Why do small objects stick to our fingers when we push down on them?

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It's weak chemical bonding which is usually the source of adhesion when any two surfaces are brought together.

The real question is, why don't our fingers pick up everything we touch? They do! Try touching some fine powder, or some liquid.  The surface of powder-grains will weakly bond to your fingers. The surface of everything will weakly bond to your fingers.  But large objects usually weigh too much to be lifted.

So, then why can we sometimes pick up much larger objects? That's simply a matter of the size of the surface area in contact. Ordinarily our fingers don't match the shape of an object's surface. Our fingertips are not smooth. If we touch an object, the total area of direct contact at the nano-scale is very, very small. With such a small contact area, the total bonding force between the surfaces is also small.

If instead your fingers were flat and polished, then they could pick up a much heavier object which also had a flat and polished surface. Or, by pushing our fingers hard against a non-flat, non-polished surface, we can cause our skin to partly conform to that object's shape, at least temporarily.  In that case the total area of contact is much larger, so the total force of the weak chemical bond between the surfaces is also much larger. Push your finger against a staple, or against hairs and large dust-grains, and suddenly they can be lifted.

Which common surface employs this trick for sticking to objects? Adhesive tape. The surface of tape is not rigid.  It easily changes shape, especially down at the nano-scale, and because of this, whenever an object is placed against it, then a quite enormous amount of the tape's surface will actually touch the object. The tape surface-molecules don't necessarily have a larger bonding force. It's just that a lot more of the molecules can actually touch the object.

And, if the dead skin on the surface of your fingers was very soft, it would easily conform to surfaces being touched, so the normal chemical-boding would make skin act sticky. With skin made of soft gel, you might be able to pick up an entire stapler, not just a staple.

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