Angle of Friction, Angle of Repose, and Rolling Friction

Angle of friction. When a body rests on a rough surface at the point of limiting equilibrium (an applied force has just reached the value at which sliding is about to begin), the two separate contact forces acting on it -- the normal reaction $N$ and the limiting static friction $f_{s(\text{max})} = \mu_s N$ -- can be combined into a single resultant contact force $R$. The angle of friction, $\phi$, is defined as the angle this resultant $R$ makes with the normal $N$. Since $f_{s(\text{max})}$ and $N$ are simply the two perpendicular sides of a right triangle whose hypotenuse is $R$,

so the tangent of the angle of friction is numerically equal to the coefficient of static friction itself.

Angle of repose. The angle of repose, $\theta$, is defined quite differently at first sight: it is the steepest angle to which an inclined plane can be tilted before a body placed on it, relying on friction alone (with no other applied force), just begins to slide down under its own weight. Consider a block of mass $m$ resting on such an incline, tilted at exactly this critical angle $\theta$. Resolving its weight along and perpendicular to the incline (exactly as in Section 4.7's free body diagram method) gives a down-slope component $mg\sin\theta$ and a perpendicular component $mg\cos\theta$ (which equals the normal reaction $N$, since there is no acceleration perpendicular to the incline). At the critical angle, the block is on the verge of sliding, so friction is acting at its limiting value, $f_{s(\text{max})} = \mu_s N$, directed up the slope, exactly balancing the down-slope component of gravity:

Comparing this directly with the earlier result $\tan\phi = \mu_s$ for the angle of friction shows immediately that

the angle of repose and the angle of friction, though defined by two seemingly quite different physical situations (a body on the verge of sliding on a tilted surface, versus the direction of the resultant contact force on a body about to slide under an applied push), turn out to be exactly the same angle for any given pair of surfaces, because both reduce, algebraically, to nothing more than $\tan^{-1}(\mu_s)$. …