Question

Small spheres of equal mass are made of lead (density \(=11.3 \mathrm{~g} / \mathrm{cm}^{3}\) ), silver \(\left(10.5 \mathrm{~g} / \mathrm{cm}^{3}\right)\), and aluminum \(\left(2.70 \mathrm{~g} / \mathrm{cm}^{3}\right)\). Without doing a calculation, list the spheres in order from the smallest to the largest.

Short answer

The spheres are in order from smallest to largest: Lead, Silver, and Aluminum.

Step-by-step solution

Understand the given information

We are given the densities of lead, silver, and aluminum. We are also told that the spheres have equal mass, and we need to determine their volumes and compare the sizes from smallest to largest.

Recall the relationship between mass, volume, and density

The relationship between mass, volume, and density is given by the formula: \(mass = density \times volume\) We can rearrange this formula to find the volume as: \(volume = \frac{mass}{density}\) Since the spheres have equal mass, we can compare their volumes by comparing their densities.

Analyze the densities

In order to compare their sizes, we do not have to calculate their exact volumes. We can compare their densities since mass is constant for all three spheres. Given densities are: - Lead: 11.3 g/cm³ - Silver: 10.5 g/cm³ - Aluminum: 2.70 g/cm³ Recall the formula for volume: \(volume = \frac{mass}{density}\) Since the mass of the spheres is the same, a higher density will result in a smaller volume, and a lower density will result in a larger volume.

Order the spheres based on their densities

Now, we will list the spheres according to their densities, from highest to lowest: 1. Lead: 11.3 g/cm³ 2. Silver: 10.5 g/cm³ 3. Aluminum: 2.70 g/cm³

Relate the density to the sphere sizes

As mentioned earlier, a higher density corresponds to a smaller volume. Therefore, we can order the spheres by size (from smallest to largest) as follows: 1. Lead (highest density) 2. Silver (middle density) 3. Aluminum (lowest density) So, the spheres are in order from smallest to largest: Lead, Silver, and Aluminum.

Key concepts

Density

Density is a fundamental concept in physics and materials science, describing how much mass is contained within a specific volume. In simple terms, it's a measure of how tightly packed the matter in a substance is. The standard unit for density is grams per cubic centimeter (g/cm³) for solids and liquids, and kilograms per cubic meter (kg/m³) for gases.

When analyzing objects with the same mass, understanding density helps us to infer their size. As a core principle, an object with high density means that there is a greater amount of mass in a smaller volume, leading to a heavier feeling object of the same size. Conversely, an object with low density has less mass in the same volume, resulting in a lighter object. Everyday examples include comparing a balloon filled with air (low density) and the same balloon filled with water (high density); despite its identical exterior, the water-filled one would feel much heavier.

Mass-Volume Relationship

The mass-volume relationship is a direct way of expressing the concept of density. It is mathematically represented by the equation: \(density = \frac{mass}{volume}\). When we discuss objects of equal mass, this relationship becomes crucial in understanding their volume differences based on their density.

For example, if we consider substances with the same mass but different densities, the substance with the higher density will have a smaller volume, while the one with the lower density will take up more space. This can be visually understood by pouring the same weight of lead and feathers into two containers. The denser lead would occupy considerably less space than the feathers. Thus, by rearranging the density equation to \(volume = \frac{mass}{density}\), one can determine the volume of an object if its mass and density are known, or vice versa.

Comparing Densities

Comparing densities allows us to make predictions about the relative sizes of objects without actual measurements. In the context of the exercise, with spheres of equal mass made from different materials, the sphere made of the material with the highest density will be the smallest. Equivalently, the material with the lowest density will have the largest sphere.

This comparison is directly linked to how much space an object occupies (volume) in relation to the amount of matter it contains (mass). To visualize this concept, imagine having equal weights of cotton, plastic, and lead. Without weighing them, we know that the cotton will appear the largest, and lead, being the densest material, will look the smallest. This principle is widely applied in various fields, spanning from material sciences to packing industries, where understanding densities can optimize space and weight management.