slipping $|\vec F_s|=|\vec N|\mu_s$.

5. May 21, 2017

### Isaac0427

What is your work for that? I got a different solution.

6. May 22, 2017

### fedecolo

I think that $\vec{N}$is always directed along the line that connect the mass to the centre of the cylinder, isn't it?

7. May 22, 2017

### haruspex

Right, but you do not know at what point in the rotation is the greatest risk of slipping. So you need an expression for that vector at a general position. My preference wouldbe to stick with the usual Cartesian coordinates. If you prefer to work with cylindrical here then the challenge, instead, is to represent gravity in those.

8. May 22, 2017

### zwierz

my result is different as well

9. May 22, 2017

### fedecolo

As zwierz did, I have $N=m \omega^2 R-mg sin \theta$ and $N= \frac{mg cos \theta}{\mu}$ and then $\mu \geq \frac{g cos \theta}{\omega ^2 R-g sin \theta}$ and so $\omega^2 R \geq g sin \theta$ because the denominator must be > 1 (otherwise the fraction would be > 1 and the block would slide)

10. May 22, 2017

### fedecolo

The final solution must be $\omega^2 R \geq g\sqrt{1+ \mu ^-2}$

11. May 22, 2017

### zwierz

sure

Last edited: May 22, 2017
12. May 22, 2017

### fedecolo

But I don't know how to get it

13. May 22, 2017

### zwierz

that is after I have posted the solution ;(

14. May 22, 2017

### fedecolo

The expressions $\omega^2 R \geq g(cos \theta \mu^-1+ sin \theta)$ and $\omega^2 R \geq g \sqrt{\mu^-2 +1}$ aren't they?

15. May 22, 2017

### zwierz

Find a maximal value of the function $f(\theta)=|\cos\theta|+\mu\sin\theta$

16. May 22, 2017

### fedecolo

Oh you are right, thank you!

17. May 22, 2017

### Isaac0427

The case for part (b) is really not all that different, in terms of how you go about the problem. Now, you just need to find the forces in a third direction; the direction along the length of the cylinder (often denoted by p). Step one is breaking up gravity into its r, θ and p components, in terms of m, g, θ and α. From there, it should be obvious how to continue.

18. May 24, 2017

### fedecolo

How can I scompose a three-dimension vector? (I never did something like that) Do you know any reference I can study about it?

19. May 25, 2017

### Isaac0427

Just curious, how did you get the |cosθ|+μsinθ? I got μsinθ-cosθ.

20. May 25, 2017

### zwierz

Well I have written everything I think about that in #8