Animated Solution for Physics - System of Particles: The centre of mass of a solid hemisphere of radius $8\text{ cm}$ is $x\text{ cm}$ from the centre of the flat surface. Then, value of $x$ is …… .

The journey to mastering the center of mass often involves visualizing standard geometric shapes and remembering their key properties. In this problem, we are dealing with a classic: the solid hemisphere.

Imagine a perfectly uniform solid sphere, like a heavy bowling ball, sliced exactly in half. The flat circular face rests on a table. Where would its balancing point be? Intuitively, because there is more mass concentrated near the flat base than near the curved top, the center of mass won't be exactly halfway up. It will be pulled closer to the base.

For a uniform solid hemisphere of radius $R$, the center of mass lies on its central axis of symmetry. The exact distance $d$ from the center of the flat base is given by a standard result derived using calculus:

This is a high-yield formula for JEE. It is crucial to distinguish this from a hollow hemispherical shell, where the center of mass is located exactly halfway up at $\frac{R}{2}$.

The problem provides us with the radius of the solid hemisphere, $R=8\text{ cm}$. We simply need to substitute this value into our formula to find the distance $x$.

The $8$ in the numerator and the denominator perfectly cancel each other out, leaving us with:

The problem states that this distance is $x\text{ cm}$. By comparing our result, we can confidently conclude that the value of $x$ is $3$.