Moment of Inertia of a Hollow Sphere

The Moment of Inertia, as well known as the mass moment of inertia, rotational inertia, or angular mass of a rigid body, is really a quantity that determines the torque needed for a desired angular acceleration around a rotational axis, in the same way, that the mass determines the force required for the desired acceleration. 

This seems to be entirely dependent on the body’s mass distribution and the axis selected, with larger moments necessitating more torque to change the body’s rate of rotation.

The moment of inertia of a hollow sphere or spherical shell has been frequently calculated using the following formula: I = MR²

Derivation of Hollow Sphere Formula

Before we start, let us recall or consider the Moment of Inertia of a Circle, which is given by,

I = mr²

When applying the differential analysis, we get;

dl = r² dm

Then, we need to find the dm value with the formula,

dm = dA

In which A is the total surface area of the shell, which is given as 4πR², and dA is said to be the area of the ring that is formed by differentiation and is expressed as;

dA = R dθ × 2πr

Here R dθ is considered the thickness and 2πr is the circumference of the ring.

Remember that, we get R dθ from the equation of arc length, S = R θ

We have, sin θ = = r = R sinθ

Then, dA becomes as below.

dA = 2πR²sinθ dθ

When substituting the equation for dA into dm, we get,

dm = dθ

At this point, we will substitute the above equation and for ‘r’ into the equation for ‘dI.’ Then we get;

dm = sin³ θ dθ

So, integrating within the limits of 0 to π radians from one end to another, we get;

I = sin³ θ dθ

Then, we need to split sin³θ into two, because it depicts the case of integral of odd powered trigonometric functions. So, we get;

I = sin² θ sin θ dθ

We have, sin² θ is given as sin² θ = 1- cos² θ.

Thus,

I = (1- cos² θ) sin θ dθ

We could use substitution after this, where u = cos θ, we will get;

I = u² – 1 du

We need to carry out the integration as:

I = u² – 1 du,

In this, the integral of u² du = u and the integral of 1 du = u

When substituting the values, we get.

I = {₁^-1 – u₁^-1

I = {[ (-1)³ -1³] − − 1−1}

I = {– −2}

I = { +2}

I = {}

I = x

I = MR²

Hollow Sphere Formula - Volume and Surface Area

Understanding a Hollow Sphere

A hollow sphere is a three-dimensional geometric shape resembling a spherical shell, characterized by an outer and an inner radius. Unlike a solid sphere, a hollow sphere has a cavity at its center, making it a common structure in various scientific and engineering applications.

Key Parameters of a Hollow Sphere

• Outer Radius (R): The distance from the center to the outer surface.
• Inner Radius (r): The distance from the center to the inner surface.
• Thickness (t): The difference between the outer and inner radii, calculated as t = R - r.

Formulas for Volume and Surface Area

1. Volume of a Hollow Sphere

The volume is determined by subtracting the volume of the inner sphere from that of the outer sphere:

V = (4/3)πR³ - (4/3)πr³

Simplifying:

V = (4/3)π(R³ - r³)

2. Surface Area of a Hollow Sphere

The total surface area comprises both the outer and inner surfaces:

A = 4πR² + 4πr²

Simplifying:

A = 4π(R² + r²)

Example Calculation

Consider a hollow sphere with an outer radius R = 5 cm and an inner radius r = 3 cm:

Volume Calculation

V = (4/3)π(5³ - 3³)

V = (4/3)π(125 - 27)

V = (4/3)π × 98

V ≈ 410.5 cm³

Surface Area Calculation

A = 4π(5² + 3²)

A = 4π(25 + 9)

A = 4π × 34

A ≈ 427.3 cm²

Applications of Hollow Spheres

• Engineering: Used in designing lightweight yet strong structures, such as pressure vessels and tanks.
• Astronomy: Models of celestial bodies like planets and stars often consider them as hollow spheres for simplification.
• Manufacturing: Hollow metal spheres are utilized in ball bearings and other mechanical components.

Key Takeaways

• The volume of a hollow sphere is calculated by subtracting the volume of the inner sphere from the outer sphere.
• The total surface area includes both the inner and outer surfaces.
• Understanding these formulas is essential in fields like physics, engineering, and manufacturing.

By mastering these concepts, one can effectively analyze and design structures involving hollow spheres, ensuring both efficiency and functionality.