Sand is pilled up on a horizontal ground in the form of a regular cone of a fixed base radius R. The coefficient of static friction between sand layers is `mu.` The maximum volume of sand that can be pilled up, without the sand slipping on the surface is

Options

A. `(mu R ^(3))/(3pi)`
B. `(mu R ^(3))/(3)`
C. `(mu R ^(3))/(3mu)`
D. `(mu pi R ^(3))/(3)`

Answer

To solve the problem of finding the maximum volume of sand that can be piled up in the form of a regular cone without slipping, we can follow these steps:

### Step 1: Understand the Geometry of the Cone
We have a cone with a fixed base radius \( R \) and height \( h \). The cone is formed by piling up sand, and we need to ensure that the sand does not slip over the slant surface of the cone.

**Hint:** Visualize the cone and identify the parameters: base radius \( R \) and height \( h \).

### Step 2: Analyze the Forces Acting on the Sand Layers
For the sand layers to remain stable and not slip, the angle of the slant surface of the cone must be such that the static friction can hold the layers in place. This means we need to relate the angle of the cone to the coefficient of static friction \( \mu \).

**Hint:** Recall that the condition for no slipping is given by the relationship between the angle of the cone and the coefficient of static friction.

### Step 3: Relate the Angle of the Cone to the Coefficient of Friction
Let \( \theta \) be the angle of the slant surface of the cone. The tangent of this angle can be expressed as:
\[ \tan(\theta) = \frac{h}{R} \]
For the sand not to slip, we need:
\[ \tan(\theta) \leq \mu \]
Thus, we have:
\[ \frac{h}{R} \leq \mu \]

**Hint:** This inequality tells us how high the cone can be relative to its radius.

### Step 4: Solve for the Maximum Height \( h \)
From the inequality \( \frac{h}{R} \leq \mu \), we can rearrange it to find the maximum height:
\[ h \leq \mu R \]

**Hint:** This gives us the maximum height at which the sand can be piled without slipping.

### Step 5: Calculate the Maximum Volume of the Cone
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi R^2 h \]
Substituting the maximum height \( h = \mu R \) into the volume formula, we get:
\[ V_{\text{max}} = \frac{1}{3} \pi R^2 (\mu R) = \frac{1}{3} \pi \mu R^3 \]

**Hint:** This is the final expression for the maximum volume of sand that can be piled up.

### Final Answer
The maximum volume of sand that can be piled up without slipping is:
\[ V_{\text{max}} = \frac{1}{3} \pi \mu R^3 \]