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To make an object on ground start moving, the force applied (lets call it $f$) must be greater than $\mu_sN$, where $\mu_s$ is the coefficient of static friction between the object and the ground, and $N$ is the normal force.

However, when the object is already moving at constant velocity with this applied force $f$, and we reduce this force so that it is lower than $\mu_s N$, the object will still remain moving. An intuitive explanation for this would be that it is easier to keep an object moving than to make it start moving.

My question will then be: Imagine an object is moving at constant velocity, and the 2 forces acting on it along the horizontal direction are $f$ and friction, is there anyway to know the boundary value for $f$, where friction turns from kinetic to static, and the object stops moving?

Thank you.

PS: Right now, I know that to find the distance for the object to stop moving, we can use the energy approach, equating the kinetic energy to the work done by friction, but I am clueless as to how exactly we can find the boundary value of $f$ before the object stops.

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  • $\begingroup$ Why are you using $\mu_k$ for static friction? That's improper considering there is also a kinetic friction coefficient. You should use $\mu_s$. Please edit! $\endgroup$
    – Bill N
    Dec 29 '20 at 15:13
  • $\begingroup$ Thanks, I have edited... $\endgroup$ Dec 29 '20 at 15:26
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My question will then be: Imagine an object is moving at constant velocity, and the 2 forces acting on it along the horizontal direction are f and friction, is there anyway to know the boundary value for f, where friction turns from kinetic to static, and the object stops moving?

Maybe you should clarify your question a bit. Friction will turn from kinetic to static if the relative velocity between the bodies is zero for some (usually negligible) amount of time. If the f force is smaller then the kinetic friction, than the relative velocity between the bodies will decrease, and in finite time decreases to zero.

If your question is whether there is a simple relation between the coefficient of static and kinetic friction, then the answer is no, altough for special material models, there may be such cases. Whether these models are accurate models of any experimentally realizable body is another matter.

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  • $\begingroup$ Hi, what do you mean by the relative velocity will decrease to zero in finite time? By F=ma, tt should be 0 when the net force acting on the object is 0, which is where kinetic friction is equal to the force that you apply. Is it possible that you reduce the magnitude of the force such that it is below the magnitude of kinetic friction, and the object is still moving? Thanks $\endgroup$ Dec 29 '20 at 15:03
  • $\begingroup$ @bobthelegend Sentence 2: NO! If the net force is zero, the acceleration is zero. If the object is moving it will keep moving. And yes, even if $f$ is less than the friction the object can be moving... until it stops. $\endgroup$
    – Bill N
    Dec 29 '20 at 15:16
  • $\begingroup$ @BillN What are the conditions for it to stop, that is what i am asking here (such as the boundary value of $f$ when it stops). Also, why did you say both NO and yes... ? It is very confusing $\endgroup$ Dec 29 '20 at 15:25
  • $\begingroup$ @bobthelegend Your sentence 2: "...it [referring to the relative velocity in sentence 1] should be 0 when the net force ... is 0". That's the big NO! Zero force does NOT mean zero velocity. It means constant velocity. $\endgroup$
    – Bill N
    Dec 29 '20 at 15:38
  • $\begingroup$ @bobthelegend There is no "condition" for it to stop other than the acceleration is in the opposite direction of velocity and enough time passes for the velocity to become zero. $\endgroup$
    – Bill N
    Dec 29 '20 at 15:39

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