Why does pushing with arms against a table (while sitting on a chair with feet not touching ground) cause the chair to move? Instead of pushing the table, I push myself(seated in chair).
I simplified it to two forces - downward weight/gravity force and horizontal arm-push force. But is the arm-push force moving my chair or the weight of the table pushing my chair?
Newton's 3rd Law tells us that when you push on the table, the table pushes on you with an equal and opposite force. If your chair had wheels, like a desk chair, you would be intending to push yourself away from the desk when you push on the desk. So it turns out that both the wheels of a desk chair and the legs of the chair at the table move away from the desk or table when you push. Both the wheels and the chair legs are willing to move away. What determines that willingness?
Friction between the wheels and floor in one case, and between the legs and the floor in the other case, determines how much resistance to moving away either chair has. With the desk chair, the friction is primarily in the axles of the wheels. I think there is no surprise that the chair rolls. So let's not carry the discussion of the desk chair any farther. The table probably has legs similar to the legs of the chair you are in. The table's legs also have friction with the floor. If both the table's and the chair's legs have equivalent surfaces, the friction characteristics of both table and chair would be equal. Because I don't know any different, I will assume those surfaces are equivalent.
The way friction works, the resistance to sliding is proportional to the force pressing the 2 surfaces together. In both cases, that is the weight of the object. Remember what I said at the beginning about Newton's 3rd Law. The force attempting to slide the table is the same as the force attempting to slide your chair. So it is a matter of whose friction is greater. If the table is heavier than you and your chair, with the assumptions I have made, the table's friction should be greater than your friction and you should be the one to slide.
I hope this helps,