Question

why does changing the volume of something --- like flattening a ball of aluminum foil into the shape of a boat -- change its density? type your answer here rewatch

Explanation:

Step1: Recall density formula

Density is defined as $
ho=\frac{m}{V}$, where $m$ is mass and $V$ is volume.

Step2: Analyze mass in the process

The mass of the aluminum foil does not change when we reshape it. The amount of aluminum (and thus its mass) remains constant.

Step3: Analyze volume in the process

When we flatten the aluminum foil ball into a boat - shaped object, the volume that is relevant for the "density" of the object - water system changes. The solid aluminum foil has a volume $V_{foil}$. When made into a boat, the boat displaces a volume of water $V_{displaced}$ which is equal to the weight of the boat (by Archimedes' principle). The boat's volume (including the air inside the boat's hull) $V_{boat}$ is larger than $V_{foil}$. If we consider the density of the boat as a system (mass of foil divided by the volume of the boat or the displaced water), since $m$ is constant and $V$ (of the system) increases, the density decreases. For the aluminum material itself, its density (mass of foil over volume of foil) is constant, but the overall density of the boat - like structure (which is what affects its ability to float) is lower because the total volume (foil + air) has increased while mass has not.

Answer:

Changing the volume (by shaping) affects density because density is mass per unit volume ($
ho = \frac{m}{V}$). When flattening aluminum foil into a boat, the mass ($m$) of the foil remains constant, but the total volume ($V$) it displaces (including the air - filled space in the boat) changes. If we consider the effective volume for density in the context of floating (where the boat displaces water equal to its weight), the apparent density (mass of foil divided by the volume of water displaced) decreases. For the foil itself, its intrinsic density (mass of foil divided by the volume of the foil material) is constant, but when shaped into a boat, the overall system (foil + air space) has a lower density than the solid foil because the total volume (foil volume + air volume) increases while the mass (only foil mass) stays the same, so $
ho=\frac{m}{V_{foil + air}}$ is less than $
ho=\frac{m}{V_{foil}}$.