Question

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

Short answer

a) Surface Area = \(4\pi a^2\), b) Centroid at \( \frac{4a}{3\pi}\), c) Moment of inertia = \(\frac{8\pi a^4\sigma}{3}\), d) Volume = \(\frac{4}{3}\pi a^3\), e) Centroid at \(\frac{3a}{8}\).

Step-by-step solution

- Surface Area of Sphere

The radius of the sphere is given as \(r = a\). The formula for the surface area of a sphere is given by \(4\pi r^2\). Substituting \(r = a\), the surface area \(A\) is:\[A = 4\pi a^2\]

- Centroid of Curved Surface Area of Hemisphere

For a hemisphere, the centroid of the curved surface area is located at a distance \( \frac{4a}{3\pi} \) from the flat base surface. Therefore, the centroid \(C\) from the base is:\[C = \frac{4a}{3\pi}\]

- Moment of Inertia of Spherical Shell

Given a constant area density \( \sigma \), the moment of inertia \(I\) of the spherical shell about a diameter can be derived from its formula for a thin spherical shell of radius \(a\): \[I = \frac{2}{3}Ma^2\] where \(M = 4\pi a^2 \sigma\) is the mass of the shell.

- Volume of the Ball

The volume \(V\) of a ball (solid sphere) of radius \(a\) is given by the formula: \[V = \frac{4}{3}\pi a^3\]

- Centroid of Solid Half Ball

For a solid hemisphere (half ball), the centroid is located at a distance \( \frac{3a}{8} \) from the flat base. Therefore, the centroid from the base is: \[C = \frac{3a}{8}\]

Key concepts

surface area of a sphere

The surface area of a sphere is a measure of how much exposed area it has. If a sphere has a radius denoted by \(r\), the surface area (A) can be given by the formula \(A = 4\pi r^2\).
To find it for a sphere where the radius \(r = a\), simply substitute \(a\) into the formula, resulting in: \[A = 4\pi a^2 \] .
This calculation is crucial in various fields including physics, engineering, and geometry as it helps determine the extent of a spherical object.

centroid of hemisphere

The centroid of a hemisphere is the average position of all the points on the curved surface, considering it is evenly distributed.
For a hemisphere with radius \(a\), the centroid of its curved surface is located at a distance from its flat base. This distance is derived using integration and is given by: \[C = \frac{4a}{3\pi} \] .
This result tells us where the ‘center of mass’ of the curved surface lies, which is valuable in design and analysis of structures and objects.

moment of inertia of spherical shell

The moment of inertia of a spherical shell relates to how the mass is distributed around its axis of rotation.
Assuming a constant area density \sigma\, the moment of inertia \(I\) for a thin spherical shell of radius \(a\) about a diameter is given by: \[I = \frac{2}{3}Ma^2 \] .
Here, \(M = 4\pi a^2 \sigma \) represents the mass of the spherical shell.
This calculation is crucial in physics and engineering as it affects how the shell will resist rotational motion, impacting designs of mechanical and structural components.

volume of sphere

The volume of a sphere is the amount of space it occupies. For a sphere of radius \r\, the volume \(V\) can be calculated using the formula: \[V = \frac{4}{3} \pi r^3 \] .
When \(r=a\), substitute \(a\) into the formula to get: \[V = \frac{4}{3} \pi a^3 \] .
This information is widely used in different fields whether it's calculating quantities, design considerations, or understanding physical properties of spherical objects.

centroid of solid hemisphere

Determining the centroid of a solid hemisphere (or half ball) involves finding the average position of its mass.
For a solid hemisphere with radius \(a\), the centroid is located at a distance from the flat base. This distance can be determined using integration and is given by: \[C = \frac{3a}{8} \] .
This tells us the exact 'balance point' or center of mass for the solid half ball, an essential concept for structural stability and balance in various applications.