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I believe that when we lift an object placed on a rough surface then initially frictional force(due to the cold welding) acts on the object (together with gravitational force) to oppose the relative motion but I'm wondering whether the force of static friction here is equal to the one in the case where relative motion is trying to occur parallel to the surface of the object.

So: Is the force of friction in upward case same as sideward one?

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Generally, no- the effort required to lift an object from a surface is not materially increased by the factors that cause resistant to motion across the surface. However, when two very flat and clean surfaces are pressed together the forces of attraction between them can exceed the effects of gravity.

To understand why this is the case you should consider that everyday frictional effects arise because of myriad random small-scale projections on surfaces. When two surfaces are brought together these projections interlock to some extent. When you try to slide the surfaces across each other the faces of the mating projections butt-up, and the normal force between them is what accounts for friction.

Cold welding arises when you bring together two faces that are sufficiently clean and flat to allow bonding between large numbers of particles on either side of the join.

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