Question

Consider these observations on two blocks of different unknown metals: \(\begin{array}{ll} & \text { Volume } \\ \text { Block A } & 125 \mathrm{~cm}^{3} \\ \text { Block B } & 145 \mathrm{~cm}^{3}\end{array}\) If block A has a greater mass than block B, what can be said of the relative densities of the two metals? (Assume that both blocks are solid.)

Short answer

Block A has a greater density than block B.

Step-by-step solution

Understand Density

Density is defined as mass per unit volume, and it is expressed as \( \rho = \frac{m}{V} \), where \( \rho \) is density, \( m \) is mass, and \( V \) is volume.

Compare the Masses

It is given that block A has a greater mass than block B. Therefore, for two volumes where \( V_{A} < V_{B} \), if \( m_{A} > m_{B} \), it implies that \( \rho_{A} \) must be greater than \( \rho_{B} \).

Conclude Based on Density and Mass

Since block A has a greater mass but a smaller volume than block B, block A must have a greater density than block B. This is because for density to be higher with a smaller volume, the mass must be relatively more concentrated.

Key concepts

Mass and Volume Relationship

Understanding how mass and volume relate to each other is crucial when exploring matter and its properties. Simply put, mass refers to how much matter is present in an object, while volume measures the amount of space that object occupies. When comparing the mass-to-volume ratio of two objects, we gain insight into their density.

Consider two blocks of metals with volumes given as 125 cm³ and 145 cm³ respectively. If the block with the smaller volume (Block A) has a greater mass than the one with the larger volume (Block B), it indicates that the material of Block A packs more matter into a smaller space. This fundamental concept allows us to infer differences in material composition, structural characteristics, and the potential identity of the two metals just by knowing their mass and volume.

Density Calculation

The calculation of density is a straightforward process, grounded in the formula:
\[ \rho = \frac{m}{V} \]
Here, \( \rho \) stands for density, \( m \) for mass, and \( V \) for volume. To assess an object's density, you determine its mass and volume and then divide the mass by the volume. Density is usually expressed in units such as grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³).

When applied to the exercise, where Block A has a smaller volume but a greater mass than Block B, we do not need the exact masses to know that Block A has a higher density. This is because the greater mass is divided by a smaller volume, resulting in a larger density value for Block A.

Comparing Densities

Comparing densities is often necessary when you want to identify materials or predict how substances will interact under various conditions. Denser materials, generally, have their particles more closely packed, which can lead to differences in durability, malleability, and many other physical properties.

In the context of our textbook exercise, the act of comparing densities allows us to conclude that Block A, with a higher mass but a lower volume, is denser than Block B. This means that if both blocks were placed in a fluid, we would expect Block A to sink faster or be less likely to float than Block B, assuming the fluid can support some degree of buoyancy for these materials. Understanding these principles is not just academically important; they're also the basis of real-world applications like material selection in engineering and design.