A solid with an irregular shape and a mass of \(11.33 \mathrm{~g}\) is added to a graduated cylinder filled with water \((d=1.00 \mathrm{~g} / \mathrm{mL})\) to the \(35.0-\mathrm{mL}\) mark, After the solid sinks to the bottom, the water level is read to be at the 42.3-mL mark. What is the density of the solid?

The displacement method is a practical technique used to measure the volume of an object, especially those with irregular shapes that would otherwise be difficult to calculate using standard geometric formulas. This method involves submerging the object into a fluid and recording the change in the fluid's level. The principle behind this method is that the object will displace a volume of fluid equal to its own volume.

For example, when a stone is placed into a graduated cylinder containing water, the water level rises. If we note the initial water level and the final level after the object is submerged, the difference in these levels represents the volume of the object. In our exercise, the water level in the graduated cylinder rose from 35.0 mL to 42.3 mL after the solid was added, indicating the solid displaced 7.3 mL of water. Therefore, we conclude that the volume of the solid is 7.3 mL.

Archimedes' principle states that when an object is fully submerged in a fluid, it experiences an upward force equal to the weight of the fluid it displaces. This is a powerful concept in physics that applies to all fluids - liquids and gases.

Applying this principle helps us to determine the volume of solids indirectly through liquid displacement. For the given exercise, the completely submerged solid pushes away 7.3 mL of water, which, according to Archimedes' principle, is equal to the volume of the solid itself. Knowing the volume through this principle makes it possible to calculate the density of irregularly shaped objects, as done in the exercise. It allows for a simple mathematical approach to finding volume, bypassing complicated geometrical calculations.

Density is defined as mass per unit volume and is expressed using the formula: \[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\].

In any density calculation, understanding the relationship between mass and volume of an object is crucial. For substances with a uniform composition, density is a constant value, meaning that no matter how much of the substance you have, its density remains the same. This concept is pivotal when comparing the density of different materials and helps in identifying substances.

In the given exercise, the mass of the solid is 11.33 g, and we calculated the volume to be 7.3 mL using the displacement method. Plugging these values into the density formula gives us a density of 1.55 g/mL for the solid. Knowing the density helps in predicting whether the object will float or sink in a given fluid, as well as determining material properties and identifying unknown materials based on their density.