expresion for hollo sphere of momen of inertia
The moment of inertia, usually denoted as "I," is a measure of how the mass in an object is distributed around its axis of rotation. For a hollow sphere, the moment of inertia depends on both the mass of the sphere and its radius.
To derive the expression for the moment of inertia of a hollow sphere, you can use an integral approach. First, let's define the following variables:
- "r" as the radius of the hollow sphere.
- "M" as the mass of the hollow sphere.
- "dm" as a small mass element on the hollow sphere's surface.
- "R" as the distance of the small mass element "dm" from the axis of rotation.
The moment of inertia of a hollow sphere can be obtained by integrating the moment of inertia of each small mass element over the entire sphere. The moment of inertia of a small mass element, "dI," can be expressed as follows:
dI = dm * R^2
The mass element "dm" can be expressed in terms of the surface area element "dA" and the mass distribution per unit area "σ." Since it is a hollow sphere, the mass distribution is constant across its surface, so "σ" can be considered a constant value:
dm = σ * dA
To calculate the surface area element "dA," we can consider an infinitesimally thin strip on the sphere's surface. The length of this strip, "dl," can be expressed as:
dl = R * dθ
The surface area element "dA" can thus be calculated by integrating the length element "dl" over the entire circumference of the sphere:
dA = ∫ dl = ∫ R * dθ = 2πR * dθ
Substituting dm and dA back into the equation for dI, we get:
dI = σ * dA * R^2
= σ * (2πR * dθ) * R^2
= 2πR^3σ * dθ
To find the moment of inertia of the hollow sphere, we need to integrate dI over the entire surface. Since the mass distribution is uniform, the value of "σ" can be expressed as the total mass of the hollow sphere divided by its surface area:
σ = M / (4πR^2)
Substituting this expression for σ, we can solve the integral:
I = ∫ dI
= ∫ 2πR^3σ * dθ
= ∫ 2πR^3 (M / (4πR^2)) * dθ
= (M/2) ∫ R^3 dθ
= (M/2) * R^3 ∫ dθ
= (M/2) * R^3 [θ]
Evaluating the integral and substituting back the values, we get the expression for the moment of inertia of a hollow sphere:
I = (M/2) * R^3 * [θ]
In this expression, [θ] represents the change in angle between the initial and final position of the hollow sphere's axis of rotation.