Some of the most important Physics Topics include energy, motion, and force.
What is Angle of Repose and its Formula?
Angle of friction: Let us consider a body placed on a rough surface. Now if a horizontal force is applied on the body, a frictional force comes into play. If the force applied on the body is increased, the friction also increases until it becomes equal to the limiting friction.
Let the limiting friction be denoted by \(\vec{f}\), the normal force acting on the body be denoted by \(\vec{R}\) and the resultant of the two forces \(\vec{R}\) and \(\vec{f}\) be denoted by \(\vec{Q}\).
Now if \(\vec{Q}\) makes an angle λ with \(\vec{R}\), then λ is called the angle of friction.
Definition: The resultant of the limiting friction and the normal force between two surfaces in contact, makes an angle with the normal force. This angle is called the angle of friction.
From Fig., R = Q cosλ, f = Q sinλ.
Again, µ = \(\frac{f}{R}\) = \(\frac{Q \sin \lambda}{Q \cos \lambda}\) = tanλ
Hence, the tangent of the angle of friction is equal to the coefficient of friction.
Also, R2 + f2 = Q2(cos2λ + sin2λ) = Q2
or, R2 + μ2R2 = Q2 or, Q = R\(\sqrt{1+\mu^2}\)
Cone of friction: Let a force be applied on a body at rest on a plane [Fig.]. The force of friction acts on the body exactly in the opposite direction. The applied force acts at point O of the body. If the direction of the applied force along the plane changes, the direction of the force of limiting friction f will accordingly change, thereby changing the direction of the resultant Q. But the direction of the normal force and the angle λ between Q and R remains the same, whatever be the direction of Q.
Hence, taking into consideration the different directions of the applied force in the same plane, it is seen that the resultant Q makes an imaginary cone, with the normal force R as its axis. This imaginary cone is called the cone of friction.
The vertex of the cone is the contact point O of the body and the plane, its axis is along the normal force R and the semi-vertical angle is the angle of friction λ.
Angle Of Repose
Consider a body of weight W, placed on a surface inclined at an angle θ with the horizontal [Fig.]. The normal force of the inclined plane on the body is R = W cosθ. In this case, the component W sinθ along the plane, tries to set up a downward motion of the body, Consequently, a frictional force develops upwards along the surface. If the inclination θ is such that the surface just prevents the downward motion of the body, then the frictional force f is the limiting friction, expressed as f = W sinθ.
Hence, the coefficient of friction,
μ = \(\frac{f}{R}\) = \(\frac{W \sin \theta}{W \cos \theta}\) = tanθ
Therefore, when the tangent of the angle of inclination equals the coefficient of friction, the body remains in limiting equilibrium on the inclined plane. If the angle of inclination θ is increased further, W sinθ also increases. Then the limiting friction f cannot balance the body and it starts to slide down the inclined plane. The angle of inclination θ in this case is called the angle of repose of the inclined plane.
Definition: The maximum value of the angle of inclination of a surface, for which an object placed on it is just on the verge of downward motion due to its own weight, is called the angle of repose.
Discussion
- For a very smooth surface, i.e., for μ ≈ 0, the angle of repose becomes approximately equal to 0°. This means that, the surface cannot prevent the downward motion of this body, even when it is very slightly inclined.
- When the angle of inclination exceeds the angle of repose, then the frictional force can no longer keep the body in equilibrium on the inclined plane.
- The angle of repose of different objects, placed on the same inclined surface, may be different.
- We know μ = tanλ, where λ is the angle of friction, and also μ = tanθ, where θ is the angle of repose, for a pair of surfaces in contact.
∴ tanλ = tanθ or, λ = θ.
Thus, the angle of repose and the angle of friction are numerically equal. However, they are two different physical quantities. The angle of friction is a quantity related to friction between any two surfaces; on the other hand, the angle of repose is a property of inclined surfaces only.