Density and Pressure — Question 2

Question 2

An exterior volume can hide an interior cavity

A sealed hollow sphere has external volume $V_o=2.00\times10^{-3}\,\mathrm{m^3}$ and mass $3.60\,\mathrm{kg}$. Its shell material has density $2700\,\mathrm{kg/m^3}$; the cavity contains negligible mass.

1. Find its bulk density and the fraction of the exterior volume occupied by the cavity.

2. If the same mass were reshaped into a solid sphere of the same material, what would be the new radius divided by the old outer radius?

Question 2 - Solution

An exterior volume can hide an interior cavity

(a) Derivation and calculation

$$\boxed{\rho_{\mathrm{bulk}}=m/V_o=3.60/(2.00\times10^{-3})=1800\,\mathrm{kg/m^3}}.$$$$V_s=m/\rho_s=3.60/2700=1.33333\times10^{-3}\,\mathrm{m^3}.$$$$\boxed{\frac{V_c}{V_o}=1-\frac{V_s}{V_o}=1-\frac{1800}{2700}=\frac13}.$$

(b) Result and interpretation

For equal material mass after reshaping,$$\frac43\pi R_{\mathrm{solid}}^3=V_s=\frac23V_o=\frac23\frac43\pi R_o^3.$$$$\boxed{\frac{R_{\mathrm{solid}}}{R_o}=\left(\frac23\right)^{1/3}=.87358}.$$Material density uses shell volume; bulk density uses the entire external volume.

Physical check

The radius decreases when empty cavity volume is removed without adding mass.