Understanding Centre of Mass for Hollow and Solid Hemispheres

The centre of mass of a hemisphere is a key concept in mechanics, determining the point at which the entire mass of the hemisphere can be considered to act for various physical calculations. The position of this point differs for solid and hollow hemispheres due to their different mass distributions.

Understanding Centre of Mass for Hemispheres

A solid hemisphere has mass uniformly distributed throughout its volume, while a hollow hemisphere has mass distributed only over its curved surface. The calculations for their centres of mass rely on symmetry and integration along the axis perpendicular to the flat base.

For both types, the reference axis is chosen from the centre of the flat base, extending upward along the symmetry axis. The nature of mass distribution directly affects the centre of mass location relative to the base.

Centre of Mass of Solid Hemisphere: Derivation

Consider a uniform solid hemisphere of radius $R$ and total mass $M$. The centre of mass lies along the symmetry axis, above the base. Using calculus, elemental discs parallel to the base are integrated to find the net position.

For a disc at height $z$ from the base, the volume element is $dV = \pi r^2 dz$, where $r^2 = R^2 - z^2$. The mass element is $dm = \rho dV$ with uniform density $\rho$.

The $z$-coordinate of the centre of mass is:

$z_{cm} = \dfrac{1}{M} \int z\, dm$

Evaluating this integral for limits $z = 0$ to $z = R$ gives:

$z_{cm} = \dfrac{3R}{8}$

Thus, for a uniform solid hemisphere, the centre of mass is at a distance $\dfrac{3R}{8}$ from the flat base along the axis of symmetry.

Concepts around the centre of mass are connected with broader mechanical topics, as explained in Centre Of Mass Explained.

Centre of Mass of Hollow Hemisphere: Derivation

For a uniform hollow hemisphere (hemispherical shell) of radius $R$ and mass $M$, all the mass resides on the curved surface. The centre of mass will also lie on the symmetry axis, but at a different distance from the base compared to the solid case.

Each elemental mass lies at a distance $z = R \cos \theta$ above the base, where $\theta$ is the polar angle with respect to the axis. Integration is carried out over the curved surface.

The $z$-coordinate of the centre of mass is:

$z_{cm} = \dfrac{1}{M} \int z\, dm = \dfrac{R}{2}$

Therefore, for a uniform hollow hemisphere, the centre of mass is located at $R/2$ above the flat base, along the axis of symmetry.

Comparison of Solid and Hollow Hemisphere

The formula for the centre of mass differs due to the mode of mass distribution. For a solid hemisphere, mass fills the volume; for a hollow hemisphere, mass is present only on the shell. This difference influences the height above the base where the centre of mass is located.

Co-ordinates and Reference System

For hemispheres placed on the xy-plane with the geometric centre at the origin, the centre of mass position in coordinates is $(0, 0, 3R/8)$ for the solid hemisphere and $(0, 0, R/2)$ for the hollow hemisphere.

Accurate reference axis and correct use of formula are essential in problems involving these shapes.

For related rotational aspects, see Moment Of Inertia Overview.

Practical Points and Common Mistakes

Students sometimes confuse the formula for the centre of mass of a full sphere with that of a hemisphere. Only half the mass distribution is present in a hemisphere, shifting the centre of mass location closer to the base.

Confusing the hollow and solid results, or using an incorrect axis of reference, are frequent errors. Always clarify whether the question refers to solid or hollow hemisphere and measure distances from the flat base.

The centre of mass always lies inside the material volume or on the shell, not outside the physical boundary of the hemisphere.

For comparison with other shells and discs, the article Moment Of Inertia Of Disc offers useful reference points.

Solved Example: Finding the Centre of Mass

A uniform solid hemisphere of radius $10\ \text{cm}$ has centre of mass at a distance $z = \dfrac{3}{8}\times10 = 3.75\ \text{cm}$ from the base. For a hollow hemisphere of the same radius, the location is $z = \dfrac{1}{2}\times10 = 5\ \text{cm}$ from the base.

Key Reminders for JEE Preparation

• Always distinguish between hollow and solid hemispheres
• Centre of mass for solid = $3R/8$ from base
• Centre of mass for hollow = $R/2$ from base
• Choose reference axis carefully
• Results apply only for uniform hemispheres

For composite systems, such as combining a solid hemisphere with a hollow cone, calculate the centre of mass of each and apply the weighted average principle. Related topics can be found in Moment Of Inertia Of Semihollow Cone.

Summary Table: Formulae for Centre of Mass

A clear understanding of the centre of mass in hemispherical shapes is necessary for tackling various JEE problems in mechanics and rotational motion, as further detailed in Moment Of Inertia Of Hollow Sphere.