Consider the two spheres shown here, one made of silver and the other of aluminum. (a) What is the mass of each sphere in kg?

Verified step by step guidance

1. Determine the volume of each sphere using the formula for the volume of a sphere: $V = \frac{4}{3}\pi r^3$, where $r$ is the radius of the sphere.
2. Identify the density of silver and aluminum. The density of silver is approximately $10.49 \text{ g/cm}^3$ and the density of aluminum is approximately $2.70 \text{ g/cm}^3$.
3. Convert the densities from $\text{g/cm}^3$ to $\text{kg/m}^3$ by multiplying by $1000$.
4. Calculate the mass of each sphere using the formula: $\text{mass} = \text{density} \times \text{volume}$.
5. Convert the mass from grams to kilograms by dividing by $1000$.

Key Concepts

Density

Density is defined as mass per unit volume and is a crucial property of materials. It is typically expressed in grams per cubic centimeter (g/cm³) or kilograms per cubic meter (kg/m³). Knowing the density of a material allows us to calculate its mass if we know its volume, which is essential for solving the problem regarding the spheres.

Volume of a Sphere

The volume of a sphere can be calculated using the formula V = (4/3)πr³, where r is the radius of the sphere. This geometric formula is fundamental for determining how much space the sphere occupies, which is necessary for calculating its mass when combined with the material's density.

Material Properties

Different materials have distinct properties, including density, which affects their mass for a given volume. Silver and aluminum have different densities (approximately 10.49 g/cm³ for silver and 2.70 g/cm³ for aluminum), which means that even if the spheres have the same volume, their masses will differ significantly. Understanding these properties is essential for accurately determining the mass of each sphere.