The centre of mass of a solid cone along the line form the center of the base to the vertex is at

Options

A. One-fourth of the height
B. One-third of the height
C. One-fifth of the height
D. None of these

Answer

To find the center of mass of a solid cone along the line from the center of the base to the vertex, we can follow these steps:

### Step 1: Understand the Geometry of the Cone
- A solid cone has a circular base and a vertex. Let the radius of the base be \( R \) and the height of the cone be \( H \).

### Step 2: Set Up the Coordinate System
- Place the cone in a coordinate system such that the vertex is at the origin (0,0) and the base is at height \( H \) along the y-axis.

### Step 3: Consider a Differential Element
- To find the center of mass, consider a thin disk of thickness \( dy \) at a height \( y \) from the vertex. The radius of this disk can be expressed in terms of \( y \).

### Step 4: Relate the Radius of the Disk to the Height
- Using similar triangles, the radius \( r \) of the disk at height \( y \) can be expressed as:
\[ r = \frac{R}{H}y \]

### Step 5: Calculate the Volume of the Differential Element
- The volume \( dV \) of the thin disk is given by:
\[ dV = \pi r^2 dy = \pi \left(\frac{R}{H}y\right)^2 dy = \frac{\pi R^2}{H^2} y^2 dy \]

### Step 6: Find the Mass of the Differential Element
- If \( \rho \) is the density of the cone, the mass \( dm \) of the disk is:
\[ dm = \rho dV = \rho \frac{\pi R^2}{H^2} y^2 dy \]

### Step 7: Calculate the Moment about the Vertex
- The moment of the differential mass about the vertex is given by:
\[ dM = y \cdot dm = y \cdot \rho \frac{\pi R^2}{H^2} y^2 dy = \rho \frac{\pi R^2}{H^2} y^3 dy \]

### Step 8: Integrate to Find Total Mass
- The total mass \( M \) of the cone is:
\[ M = \int_0^H dm = \int_0^H \rho \frac{\pi R^2}{H^2} y^2 dy = \rho \frac{\pi R^2}{H^2} \left[\frac{y^3}{3}\right]_0^H = \rho \frac{\pi R^2 H}{3} \]

### Step 9: Integrate to Find the Total Moment
- The total moment \( M_{cm} \) about the vertex is:
\[ M_{cm} = \int_0^H dM = \int_0^H \rho \frac{\pi R^2}{H^2} y^3 dy = \rho \frac{\pi R^2}{H^2} \left[\frac{y^4}{4}\right]_0^H = \rho \frac{\pi R^2 H^3}{4} \]

### Step 10: Find the Center of Mass
- The center of mass \( y_{cm} \) is given by:
\[ y_{cm} = \frac{M_{cm}}{M} = \frac{\rho \frac{\pi R^2 H^3}{4}}{\rho \frac{\pi R^2 H}{3}} = \frac{3H}{4} \]

Thus, the center of mass of the solid cone along the line from the center of the base to the vertex is located at \( \frac{H}{4} \) from the base.