What Factors Influence Frictional Force in Different Scenarios?

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Frictional force is a macroscopic effect resulting from atomic electromagnetic forces, acting to decrease velocity and total energy in a system. It is classified into two types: dry friction, which increases with surface roughness and normal force, and fluid friction, which is influenced by factors such as fluid viscosity, density, and the shape and speed of the object moving through the fluid. Dry friction is path-dependent and nonconservative, while fluid friction's behavior varies with velocity, especially at high speeds where it becomes proportional to the square of the velocity. Understanding these factors is crucial for analyzing motion in different scenarios. The interplay of these elements determines the overall frictional force experienced in various contexts.
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What is frictional force and its causes to increase or decrease?
 
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frictional force is a macroscopic effect of atomic size electromagnetic forses.
It usually causes the decreasion of the velocity and because of being a nonconservative force, has a path dependent integral. (I mean the work) and decreases the total energy of the body
 
There are two kinds of friction: dry and fluid. Dry friction increases with the roughness of the two surfaces in contact and the normal force between the surfaces.
Fluid friction depends on the viscosity and density of the fluid (air, water, oil,...), the cross section of the body perpendicular to the direction of movement, the shape of the body, the angle of incidence (angle between the velocity vector and the symmetry axis of the body) and to the velocity (for high speeds it is proportional to the square of the velocity).
 

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I have recently been really interested in the derivation of Hamiltons Principle. On my research I found that with the term ##m \cdot \frac{d}{dt} (\frac{dr}{dt} \cdot \delta r) = 0## (1) one may derivate ##\delta \int (T - V) dt = 0## (2). The derivation itself I understood quiet good, but what I don't understand is where the equation (1) came from, because in my research it was just given and not derived from anywhere. Does anybody know where (1) comes from or why from it the...
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