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I came across the following in Goldstein Book:

The frictional force is proportional to the velocity of the particle. Thus its $x$ component has the form $F_{fx}=-k_x v_x$ .

Wikipedia says,

frictional force on a particle with velocity ${\displaystyle {\vec {v}}}$ can be written as ${\displaystyle {\vec {F}}_{f}=-k\cdot {\vec {v}}}$

I know the definition of frictional force and velocity, but dont know the meaning of this line. Can anyone explain with an example?

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  • $\begingroup$ $k$ is a constant not vector, I edited $\endgroup$
    – LearneR
    Sep 2, 2021 at 23:29
  • $\begingroup$ It is pretty straightforward, I am not sure what you don't exactly understand. Write the second equation in components and the x-component becomes the first equation (it should be $k$ instead of $k_x$). $\endgroup$
    – user65081
    Sep 2, 2021 at 23:30
  • $\begingroup$ I mean, an example which explains the equation $\endgroup$
    – LearneR
    Sep 2, 2021 at 23:32
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    $\begingroup$ This equation specifically holds true for fluid friction at low velocities. Such as when you drop a marple in a bowl of oil. At higher velocities, the friction force (in that case often called drag force) will depend on the velocity squared instead. Such as the air friction on an airplane. $\endgroup$
    – Steeven
    Sep 2, 2021 at 23:35
  • $\begingroup$ Is it the small arrows above the variables that you are asking about? $\endgroup$ Sep 2, 2021 at 23:46

4 Answers 4

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Suppose you give a block an initial velocity $u$ along $x-$axis on such a surface where $\vec{f}=-k\vec{v}$, then magnitude of frictonal force is $f=kv$.

As body moves forward , frictional force opposes it's motion decreasing the speed of body. As speed decreases so does the magnitude of force therefore speed decreases but rate of decrease in speed also decreases.

Mathematically ,

$$f=-kv$$

$$a=-\frac{k}{m}v$$

$$v\frac{dv}{dx}=-\frac{k}{m}v\implies\frac{dv}{dx}=-\frac{k}{m}$$

$$m\int_{u}^{v} dv=-k\int_{0}^{x}dx\implies v=u-\frac{k}{m}x$$

To get position as a function of time

$$\int_{0}^{x}\frac{dx}{u-\frac{k}{m}x}=\int_{0}^{t}dt$$

Here I assumed one dimensional motion of body, If we have a two dimensional motion then we can break the motion into two parts ( along $x-$axis and along $y-$axis), write corresponding equations and then add the result.

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I'm assuming it is the intuition behind these equations you are looking for? as in why the friction is related to velocity?

let's imagine you have a ball falling down, and look at friction on the "semi" macroscopic level in it's classical/Newtonian understanding. There are air particles in the air which collide with the falling ball restricting it's natural motion. As one can intuitively understand that the higher the velocity, the higher the momentum and thus bigger force because of the collision, so you can see how friction in this setup has a relationship with velocity.

In my limited experience, when analyzing analytically how the speed of the falling ball varies in a fluid of sort, with respect to time it tends to stabilize around a certain value. Depending on how accurate the calculations are required to be, estimations of effects of higher order dependencies would be studied.

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Friction Resisting Velocity

Usually when friction is proportional to velocity, it’s due to a liquid, from viscous damping. Examples are shock absorbers, dashpots, and laminar drag forces.

Friction force between solids, such as a mass on a spring, or a block on a table, is generally proportional to the normal force, not velocity. This is usually how people are introduced to friction, proportional to normal force only. Your case just means friction force pointing opposite to the velocity.

Most people first encounter such friction in SHM, with the classic component in harmonics being the “dashpot”, covered below.

Notation and Direction

The location $x$ is technically a vector $\vec{x}$, has an $x,~y,$ and in some problems a $z$, component. People don’t always use different notation for vectors (I don’t below). So $v$, $a$ and $F$ are vectors also. The negative sign in $F=-kv$ (and in other terms below) are because the resistance is in the direction opposite to the acceleration, displacement, and velocity respectively.

The direction is exactly opposite to the direction of the velocity and proportional to its speed (“speed” just means the magnitude of velocity vector). This is expressed as $\vec{F}= -k\vec{v}$ and the $x$-component of $\vec{F}$. $F_{fX}$ is $-k$ times the $x$-component of the velocity, same for $y$ and $z$ directions (for $F_{fY}$ and $F_{fZ}$) if there are three dimensions.

Simple Harmonic Motion

The simple harmonic motion (SHM) would then be a mass (which resists acceleration with a term $-m~ \tfrac{d^2x}{dt^2}$), a spring (which resists displacement with a term $-k x$), and a dashpot (which resists velocity with a term $-c~ \tfrac{dv}{dt}$), and table friction (which resists always, any time there’s motion with a constant term $-\mu N$). The dashpot and table are just two different types of friction. $F=ma \implies$

$$~-\mu N ~(1)~-k ~(x)~-c~ (\tfrac{dx}{dt}) ~-m ~(\tfrac{d^2x}{dt^2})~ =0$$

Other Example

On a car, shock absorbers, as opposed to springs or struts, are viscous dampers that counter the motion with a force proportional to velocity, or sometimes proportional to $cv^n$ where $n>1$, depending upon how the fluid is constrained and the speed being operated at. When the fluid is unconstrained or is in Laminar flow resistance (which happens at lower speeds), the resistance is proportional to velocity as in your example.

You may have exercised on machines with pistons, which are in some gyms, designed to increase explosiveness. That’s what friction force proportional to velocity feels like.

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Example for Frictional force that proportional to the velocity

Vehicle suspension

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shock absorbers

enter image description here

The shock absorbers force is proportional to the relative velocity between Point A and B and the damping coefficient k.

$$\vec F_k= k\,\vec v_{AB}$$

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