Angle of Friction and Angle of Repose

Table of Contents

  1. Angle of Friction
  2. Angle of Repose
  3. Relation between Angle of Friction and Angle of Repose

Angle of Friction

Angle of friction is defined as the angle made by the resultant of normal reaction and frictional force with the normal reaction. 
Consider a body lying on a horizontal surface. Let $R$ be the normal reaction and $F_c$ be the frictional force. If $\alpha$ is the angle made by the resultant of normal reaction and frictional force with the normal reaction, then, $\alpha$ is the angle of friction.

Angle of friction

​Here, \[\tan\alpha = \frac{BC}{OB}\] \[\tan\alpha = \frac{OA}{OB}\] \[\tan\alpha = \frac{F_c}{R}\] \[\therefore \tan\alpha=\mu\text{ ___(1)}\] Therefore, coefficient of limiting friction is equal to the tangent of angle of friction.

Angle of Repose

Angle of repose is defined as the angle of an inclined plane, at which a body placed on it just begins to slide. 
Place a body of mass $m$ on an inclined plane. Increase the angle of inclined plane till the body just begins to slide. If $\theta$ is the angle of inclination at which the body just begins to slide, then $\theta$ is the angle of repose.

Angle of Repose

The weight $mg$ of the body acts vertically downwards. Because of the angle of inclination, this weight can be resolved into two components; $mg\cos\theta$ and $mg\sin\theta$ as shown in fig. 
If the body just begins to slide, then 
1. $mg\sin\theta$ is equal to the frictional force \[mg\sin\theta=F_c\text{ __(a)}\] 
2. $mg\cos\theta$ is equal to the normal reaction \[mg\cos\theta=R\text{ __(b)}\] Dividing $\text{(a)}$ by $\text{(b)}$, \[\frac{mg\sin\theta}{mg\cos\theta}=\frac{F_c}{R}\] \[\tan\theta=\frac{F_c}{R}\] \[\tan\theta=\mu\text{ ___(2)}\] Thus, coefficient of limiting friction is equal to the tangent of the angle of repose.

Relation between Angle of Friction and Angle of Repose

​From $\text{(1)}$ and $\text{(2)}$, \[\tan\alpha=\mu=\tan\theta\] \[\tan\alpha=\tan\theta\] \[\alpha=\theta\] Thus, angle of friction is equal to the angle of repose.

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