## Posts Tagged ‘friction’

### Shaking The Right Way

April 30, 2010

The other day I was working at a job – a real one, like grownups have. I used to think this job was boring. But things aren’t boring; they’re just what they are. People can be bored, but that’s their business. Don’t blame the thing!

I work at a place where they make widgets. One of the pieces for the widgets is a little, flat, plastic rod. There’s an assembly line with a robot that puts the rods in the partially-assembled widgets. But before that, the rods have to be fed into the robot one by one. The rods come from the rod factory in a big pile, though, and the robot can’t reach in a pick one out. We need a way to take the rods from a jumbled mess and feed them one-at-a-time into the robot.

To do that we dump the plastic rods in a bowl about a meter across. There’s a ramp winding up along the edge of it. We vibrate the bowl, and the rods march up the ramp and into the rod-eating widget-making robot. The bowl looks a little like this:

There was a library nearby when I was a kid that had a ramp like this in it. I always wanted to go to that library. I never read any of its books.

This is supposed to be a cut-away view of the side of the bowl. The ramp is on the inside of the bowl, angled up slightly to keep the rods from falling off.

When we turn it on, the bowl starts vibrating fast enough that it’s just a blur (60 Hz seems plausible). And the rods walk their way up the ramp at a very even pace. The rods will go up the ramp whether there are one thousand or just one of them in the bowl (although if there are lots of them some will get pushed off the ramp, fall to the bottom, and start over).

This is really crazy! How do the rods go up the ramp, not down? Things are supposed to go down, generally speaking. It’s pretty freaky to watch a single rod climb right up this long winding ramp as the bowl vibrates. The bowl doesn’t spin. It doesn’t visibly tilt. It works for different sizes of rods and even for a penny (I think. I haven’t found an opportunity to slip one in yet, but I’m pretty sure it’ll work for a penny. Not a marble, though.)

I asked the technician who works with the bowl, and he said he wasn’t sure, but that the rods don’t always go up. The people who make the bowl can make some adjustments to it, and then the rod will go up slower, then stop, then march their way back down as the workers keep adjusting.

I couldn’t figure it out, so I asked my boss, who’s mechanically-minded, and we worked out a theory. In short, it goes like this: the bowl can vibrate in two different ways. It can bob up and down, and it can rotate back and forth. It can move a couple of millimeters in each direction. The bowl makes the rods go up the ramp by doing both of these at the same time, and we can adjust the speed of the rods up the bowl’s ramp by adjusting the phase lag between the two oscillation modes.

Draw a little dot on the side of the bowl. Since a few millimeters (the amplitude of the bowl’s vibrations) is small compared to a meter (the bowl’s diameter), for any given point vibrating the bowl by rotation is the same as shaking back a forth horizontally in the direction tangent to the edge of the bowl. So we’ve got a sine wave motion horizontally.

Close-up of one part of the ramp. The red arrows show the motion of the the red dot, although any other point on the ramp would move the same way.

If the bowl instead bobs up and down, the motion of the dot is like this:

Same as last time, but the bowl bobs up and down.

Now make the bowl rotate and bob simultaneously. There are different ways we could do this. Assuming we give both modes the same frequency (easier to engineer, I’d think), then we can describe the way these modes are combined with a single parameter – the phase lag between them. When the phase lag is zero the bowl moves all the way forward at the same time it’s all the way up, and all the way back at the same time it’s all the way down. The red dot traces out a diagonal line, like this:

Moving forward/backward and up/down in sync.

If the phase lag is 180 degrees, the diagonal line switches directions, like this:

But if the phase lag is 90 degrees, the dot instead traces out a circle. Make it -90 degrees and the circle goes the other direction.

Combining back/forth and up/down just so, we get a circle.

We can also make the circle go the other way, if we want.

This is just right for moving a rod up a ramp. Remember that when something moves in circle, its acceleration points towards the center of the circle. So imagine the ramp shaking this way when it’s at the bottom of the circle. The ramp is accelerating up, pushing into the rod. That means there’s a big normal force on the rod, and a big normal force means more friction. To take advantage of this high friction, the ramp should move forward, pushing the rod ahead in space.

The CCW circle moves forward at the bottom of its loop (green arrow). The ramp is accelerating up (blue arrow), jamming into the rod and carrying it forward along with it.

Now the red dot has reached the high point of the circle. It’s accelerating down. That means it’s falling away from beneath the rod, so the normal force goes down. If the acceleration is high enough, the ramp may even pull away from the bottom of the rod completely (I’m not sure whether it actually does this). Now the friction is low, and the ramp can slide backwards, leaving the rod where it is. The net result is that the rod is further along the ramp at the end of the cycle than at the beginning.

Same circle, but now the ramp is at the top of its motion. The rod will get left behind as the ramp pulls away from underneath.

This really works, even without a fancy industrial bowl. I did it just now. I found the longest hardback book in my room (The Feynman lectures) and an appropriate rod substitute (a Rubik’s Cube), and I easily sent the cube uphill on the tilted book by rotating circularly one direction with my hands, and brought it back down by rotating circularly the other direction.

The book was still pretty short, but I have a longer flat, mobile surface – a white board. I’m a bit too clumsy to shake it in a circle with my hands, though. But I have a bicycle…

I duct taped my white board to the pedal of my bike in an attempt to recreate the circular motion I posited for the ramp. I propped the other end of the white board up against a table,

This didn’t work well. The board was tilting during the cycle because the far end, leaning on the table, wasn’t going up and down the same way as the near end. I tried propping the table up at an angle in hopes of keeping the white board close to flat, but that didn’t help much.

Time to recruit the neighbors. It was 11:20 PM so my neighbors, who are college students, were awake and drunk enough to agree to anything that sounded weird or stupid. With a human at the other end of the white board, I could keep the board pretty much flat, or pretty close to tilted at a constant angle as it went around.

The results were that as long as I kept the white board flat, my Rubik’s Cube would go forward and backward the way I predicted, but I couldn’t get it to climb any significant hill. I tried switching out the cube for a high-friction rock-climbing shoe, but that still didn’t make much progress. I conclude the bowl at my job relies on high-speed, high frequency vibrations, the opposite of my bicycle.

Finally, in order to be a good scientist I needed a more objective test of my theory. I was able to get my conveyor belt to work roughly, but I already knew the result I wanted. Maybe I was subconsciously tipping the book/board the direction that I wanted things to go?

To test this, I called a friend and asked him to replicate my experiment with a book and a penny, without telling him what I expected to happen. He reported the same result! We can indeed make things move forward or backward, even up slight inclines, just by shaking the right way.

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### New Problem: Spool of String

December 27, 2008

Five and a half years ago I saw a simpler version of this question on the physics quiz Caltech sends to incoming freshman so they can prove how smart they are before arriving. This generalization appears in the first chapter of Motion Mountain, a free PDF textbook by Christoph Schiller, which purports to cover all of modern physics, integrating the theoretical and empirical viewpoints. I was linked here by a blog post from by Bee at Backreaction.

You have a spool of thread resting on a table, like so.

Picture of spool problem. There is friction between the spool and table, and gravity is as normal.

You pool on the thread very gently. What is the behavior of the spool as a function of the angle the thread makes with the horizontal? Assume the spool rolls without slipping. You should find two separate domains of behavior (one where to spool rolls forward, one where it rolls backward). What is the critical angle separating them? Provide a geometric interpretation of this result. Or, if you are sufficiently clever (I was not), skip right to the geometric interpretation, and use this to derive the critical angle.

What if the table is inclined with respect to the horizontal?

I only read as far as this first problem, so I don’t know yet whether the book actually provides the answer or simply asks the question. I don’t mind if you cheat, honestly. So long as you think the answer is pretty, however you get it.

### New Problem: Boa Constrictor

October 21, 2008

Text from my brain. Physics from:
Introduction to Classical Mechanics: With Problems and Solutions

A boa constrictor is squeezing you to death. By the way, you are a cylinder. The constrictor has wrapped around you by some angle $\theta$. Your friend Joe grabs the tail of the boa constrictor and starts pulling. He is trying to save your life. Your other friend Debbie grabs the head of the boa constrictor and starts pulling. She was just feeling left out of the excitement.

Debbie is pulling harder than Joe, because Joe is a midget. He also has Down Syndrome. Question: given that Joe is pulling on the tail of the boa constrictor with a force $F_{Joe}$, how hard must Debbie pull, as a function of $\theta$ and the coefficient of static friction, $\mu$, between you and the snake, in order to get the snake to slip across your body? (see awesome diagram for purposes of clarification/divine afflatus (I got that word from the thesaurus so I would sound smart))

You, being semi-squeezed to death by your snakes and friends.

### New Problem: The Kepler Exhibit

October 17, 2008

At the Exploratorium in San Francisco, you can play with this exhibit:

What should the height profile of the bowl be so that balls that roll without slipping (or, so that blocks sliding without friction) would reproduce 2-D Kepler orbits when viewed in projection from above?