In this discussion, we explore the concept of moment of inertia, particularly for a solid sphere, which is characterized by a continuous mass distribution. A solid sphere is defined as having mass evenly distributed throughout its volume, unlike a hollow sphere where mass is concentrated at the edges. The moment of inertia (I) for a solid sphere is given by the formula:

I = \frac{2}{5} m r^2

where m is the mass and r is the radius of the sphere. In this scenario, we are provided with a radius of r = 8 \times 10^7 meters and a density of \rho = 10,000 kilograms per cubic meter. To find the mass m, we first need to calculate the volume V of the sphere using the formula:

V = \frac{4}{3} \pi r^3

Substituting the given radius into this equation allows us to compute the volume. Once we have the volume, we can find the mass using the relationship:

m = \rho V

By substituting the volume expression into the mass equation, we can express mass in terms of the radius:

m = 10,000 \left(\frac{4}{3} \pi r^3\right)

Next, we substitute this expression for mass back into the moment of inertia formula:

I = \frac{2}{5} \left(10,000 \left(\frac{4}{3} \pi r^3\right)\right) r^2

After simplifying, we find:

I = \frac{80,000 \pi}{15} r^5

Substituting the radius value into this equation yields a moment of inertia of approximately I \approx 5.49 \times 10^{43} kilograms meter squared. This calculation illustrates the process of deriving moment of inertia for a solid sphere by integrating concepts of density, volume, and mass distribution.