Animated Solution for Physics - System of Particles: 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

Imagine a solid uniform cone resting on its flat circular base. Let's mark its highest point, the vertex, as $Y$, and the center of its circular base as $O$. The total vertical height of this cone is given as $h$. Our goal is to find the exact location of its center of mass (CM) and specifically measure its distance from the vertex, which the problem denotes as $z_0$.

For standard geometric shapes, it is incredibly useful to memorize the positions of their centers of mass. Because a solid cone is perfectly symmetric around its central vertical axis, its center of mass must lie somewhere along this axis.

Through the process of integration (summing up infinitesimally thin circular disks from the base to the vertex), we find that the center of mass of a solid uniform cone is located at a height of $\frac{h}{4}$ from its flat base.

Now, let's look at the geometry of the situation. The total height $h$ of the cone is simply the sum of two segments along the central axis: the distance from the vertex to the center of mass ($z_0$), and the distance from the center of mass to the base.

Mathematically, we can write this as:

Substituting the standard result we just recalled:

To isolate $z_0$, we simply subtract $\frac{h}{4}$ from the total height $h$:

Solid vs Hollow

A classic trap in exams like JEE is swapping the word "solid" for "hollow". If the cone were hollow (like an empty ice cream cone), the mass would only be distributed along its slanted surface. In that case, the center of mass is located slightly higher, at a distance of $\frac{h}{3}$ from the base. Consequently, its distance from the vertex would be $h-\frac{h}{3}=\frac{2h}{3}$. Always read the problem statement carefully to identify the exact nature of the object!