In one line
The angle of friction is the angle between the total reaction and the normal at limiting friction; the angle of repose is the steepest incline on which a body stays at rest; and the two are equal.
Explained simply
Tilt a book with an eraser on it slowly. At some tilt the eraser starts to slide: that tilt is the angle of repose. It tells you how 'grippy' the surface is, just like the coefficient of friction does.
Technically, at limiting equilibrium, the normal reaction N and limiting friction F combine into a resultant reaction R that leans away from the normal by the angle of friction \( \phi \), with \( \tan\phi = F/N = \mu \).
How it works
- Angle of friction \( \phi \): \( \tan\phi = \mu \).
- Angle of repose \( \alpha \): put a block of weight W on an incline at \( \alpha \). Along the plane: \( W\sin\alpha = F \). Normal: \( N = W\cos\alpha \).
- At the point of sliding, \( F = \mu N \Rightarrow W\sin\alpha = \mu W\cos\alpha \Rightarrow \tan\alpha = \mu = \tan\phi \), so \( \alpha = \phi \).
- Cone of friction: rotate the limiting resultant R about the normal; it sweeps a right circular cone with semi-vertex angle \( \phi \). If the resultant of the applied forces falls inside the cone, the body stays at rest; on the surface of the cone, motion is impending.
Key formula
\( \tan\phi = \mu \); angle of repose \( \alpha = \phi = \tan^{-1}\mu \); \( R = \sqrt{N^2 + F^2} \).
Quick example
A block just starts to slide when a plank is tilted to 22°. Then \( \mu = \tan22^\circ = 0.404 \) and the angle of friction is also 22°. Heaps of sand or grain form slopes at their angle of repose, which engineers use when designing storage bins and hoppers.
Common mistakes
- Saying the angle of repose equals \( \mu \) itself; it is \( \tan^{-1}\mu \).
- Measuring \( \phi \) from the surface instead of from the normal.
- Describing the cone of friction without saying its semi-vertex angle is \( \phi \).
How MU asks it
A frequent 3 to 5-mark theory part: 'Define angle of friction, angle of repose and cone of friction; show that the angle of repose equals the angle of friction'. Give definitions, the incline sketch, the 3-line proof and a sketch of the cone.