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

Options

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

Answer

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 height \( h \) and a base radius \( R \). The center of mass will lie along the axis of symmetry of the cone.

### Step 2: Set Up the Elemental Volume
Consider a thin disk of thickness \( dy \) at a distance \( y \) from the vertex of the cone. The radius of this disk, denoted as \( r \), can be determined using similar triangles.

### Step 3: Relate the Radius of the Disk to the Height
From the geometry of the cone, we can establish the relationship:
\[ \frac{r}{y} = \frac{R}{h} \implies r = \frac{R}{h} y \]

### Step 4: Calculate the Volume of the Disk
The volume \( dV \) of the thin disk is given by:
\[ dV = \pi r^2 dy = \pi \left(\frac{R}{h} y\right)^2 dy = \pi \frac{R^2}{h^2} y^2 dy \]

### Step 5: Calculate the Mass of the Disk
Assuming the density of the cone is \( \rho \), the mass \( dm \) of the disk is:
\[ dm = \rho dV = \rho \pi \frac{R^2}{h^2} y^2 dy \]

### Step 6: Find the Center of Mass Contribution
The y-coordinate of the center of mass of the disk is simply \( y \). Therefore, the contribution to the center of mass \( d(y_{cm}) \) from this disk is:
\[ d(y_{cm}) = y \cdot dm = y \cdot \left(\rho \pi \frac{R^2}{h^2} y^2 dy\right) \]

### Step 7: Integrate to Find the Total Center of Mass
The total center of mass \( y_{cm} \) of the cone is given by:
\[ y_{cm} = \frac{\int_0^h y \cdot dm}{\int_0^h dm} \]
We need to compute both the numerator and the denominator.

#### Numerator:
\[ \int_0^h y \cdot dm = \int_0^h y \cdot \left(\rho \pi \frac{R^2}{h^2} y^2 dy\right) = \rho \pi \frac{R^2}{h^2} \int_0^h y^3 dy = \rho \pi \frac{R^2}{h^2} \cdot \frac{h^4}{4} = \frac{\rho \pi R^2 h^4}{4h^2} = \frac{\rho \pi R^2 h^2}{4} \]

#### Denominator:
\[ \int_0^h dm = \int_0^h \rho \pi \frac{R^2}{h^2} y^2 dy = \rho \pi \frac{R^2}{h^2} \cdot \frac{h^3}{3} = \frac{\rho \pi R^2 h}{3} \]

### Step 8: Combine Results to Find \( y_{cm} \)
Now substituting back into the formula for \( y_{cm} \):
\[ y_{cm} = \frac{\frac{\rho \pi R^2 h^2}{4}}{\frac{\rho \pi R^2 h}{3}} = \frac{3h}{4} \]

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