Unit 3.7 – Angle of Repose

What is the Angle of Repose?

• Key Idea: The angle of repose is determined by the balance between the force of gravity pulling the material downward and the frictional force resisting its motion.

Factors Affecting the Angle of Repose

1. Nature of the Material:
   • Materials with rougher surfaces (e.g., sand) have a higher angle of repose because they experience more friction.
   • Smooth or slippery materials (e.g., fine powder) have a lower angle of repose.
2. Particle Size and Shape:
   • Larger, irregularly shaped particles tend to interlock, increasing the angle of repose.
   • Smaller, spherical particles slide more easily, reducing the angle of repose.
3. Moisture Content:
   • Dry materials have a lower angle of repose.
   • Adding moisture increases cohesion between particles, raising the angle of repose. However, too much moisture can reduce friction and cause the material to flow like a liquid.
4. External Forces:
   • Vibrations or disturbances can reduce the angle of repose by breaking the frictional bonds between particles.

Formula for the Angle of Repose

• $\theta$ = Angle of repose.
• ${\mu }_s$ = Coefficient of static friction.

Real-Life Examples of the Angle of Repose

1. Sand Piles:
   • When sand is poured onto a flat surface, it forms a conical pile. The slope of the pile corresponds to the angle of repose of the sand.
2. Landslides:
   • The angle of repose determines the stability of slopes in mountains or hills. If the slope exceeds the angle of repose, a landslide may occur.
3. Agriculture:
   • Grains stored in silos or bins form a natural slope at the angle of repose.
4. Construction:
   • Engineers consider the angle of repose when designing embankments, dams, and retaining walls to ensure stability.

Examples to Understand the Angle of Repose

Example 1:

• Solution: Using the formula:
  $$\theta ={\tan }^{-1}({\mu }_s)$$
  Substituting values:
  $$\theta ={\tan }^{-1}(0.6)\approx {31}^{\circ }\mathrm{.}$$
  The angle of repose is approximately ${31}^{\circ }$.

Example 2:

• Solution: The angle of repose is the angle between the sloping side of the cone and the horizontal base. Using trigonometry:
  $$\tan \theta =\frac{\text{Height}}{\text{Radius}}$$
  The radius of the base is half the diameter:
  $$\text{Radius}=\frac{6}{2}=3\text{m}\mathrm{.}$$
  Substituting values:
  $$\tan \theta =\frac{2}{3}\mathrm{.}$$
  Taking the inverse tangent:
  $$\theta ={\tan }^{-1}\left(\frac{2}{3}\right)\approx {33.7}^{\circ }\mathrm{.}$$
  The angle of repose is approximately ${33.7}^{\circ }$.

Applications of the Angle of Repose

1. Civil Engineering:
   • Used to design stable slopes for roads, railways, and buildings.
2. Mining:
   • Helps determine the safe stacking of mined materials like coal or ore.
3. Geology:
   • Predicts the likelihood of landslides based on the angle of repose of soil and rock.
4. Agriculture:
   • Ensures proper storage of grains and other materials in silos and bins.

Practice Questions

1. The coefficient of static friction between two surfaces is $0.8$. What is the angle of repose?
   • Answer:
   $$\theta ={\tan }^{-1}(0.8)\approx {38.7}^{\circ }\mathrm{.}$$
2. Why does adding water to sand increase its angle of repose?
   • Answer:
     Water increases cohesion between sand particles, enhancing friction and raising the angle of repose.
3. A pile of soil forms a cone with a base radius of $4\text{m}$ and a height of $3\text{m}$. What is the angle of repose?
   • Answer: Using $\tan \theta =\frac{\text{Height}}{\text{Radius}}$:
     $$\tan \theta =\frac{3}{4}\mathrm{.}$$
     $$\theta ={\tan }^{-1}\left(\frac{3}{4}\right)\approx {36.9}^{\circ }\mathrm{.}$$
4. Why is the angle of repose important in landslide prevention?
   • Answer:
     If the slope of a hill exceeds the angle of repose of the soil, it becomes unstable and may result in a landslide.

Quick Recap

1. Angle of Repose: Maximum angle at which a material can rest without sliding.
2. Formula: $\theta ={\tan }^{-1}({\mu }_s)$.
3. Factors: Material type, particle size, moisture content, external forces.
4. Applications: Civil engineering, mining, geology, agriculture.