Can we "pick up" an absolutely smooth cone off the ground?

Question

I came up with the following problem: imagine an absolutely smooth cone with its circular face in contact with the ground. Is it possible to pick it up? Here are the detailed requirements:

1. The cone is an even rigid body;

2. Do not exert forces without contact (e.g. gravity, magnetic force);

3. To pick up the cone means to hold it such that it is totally above the ground.

I suppose that we need to let the cone topple over first, so that we could hold it somewhere related to the point of mass, but I cannot think of a full solution.

Answer 1

In practice glue will work. But let's consider an artificial problem where no glue will attach. That amounts to saying we are not allowing chemical bonds etc. But we can, I suppose, suggest the cone and the ground can be compressed a little. If so then, to get the cone to leave the ground momentarily, strike the top of it with a hammer. The cone and the surface underneath it will be compressed. As they return to their original shape the cone might just hop off the ground a little.

If this succeeds, then while the cone is in the air, slip a paper (or your hand if there is room) underneath it. You can then lift it.

Second method: knock the cone over by applying horizonal force to the top and the base in opposite directions. Catch the cone as it falls. Then you can lift it.

Answer 2

If by applying a horizontal force to the cone, with a suitable downward component to prevent toppling, I can make it move with an arbitrarily large speed relative to the ground, then I should be able to catch it via this cage/box contraption:

then quickly close the lid (before the cone can bounce off the wall of the box and escape), and lift the box above the ground as far as you want.

Answer 3

As per example :

Constraints :

• Cone must be placed in a non-negligible pressure environment, such as under $1~\text{atm}$ Earth atmosphere pressure (can be bigger depending on cone mass).
• Crane material must have enough elasticity, so that after sealing and making vacuum inside it,- crane walls could deform enough pressing upon cone surface which acts "like a glue" (rubber, plastic, etc, depends on cone mass too)

Answer 4

Let's put some numbers to the problem. Call the radius of the base $R$, and the height of the cone $h$. The center of mass sits at $h/3$, measured from the base up. Let us call the radius at this point $R^\prime$.

As long as the cone is made from fermionic matter, this would work even for a perfectly frictionless cone in exactly the way you describe: topple over the cone, and then take prongs with a circular profile of diameter $r$, where $R^\prime < r < R$. This ensures that the center of mass is below the point of attachment, and if cone and prongs are both made from fermionic matter, they cannot pass through each other and therefore no friction is required. The only tricky part is applying the force to topple over the cone, but I guess using some kind of "hat" to put on top of the cone would work for that.

Answer 5

Solution without toppling: use buoyancy.

1. If the cone is sufficiently light or flat (e.g. R≫h case you ask about in the comments), use suction: on top of the cone place a circular pipe with vacuum, diameter slightly smaller than the base of the cone. The force would press the cone into the pipe, ensuring the seal. Maybe some engineering might need to be invested in a properly shaped seal. This can work for $mg < P_{atm}\pi R^2$.

2. If the cone is made from a sufficiently low-density material, build walls around it and pour a liquid, e.g. water (1 $g/cm^3$) Mercury (13.6 $g/cm^3$), so the cone floats.

3. If the cone is heavy, dense or narrow (h>R), combine both methods: surround it with a pressurized gas or liquid (e.g. sufficient depth of water) and use the suction pipe.

Answer 6

Force applied very close to the tip, even perpendicular to the surface (as, under "ideal frictionless-ness", parallel component disappears) causes a rotational momentum away from the floor (because CoM is on the symmetry axis deep inside, while force would get applied above and not through it)

Apply force strong enough - and any cone can be rotated up from "standing on its round side" position.

The moment that happens - wedge + force to keep the cone on it should be enough to control position in any way you want

And after flipping it all the way - any cone can be held by a ring, if you put the tip downwards through it

Answer 7

Use three hands.

If you cup one hand at the base of the cone on one side and cup one hand near the tip of the cone from the other side, you can use opposing forces to force the cone to rotate about bottom edge near your hand. Tip it far enough such that one side of the cone is past 90 degrees. Once the cone is on edge, you can easily fit a 3rd hand underneath it, against the flat surface, and lift it.

Because your hand can curve around the cone a bit, both motions (tipping and lifting) are stable, even though the cone is smooth.

Interestingly, this could be more difficult if the surface the cone is resting on is perfectly flat too. Two extremely flat surfaces (so called "optically flat") can be difficult to separate because it's hard to get air between them. This creates a partial vacuum. You may need considerable force to move the cone, but not more than 1 atmosphere times the area of the cone's base.