Question

what mass of a material with density ρ is required to make a hollow spherical shell having inner radius r1 and outer radius r2? (use any variable or symbol stated above as necessary.)

Explanation:

Step1: Find the volume of the shell

The volume of a sphere is $V = \frac{4}{3}\pi r^{3}$. The volume of the outer - sphere with radius $r_2$ is $V_2=\frac{4}{3}\pi r_{2}^{3}$, and the volume of the inner - sphere with radius $r_1$ is $V_1=\frac{4}{3}\pi r_{1}^{3}$. The volume of the hollow spherical shell $V = V_2 - V_1=\frac{4}{3}\pi(r_{2}^{3}-r_{1}^{3})$.

Step2: Use the density - mass relationship

The density formula is $
ho=\frac{m}{V}$, where $
ho$ is density, $m$ is mass, and $V$ is volume. Rearranging for mass gives $m=
ho V$. Substituting the volume of the shell into the mass formula, we get $m = \frac{4}{3}\pi
ho(r_{2}^{3}-r_{1}^{3})$.

Answer:

$m=\frac{4}{3}\pi
ho(r_{2}^{3}-r_{1}^{3})$