Coefficient of Friction on an Inclined Surface

An interesting relationship between the coefficient of kinetic friction $\mu_k$ between a wood block and an inclined surface and the angle of incline when the block travels at a constant speed down the incline.

The key requirement for this scenario is to have the block travel at a constant speed down the incline, which means that the acceleration is equal to zero. If the acceleration is zero, that means the net force $F_{net} = ma$ is also zero.

So, we can draw out the forces with a free-body diagram as in the diagram below:
We have three forces acting on the block as it slides down the incline: $F_g$, the force of gravity which pulls the block toward the Earth (or ground); $F_N$, the normal force which acts on the block perpendicular to the incline; and $F_{fr}$, the force of kinetic friction which acts opposite the block's direction of motion.

Since the net force on the block is zero, the force of gravity parallel to the incline must equal the force of friction and the force of gravity perpendicular to the incline must equal the normal force on the block. This gives us $F_{g\parallel} = F_{fr}$ and $F_{g\perp} = F_N$.

With $F_{fr} = \mu_k F_N$, we can relate the two equations. Since $F_g=mg$, we have that $F_{g\parallel} = mg\sin\theta$ and $F_{g\perp} = mg\cos\theta$ where $\theta$ is the angle of the incline. This is because the angle between the $F_g$ and $F_{g\perp}$ is equal to the angle of the incline by angle chasing using right triangles. 

So we can substitute these values to find $F_N = F_{g\perp} = mg\cos\theta$. Substituting this expression of $F_N$ into $F_{g\parallel} = F_{fr}$, we get $mg\sin\theta = \mu_k F_N = \mu_k mg\cos\theta$. Simplifying, we get $\mu_k = \frac{\sin\theta}{\cos\theta}$, which means $\mu_k = \tan\theta$.

This result $\mu_k = \tan\theta$ connects two seemingly unrelated variables together: the coefficient of kinetic friction and the angle of the incline. This result can be used to determine the coefficient of friction between two objects on an adjustable incline (whose angle of inclination can change) as long as you adjust the incline so that the block (or object) slides down at a constant velocity.

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