Water molecules have a network of forces attracting them to each other. At the surface, this network of force resists penetration and can support objects which will sink if pushed through the...

Water molecules have a network of forces attracting them to each other. At the surface, this network of force resists penetration and can support objects which will sink if pushed through the "skin" or "membrane"

I don't quite understand this explanation

An explanation to the reason how viscosity works in water?

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You need to understand the difference between viscosity and surface tension first. The first part of your query pertains to surface tension, while the second part seems to deal with viscosity of liquids.

Viscosity of a liquid is the property by virtue of which it tends to oppose the relative motion between its two adjacent layers. All liquids exhibit resistance to flow. This internal resistance to flow arises out of intermolecular forces of attraction in liquids.

Surface tension, on the other hand, has more to do with wetting (or otherwise) of certain substances on liquid surfaces, sinking (or floating) of light objects in it, walking of insects on water surface and spherical shapes of liquid drops.  Surface tension arises out of the unbalanced forces on the surface molecules compared to those in the bulk. It is due to such forces that the liquid surface behaves like a stretched sheet of rubber.

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