= Coefficient of kinetic friction
{title2=$\mu_k$}
The <coefficient of kinetic friction> is the <dimensionless quantity> $\mu_k=|f_k|/N$ for sliding contact with positive <normal force> $N$ in the ideal <kinetic friction> model. It characterizes the pair of contacting surfaces and their condition. A constant $\mu_k$ means the model neglects dependence on <contact slip> speed; unlike the <coefficient of static friction>, it specifies a sliding <force> magnitude rather than a no-slip upper bound.
The ratio compares the tangential <kinetic friction> magnitude with the perpendicular <normal force>:
$$
\mu_k=\frac{|f_k|}{N},\qquad N>0,\quad |\mathbf w|>0.
$$
Here $\mathbf w$ is the tangential <relative velocity> defining <contact slip>. Both numerator and denominator have the same <physical dimension>, so the <coefficient of kinetic friction> has no units. It belongs to a contact pair under specified conditions, rather than to one material alone. The <coefficient of static friction> instead describes the threshold $|f_s|_{\max}/N$ before <contact slip>; a pull required to start motion cannot be substituted for a sliding measurement.<University Physics Volume 1 by William Moebs and others>[§6.2, “Magnitude of Static Friction” and “Magnitude of Kinetic Friction”, equations 6.1–6.2; unitless coefficient and dependence on the two surfaces]{ref}
As an original model example, stipulate $\mu_k=0.30$ and <gravitational acceleration> $g=9.8\,\mathrm{m/s^2}$. An already sliding, nonrotating block of total <mass> $m=2\,\mathrm{kg}$ is pulled horizontally across a stationary horizontal surface at constant <velocity>. With no other vertical <force>, <Newton's second law> gives $N=mg=19.6\,\mathrm N$. Zero horizontal <acceleration> then requires a pulling <force> $F=|f_k|=\mu_kmg=5.88\,\mathrm N$. Doubling the total <mass> to $4\,\mathrm{kg}$ gives $N=39.2\,\mathrm N$ and $F=11.76\,\mathrm N$. The <force> doubles while the <coefficient of kinetic friction> stays $0.30$ by assumption. These numbers are model predictions, not observations or a calibration for a particular material pair.
\Image[/kinetic-friction-constant-speed-force-balance.png]
{title=Original horizontal sliding model: a 2 kg block has balanced forces at constant speed with a stipulated kinetic friction coefficient of 0.30}
A constant <coefficient of kinetic friction> is useful because one ratio can summarize many loads, as tested by <load dependence of kinetic friction>. It is an empirical approximation over a stated range of <normal forces>, <speeds> and surface conditions. It implies no universal independence from <speed> or apparent contact <area>: a change of face or sliding regime can change the result. <Temperature>, wear and lubrication are further conditions to report. <Newton's second law> still applies when the constant-coefficient approximation fails.<University Physics Volume 1 by William Moebs and others>[§6.2, approximate empirical model; small nonzero speeds; exceptions for lubricated surfaces and high speeds; “Atomic-Scale Explanations of Friction”, apparent versus actual contact area]{ref}
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