Why does a pen fall over when placed on its tip?

[autodidude]

And if we knew everything about the pen and the surface on which its placed, could we predict the direction in which it would fall? Could you then somehow put the pen so that it won't fall over?

It doesn't have to be a pen by the way, could be a pencil or similar things that tend not to topple over without support

 

[bahamagreen]

It falls because it is subject to an imbalance of forces. In order for it to stay up the forces would need to be balanced all around. But you can't place it in a balanced position.
Consider that at the resolution at which the balance would have to occur, you have the effects of heat, molecular motion, and on down to quantum uncertainty... even if you could mathematically define the balanced position, in reality you have a physical object with small internal motions, so that balance point changes with time, so once it is off balance it falls.

 

[sophiecentaur]

If it is placed 'exactly' right (on its infinitely small tip) then it will be in equilibrium because the moments will balance exactly. But this is an Unstable Equilibrium. Any perturbation will cause the centre of mass to fall (Gravitational Potential will decrease) and there is no restoring force; in fact the resulting force is away from the equilibrium position.

For an object with a broad base, the equilibrium is Stable because a small perturbation will cause the CM to rise (Potential will increase) and there will be a restoring force towards the equilibrium position.

The third case is when the CM stays at the same height (as with a ball or cylinder on its side) when there is a perturbation. There is no restoring force and no change of Potential. This is called Neutral Equilibrium.

 

[mfb]

And if we knew everything about the pen and the surface on which its placed, could we predict the direction in which it would fall?

At the level of precision you can achieve manually: Yes. Do not forget air movement and the precise way the pen is released (it might have an initial velocity).

Could you then somehow put the pen so that it won't fall over?

Without any air movement, I think this should be possible with a lot of pens and surfaces.

 

[sophiecentaur]

But for how long?
Even the molecules at the point of the pen would be in thermal motion and, in a vacuum, there would still be random molecules leaving the surface of the plastic. The momentum change would result in the pen moving a bit. Then, the instability would take over.

 

[mfb]

No pen tip is perfect, neither are surfaces. Just by placing the object on the surface, you flatten both a bit. Two flat surfaces of finite size with the right orientation are sufficient for a stable equilibrium.

 

[Khashishi]

This is called an unstable equilibrium point. At an equilibrium point, the forces are perfectly balanced, so there is no motion in the ideal case. In a stable equilibrium, small shifts away from the equilibrium point cause motions in the system toward the equilibrium point, so the system will bob around and settle down toward the equilibrium. In an unstable equilibrium, small shifts away from the equilibrium point give rise to motions away from the equilibrium point. Such a system is unstable--it won't stay in the equilibrium because small motions of the air or something will cause the system to topple.

 

[sophiecentaur]

No pen tip is perfect, neither are surfaces. Just by placing the object on the surface, you flatten both a bit. Two flat surfaces of finite size with the right orientation are sufficient for a stable equilibrium.

But the two flat surfaces are not rigid. They are not like 'engineered facets' but caused by the materials pressing on each other and deforming. There will be no restoring force, the flats will just change to allow the pen to fall over, just the same as if there were a point or a spherical interface. (It is true that there may be some slowing down, die to the 'viscosity' of the materials but it is still an unstable situation.

 

[autodidude]

Wow, that's interesting, thanks everyone.

So I guess that means we wouldn't be able to predict exactly in which direction it'll fall? (I saw molecules in there so I'm guessing we would be invoking QM and all that fun stuff)

 

[pgardn]

It is interesting.

I knew about the moments and such but never thought much about all the other variables that would come into such a balanced situation.

I will now attempt to balance a pen on my finger by constantly moving my finger so that the CM of the pen is directly over a tiny spot on my finger.

Oops. Cant do it. The pen keeps wanting to rotate, I keep moving my finger so that the normal force acting on the pencil from my finger is always at an angle not perpendicular to the ground so that it acts as a momentary moment counteracting the torque produced by gravity at the CM because the CM of the pen is rarely exactly above the tiny spot on my finger. So I have to keep moving quickly and the pen is always tipping this way and that.

I think I got it right. A noble attempt to explain how a seal is able to balance a ball on its nose. They kinda do the same thing. It much easier with a meter stick on a finger. You can be fairly clumsy and still do a decent job. I would say the CM of the meter stick is over the place on my finger more often but I still got to move my finger and body.

 

[mfb]

The meter stick is longer, and therefore has a longer timescale of the movement. This makes it easier to balance it. The tip size does not matter if you actively control it anyway.

 

[pgardn]

I would probably say the moment of inertia is larger, but not too large so that I can apply a torque easily to the bottom to attempt to stop the rotation. But I cannot apply just the right amount of torque so the stick rotates in another direction.

The pen would be more like trying to adjust the motion of a very small mass with your hand in a predictable manner. One has a very difficult time applying just the right amount of force. The hippo tries to gently pet a mouse.

Now if someone could make a robot that could readjust the normal and keep readjusting the normal force(torque) to keep the pen from falling to the ground while constantly readjusting, well, that would be cool.

 

[mfb]

Already done ;)

"robot balancing pen", first google hit and second google hit

 

[pgardn]

Already done ;)

"robot balancing pen", first google hit and second google hit

OMG :)

I don't need Christmas.

thanks for those... that is so cool!

Wait. I do need Christmas. I want one of those.

 

[the_emi_guy]

Notice in the videos that the image sensors are mounted up high looking at the eraser end of the pencil. If you want to impress your friends at a party by balancing a yardstick or some other object on your fingertip, the trick is to focus your attention on the highest point. Most people will either focus on their finger, or the center of the object which makes it very hard.

 

[jbriggs444]

I am many years out of second year calculus, but I believe that the relevant equation of motion is d^2x/dt^2 = kx

By inspection, that equation has solutions x(t) = e^(sqrt(k))t, x(t) = x(0) * e^-(sqrt(k))t along with any linear combinations thereof.

Unless x(0) = 0 and x'(0) = 0 identically, the current position will depart from the equilibrium position exponentially both forward and backward in time.


In order to stay upright for an hour on a pencil with a time constant of 1 second, you're going to need initial positioning that is accurate to a thousand or so significant digits. That's somewhat impractical.

 

[pgardn]

Notice in the videos that the image sensors are mounted up high looking at the eraser end of the pencil. If you want to impress your friends at a party by balancing a yardstick or some other object on your finger tip, the trick is to focus your attention on the highest point. Most people will either focus on their finger, or the center of the object which makes it very hard.

Yep.

After I saw the videos and the cameras I tried to balance by just using the feel in my finger and closing my eyes. I used a fireplace implement, the poker. Its easy when viewing but when I closed my eyes, no way.

Very cool stuff though.