Derivation of the Center of Mass of a Hollow Hemisphere

The center of mass of a hollow hemisphere is an important concept in physics and engineering. In this article, we will explore how to derive this point using the principles of symmetry and calculus. We will break down the derivation into several steps, making the process clear and understandable.

Understanding the Geometry

A hollow hemisphere can be visualized as a thin shell with a uniform thickness.

A hollow hemisphere of radius R centered at the origin with the flat face lying on the XY-plane.

Symmetry Considerations

Due to the symmetry of the hollow hemisphere about the z-axis, the center of mass will lie along the z-axis. Therefore, we only need to find the z-coordinate of the center of mass, denoted as zcm.

Element of Mass

To find the center of mass, we can consider a differential mass element dm on the surface of the hemisphere. Using spherical coordinates, we have: x R sin θ cos φ y R sin θ sin φ z R cos θ where θ is the polar angle from the positive z-axis, 0 ≤ θ ≤ (frac{π}{2}), and φ is the azimuthal angle around the z-axis, 0 ≤ φ ≤ 2π.

Surface Area Element

The differential surface area element dA on the surface of the hemisphere is given by:

dA R^2 sin θ dθ dφ

Differential Mass Element

If the surface density of the hollow hemisphere is σ, the differential mass element dm is:

dm σ dA σ R^2 sin θ dθ dφ

Calculating the z-coordinate of the Center of Mass

The z-coordinate of the center of mass zcm is given by the formula:

z_{cm} (frac{1}{M}) (int z dm)

where M is the total mass of the hemisphere. Substituting for z and dm gives us:

z_{cm} (frac{1}{M}) (int_0^{2π} int_0^{frac{π}{2}} R cos θ (σ R^2 sin θ dθ dφ))

Total Mass

The total mass M of the hemisphere is:

M (int_0^{2π} int_0^{frac{π}{2}} σ R^2 sin θ dθ dφ)

Calculating this integral:

M σ R^2 (int_0^{2π} dφ int_0^{frac{π}{2}} sin θ dθ) σ R^2 2π (left[-cos θright]_0^{frac{π}{2}} σ R^2 2π 1 2π σ R^2)

Evaluating the Integral for zcm

Now we return to our integral for zcm:

z_{cm} (frac{1}{M}) (int_0^{2π} dφ int_0^{frac{π}{2}} R cos θ (σ R^2 sin θ dθ))

Calculating the integral:

z_{cm} (frac{1}{2π σ R^2}) (int_0^{2π} dφ int_0^{frac{π}{2}} R cos θ (σ R^2 sin θ dθ))

(frac{R^3}{2π σ R^2}) (2π left(frac{1}{2}right))

Thus, z_{cm} (frac{R}{4}).

Final Result

Therefore, the center of mass of a hollow hemisphere is located at:

(boxed{frac{R}{2}}) above the base of the hemisphere along the z-axis.

This result provides a clear understanding of where the center of mass of a hollow hemisphere is located, which is an important consideration in various applications of physics and engineering.