Mass — Question 5

Question 5

Bulk density does not uniquely identify an interior

A specimen has a cubic outer boundary of edge length $4.00\ \mathrm{cm}$ and mass $162.0\ \mathrm{g}$. Use the displayed values as exact for this ideal model. Density is defined by $\rho=m/V$.

A manufacturer states that the specimen contains only a material of uniform density $2.700\ \mathrm{g\,cm^{-3}}$ plus an internal sealed cavity of negligible contents mass. All external faces are intact. Ignore air buoyancy.

1. Find the specimen’s external volume and bulk density, where bulk density uses the entire external volume, including any cavity.

2. Under the manufacturer’s model, determine the volume occupied by material, the cavity volume, and the fraction of the external volume occupied by the cavity.

3. The material is reshaped without loss into a fully solid body at the same material density. Find its new volume. Explain why the mass remains the same while bulk density changes.

4. Suppose the manufacturer’s material claim is unverified. Give a fully solid alternative consistent with the original mass and outer dimensions. Can these two measurements alone prove the cavity exists? State one additional type of evidence that would help.

Question 5 - Solution

Bulk density does not uniquely identify an interior

(a) Distinguish external volume from occupied material volume.

$$V_{\mathrm{ext}}=(4.00\ \mathrm{cm})^3=\boxed{64.00\ \mathrm{cm^3}},\qquad \rho_{\mathrm{bulk}}=\frac{162.0}{64.00}=\boxed{2.53125\ \mathrm{g\,cm^{-3}}}.$$

(b) Apply the stated material density only to the material.

$$V_{\mathrm{cavity}}=V_{\mathrm{ext}}-V_{\mathrm{material}}=4.00\ \mathrm{cm^3},\qquad\phi=\frac4{64}=0.0625=6.25\%.$$

$$V_{\mathrm{material}}=\frac{162.0\ \mathrm{g}}{2.700\ \mathrm{g\,cm^{-3}}} =\boxed{60.00\ \mathrm{cm^3}}.$$ The remaining volume is $64.00-60.00=\boxed{4.00\ \mathrm{cm^3}}$, so the cavity occupies $4/64=\boxed{6.25\%}$ of the external volume. This inference uses the known material density and negligible cavity contents mass.

(c) Removing empty space changes volume, not material mass.

The fully solid body occupies $\boxed{60.00\ \mathrm{cm^3}}$ and still has mass 162.0 g. Its bulk density now equals the material density, 2.700 g/cm$^3$. The original lower bulk density reflected empty space included in the denominator, not lighter material within the occupied region.

(d) The same bulk data admit another model.

A homogeneous, fully solid material of density $\boxed{2.53125\ \mathrm{g\,cm^{-3}}}$ would have the same mass and external volume. Those measurements alone therefore do not prove a cavity. Independent material identification/density measurement or suitable interior imaging could discriminate between the models. A bulk density is an aggregate property, not a unique map of the interior.