The greatest height h of the sand pile that can be erected without spilling the sand onto the surrounding circular area of radius R (If `mu` is the coefficient of friction between sand particles) is

Options

A. R
B. `(R )/(mu)`
C. `muR`
D. `mu^(2)R`

Explanation

To find the greatest height \( h \) of the sand pile that can be erected without spilling the sand onto the surrounding circular area of radius \( R \), we can use the relationship between the height of the pile, the radius, and the coefficient of friction \( \mu \).

### Step-by-Step Solution:

1. **Understanding the Geometry**:
The sand pile forms a conical shape. The height of the cone is \( h \), and the radius of the base is \( R \).

2. **Identifying the Angle of the Cone**:
The angle \( \theta \) of the cone can be related to the height and radius using the tangent function:
\[ \tan \theta = \frac{h}{R} \]

3. **Using the Coefficient of Friction**:
The coefficient of friction \( \mu \) between the sand particles gives us the maximum angle of repose, which is the angle at which the sand can be piled without sliding. This is given by:
\[ \tan \theta = \mu \]

4. **Setting the Two Equations Equal**:
Since both expressions represent \( \tan \theta \), we can set them equal to each other:
\[ \frac{h}{R} = \mu \]

5. **Solving for Height \( h \)**:
Rearranging the equation to solve for \( h \):
\[ h = \mu R \]

### Final Answer:
The greatest height \( h \) of the sand pile that can be erected without spilling the sand onto the surrounding circular area of radius \( R \) is:
\[ h = \mu R \]