Distance of the centre of mass of a solid uniform cone from its vertex is `z_0` . If the radius of its base is R and its height is h then `z_0` is equal to:

Options

A. `(h^2)/(4R)`
B. `(3h)/(4)`
C. `(5h)/(8)`
D. `(3h^2)/(8R)`

Explanation

To find the distance of the center of mass of a solid uniform cone from its vertex, we can follow these steps:

### Step 1: Understand the Geometry of the Cone
We have a solid uniform cone with a base radius \( R \) and height \( h \). The vertex of the cone is at the origin, and we need to find the distance \( z_0 \) from the vertex to the center of mass.

### Step 2: Set Up the Integral for Center of Mass
The center of mass \( z_0 \) can be calculated using the formula:
\[ z_0 = \frac{1}{M} \int z \, dm \]
where \( M \) is the total mass of the cone and \( dm \) is the mass element.

### Step 3: Define the Mass Element
To find \( dm \), we consider a thin horizontal disk of thickness \( dy \) at a height \( y \) from the vertex. The radius of this disk \( r \) can be expressed in terms of \( y \):
\[ r = \frac{R}{h} y \]
The volume \( dV \) of the disk is:
\[ dV = \pi r^2 dy = \pi \left(\frac{R}{h} y\right)^2 dy = \frac{\pi R^2}{h^2} y^2 dy \]
The mass element \( dm \) is then given by:
\[ dm = \rho dV = \rho \frac{\pi R^2}{h^2} y^2 dy \]
where \( \rho \) is the density of the cone.

### Step 4: Calculate the Total Mass \( M \)
The total mass \( M \) of the cone is given by:
\[ M = \rho V = \rho \left(\frac{1}{3} \pi R^2 h\right) \]

### Step 5: Substitute \( dm \) and \( M \) into the Center of Mass Formula
Now we can substitute \( dm \) and \( M \) into the center of mass formula:
\[ z_0 = \frac{1}{M} \int_0^h y \, dm = \frac{1}{\rho \left(\frac{1}{3} \pi R^2 h\right)} \int_0^h y \left(\rho \frac{\pi R^2}{h^2} y^2 dy\right) \]
This simplifies to:
\[ z_0 = \frac{3}{h} \int_0^h y^3 dy \]

### Step 6: Evaluate the Integral
Now we evaluate the integral:
\[ \int_0^h y^3 dy = \left[\frac{y^4}{4}\right]_0^h = \frac{h^4}{4} \]
Substituting this back gives:
\[ z_0 = \frac{3}{h} \cdot \frac{h^4}{4} = \frac{3h^3}{4} \]

### Step 7: Final Calculation
Now, simplifying gives:
\[ z_0 = \frac{3h}{4} \]

Thus, the distance of the center of mass of the solid uniform cone from its vertex is:
\[ z_0 = \frac{3h}{4} \]

### Final Answer
\[ \boxed{\frac{3h}{4}} \]