Question

For a sphere of radius \(a\) find by integration (a) its volume; (b) its surface area: (c) the centroid of a solid hemisphere; (d) the centroid of the curved surface area of the hemisphere; (e) the moment of inertia of the whole spherical shell (that is, surface) about a diameter. (assuming constant density).

Short answer

Volume: \( \frac{4}{3} \pi a^3 \), Surface Area: \( 4 \pi a^2 \), Centroid of Solid Hemisphere: \( \frac{3a}{8} \), Centroid of Curved Surface Hemisphere: \( \frac{a}{2} \), Moment of Inertia: \( \frac{2}{3} M a^2 \).

Step-by-step solution

Volume of the Sphere

To find the volume of the sphere, use the volume integral in spherical coordinates. \[ V = \frac{4}{3} \pi a^3 \]

Surface Area of the Sphere

To find the surface area of the sphere, use the surface area formula: \[ A = 4 \pi a^2 \]

Centroid of a Solid Hemisphere

To find the centroid of a solid hemisphere of radius \(a\), integrate considering the symmetric axis. The centroid is located at \( \frac{3a}{8} \) from the center along the vertical axis.

Centroid of the Curved Surface Area of the Hemisphere

To find the centroid of the curved surface area of a hemisphere, consider the integration over the hemisphere surface. The centroid is at \( \frac{a}{2} \) from the flat surface along the vertical axis.

Moment of Inertia of the Whole Spherical Shell About a Diameter

For a spherical shell of constant density \( \rho \), use the moment of inertia formula for a spherical shell. The moment of inertia about a diameter is \( \frac{2}{3} M a^2 \) where \(M\) is the mass of the sphere.

Key concepts

Volume Integral

To find the volume of a sphere, we typically use integral calculus. In spherical coordinates, the volume integral helps to account for the symmetrical nature of the sphere. The general formula for the volume of a sphere of radius \(a\) is:

\[ V = \frac{4}{3} \pi a^3 \]

Here’s a reason why this formula works:

• We start with the basic volume element in spherical coordinates, which is \( dV = r^2 \sin \theta \ dr \ d\theta \ d\phi \).


• Integrating for \(r\) from 0 to \(a\), for \(\theta\) from 0 to \(\pi\), and for \(\phi\) from 0 to \(2\pi\), we calculate the sum of those infinitesimal volumes, giving us the total volume.

Integrating all together leads to the familiar formula. While it may be more complex than using the known formula, understanding the integral provides a deeper insight into the nature of the sphere.

Surface Area Formula

The surface area of a sphere is another common calculation in geometry and calculus. It can be thought of as integrating along the surface to account for its curvature. The formula for the surface area of a sphere of radius \(a\) is:

\[ A = 4 \pi a^2 \]

Here’s why this formula comes out this way:

• The surface element in spherical coordinates is \( dA = r^2 \sin \theta \ d\theta \ d\phi \).


• Since we are dealing with a radius \(a\), substitute \(r\) with \(a\) and integrate \(\theta\) from 0 to \(\pi\) and \(\phi\) from 0 to \(2\pi\).

This integral will lead to a result of \(4\pi a^2\), giving us the complete surface area of our sphere.

Centroid Calculation

A centroid represents the center of mass or balance point of an object. For a solid hemisphere, finding the centroid involves a bit more intricate integration over its volume.

For a solid hemisphere of radius \(a\), the centroid lies along the vertical axis passing through the sphere’s flat base.

• By symmetry, the horizontal center of mass is directly at the centerline, so we consider the vertical axis.


• The vertical coordinate of the centroid for a solid hemisphere can be derived to be \( \frac{3a}{8} \) from the flat base along the vertical axis.

Similarly, for the curved surface area of the hemisphere,

• Considering symmetry and distributing the surface uniformly, the centroid is \( \frac{a}{2} \) from the flat base along the vertical axis.

These centroids are vital in understanding the distribution of mass within hemispherical objects.

Moment of Inertia

The moment of inertia (MOI), also known as rotational inertia, quantifies the rotational counterpart to mass. For a spherical shell, it’s about how the mass is distributed relative to its rotational axis.

• For a solid sphere of radius \(a\) and mass \(M\), the moment of inertia about a diameter is derived from integrating rotational contributions of infinitesimal mass elements.


• The formula for the moment of inertia is:
  
  \[ I = \frac{2}{3} Ma^2 \]

This integral essentially sums up the contributions from every possible rotational element.

• The constant \(\frac{2}{3}\) comes from these calculated integration limits.


• Understanding the MOI helps in predicting the sphere's angular acceleration under various forces.

Calculating moments of inertia is crucial in fields like mechanical engineering and physics where rotational motions are analyzed in depth.