Question

A sphere with radius 0.210 m has density rho that decreases with distance r from the center of the rho=3.25×10³kg/m³ −(9.50×10³kg/m⁴)r

Calculate the total mass of the sphere

Answer 1

To calculate the mass of a sphere with a radius of 0.210 m and a density that decreases with distance from the center, one must integrate the given density function over the volume of the sphere using spherical coordinates.

Step-by-step explanation:

The student has asked to calculate the total mass of a sphere where the density decreases with the distance from the center. The provided density function is ρ=3.25×10³ kg/m³ − (9.50×10³ kg/m⁴)r. To find the mass, we integrate the density function over the volume of the sphere.

Mass Calculation

We start by expressing the mass as an integral:

M = ∫∫∫ ρ dV, where dV is the differential volume element.

For a sphere, dV can be represented in spherical coordinates as dV = r² sin(θ)dr dθ dφ, and the limits of integration are from 0 to R for r, 0 to π for θ, and 0 to 2π for φ. Substituting the density function, we have:

M = ∫⁰ˣ (3.25×10³ - 9.50×10³ r) r² dr

After evaluating the integral, we obtain the total mass of the sphere.