Short Answer:

The angle of repose is the maximum angle of an inclined surface at which a body placed on it just begins to slide down due to gravity. It represents the steepest slope that a material or body can maintain without sliding. At this angle, the downward component of the body’s weight is exactly balanced by the limiting frictional force. Mathematically,

where θ is the angle of repose and μ is the coefficient of friction between the surfaces in contact.

Detailed Explanation :

Angle of Repose

In Engineering Mechanics, the angle of repose is an important concept that helps describe the relationship between friction and the tendency of motion of a body resting on an inclined plane. It is used to determine how steep a surface can be before an object or material begins to slide under the influence of gravity.

When a body is placed on an inclined plane, two main forces act on it — the component of its weight (W sin θ) parallel to the plane that causes motion, and the frictional force (F) that resists motion. The body remains at rest as long as the frictional force is strong enough to counteract the component of weight. However, when the slope of the plane is gradually increased, a stage comes when the body is just about to slide — this particular angle is called the angle of repose.

Definition

The angle of repose is defined as:
“The maximum angle of an inclined plane with the horizontal at which a body placed on it is just about to slide down due to its own weight.”

In other words, it is the angle between the inclined surface and the horizontal at which the body is on the verge of motion but not actually moving.

Mathematically, the angle of repose is related to the coefficient of friction (μ) as follows:

where,

θ = Angle of repose
μ = Coefficient of friction

This equation shows that the angle of repose depends entirely on the nature of the surfaces in contact and the coefficient of friction between them.

Derivation of the Relation Between Angle of Repose and Coefficient of Friction

Let a body of weight W be placed on an inclined plane making an angle θ with the horizontal.

The forces acting on the body are:

Weight (W) acting vertically downward.
Normal reaction (N) acting perpendicular to the plane.
Frictional force (F) acting parallel to the plane and opposite to the direction of motion.

Resolving the weight into two components:

W sin θ (parallel to the plane) — tends to move the body downward.
W cos θ (perpendicular to the plane) — acts towards the plane and is balanced by the normal reaction (N).

At the limiting condition (when the body is about to slide):

From equilibrium of forces,

and

Substituting the value of F = μN, we get,

Simplifying,

Hence,

Therefore, the angle of repose depends only on the coefficient of friction between the two surfaces.

Physical Meaning of Angle of Repose

The angle of repose gives a measure of how rough or smooth a surface is:

A higher angle of repose means a higher coefficient of friction, implying rough surfaces.
A smaller angle of repose means smoother surfaces with low friction.

For example:

A rough wooden block on another wooden surface has a larger angle of repose.
A steel block on a smooth steel plate has a smaller angle of repose.

The angle of repose determines the stability of materials or objects resting on inclined surfaces.

Example

If the coefficient of friction between a wooden block and an inclined plane is 0.5, find the angle of repose.

Given,

Thus, the angle of repose is 26.57°.

This means that if the inclined plane is tilted more than 26.57°, the block will start sliding down.

Applications of Angle of Repose

The concept of angle of repose is widely used in mechanical, civil, and material handling engineering. Some important applications are:

Design of Slopes and Embankments:
Used to determine the maximum safe slope of soil, sand, or gravel before it starts sliding.
Storage of Granular Materials:
When materials like coal, grains, or sand are poured, they naturally form a cone shape whose slope equals the angle of repose.
Inclined Plane Problems:
Helps in calculating the limiting condition for motion of a body on a slope.
Transportation Systems:
Used in conveyor design and chute systems to ensure material flow without blockage.
Mining and Earthwork:
Determines the stability of slopes and excavation walls.

Relation Between Angle of Repose and Angle of Friction

The angle of repose (θ) and the angle of friction (φ) are numerically equal.

From their definitions:

The angle of friction is the angle between the resultant of limiting friction and normal reaction.
The angle of repose is the maximum angle of inclination at which the body just begins to slide.

For both conditions, the body is on the verge of motion, and hence,

Thus, the angle of friction equals the angle of repose.

Factors Affecting the Angle of Repose

Nature of Material: Rougher materials have higher angles of repose.
Moisture Content: In granular materials, moderate moisture increases cohesion and the angle of repose.
Particle Size and Shape: Larger or irregular particles have greater friction and hence larger angles.
Surface Roughness: Smooth surfaces reduce the angle of repose.
Vibration and Handling: External disturbances can reduce the angle as particles rearrange and slide.

Practical Importance in Engineering

Ensures stability of slopes in civil constructions.
Helps determine the safe angle for dumping materials like sand or gravel.
Used in hopper and silo design to avoid clogging of materials.
Provides data for vehicle traction and road gradient design.

Conclusion

In conclusion, the angle of repose is the maximum inclination of an inclined plane with the horizontal at which a body just begins to slide. It represents the critical condition where the force of gravity equals the limiting frictional force. The relationship tan θ = μ shows that the angle of repose depends directly on the coefficient of friction. The concept is vital in mechanical, civil, and mining engineering for ensuring stability and controlling motion on inclined surfaces.