Problem 4 - University Physics with Modern Physics

Question

The density of gold is 19.3 g/cm\(^3\). What is this value in kilograms per cubic meter?

Step-by-Step Solution

Answer

The density of gold is 19,300 kg/m³.

Step 1: Understand the Conversion

The density of gold is given as 19.3 grams per cubic centimeter (g/cm³). We need to convert this to kilograms per cubic meter (kg/m³). This involves converting grams to kilograms and cubic centimeters to cubic meters.

Step 2: Convert Grams to Kilograms

Since there are 1000 grams in a kilogram, we convert the density of gold from grams to kilograms. \[ 19.3 ext{ g/cm}^3 = \frac{19.3}{1000} ext{ kg/cm}^3 \] This simplifies the conversion factor for mass.

Step 3: Convert Cubic Centimeters to Cubic Meters

There are 100 centimeters in a meter, and since we are dealing with volumes, we cube the conversion factor.\[ 1 ext{ cm}^3 = (\frac{1}{100})^3 ext{ m}^3 = \frac{1}{1,000,000} ext{ m}^3 \] We need to use this factor when converting the volume part of the density.

Step 4: Apply Both Conversions to the Density

Now apply both the mass and volume conversions to the original density value:\[ 19.3 ext{ g/cm}^3 = 19.3 \times \frac{1000}{1} \text{ kg/m}^3 \]This simplifies to:\[ 19.3 \times 1,000,000 \text{ kg/m}^3 = 19,300 ext{ kg/m}^3 \] Thus, the density of gold in kilograms per cubic meter is 19,300 kg/m³.

Key Concepts

Density of Gold

Density is a measure of how much mass is contained in a certain volume. For gold, this density is 19.3 grams per cubic centimeter (g/cm³). If you think about density, imagine packing as much gold as possible into a small box. In one cubic centimeter, which is about the size of a small sugar cube, you would have 19.3 grams of gold. This property tells us about the richness of gold, meaning it is very dense compared to other materials like water (which has a density of 1 g/cm³). Understanding the density of gold can help determine its purity and even its economic value. When dealing with gold, knowing its density allows jewelers to verify its authenticity and quality.
Different contexts, like physics or engineering, may require density measurements in different units, such as kilograms per cubic meter (kg/m³). This necessitates an understanding of how to convert these units accurately.

Unit Conversion

Unit conversion is the process of converting a quantity from one unit to another. In everyday life, this might mean converting inches to centimeters or pounds to kilograms. In scientific contexts, unit conversions ensure consistency and accuracy in measurements.
Converting the density of gold from g/cm³ to kg/m³ involves converting both mass and volume units. The change of mass from grams to kilograms involves a simple conversion: 1 kilogram equals 1,000 grams. Therefore, simply divide the grams by 1,000 to obtain kilograms.
Changing the volume units is a little trickier because of the cubic measurement. There are 100 centimeters in one meter, but when converting cubic centimeters to cubic meters, you have to cube the conversion factor. This means multiplying 100 by itself twice, resulting in 1,000,000.
These conversions are essential in fields like engineering and physics, where precision is crucial and measurements often vary in scale.

Volume Conversion

Volume conversion might seem challenging, but it becomes straightforward once you grasp the concept of "cubing." When converting dimensions, be mindful of the units. Volume units grow quickly because they involve three dimensions (length, width, height). Thus, understanding how to convert these measurements is crucial.
For example, let's trace the conversion from cubic centimeters (cm³) to cubic meters (m³). Consider that 1 cm equals 0.01 meters, so when converted to meters, one cubic centimeter is equivalent to \[(\frac{1}{100})^3 m³ = \frac{1}{1,000,000} m³.\]
Volume conversion becomes particularly relevant when dealing with large quantities or specific applications. For instance, in construction or material science, correctly understanding and applying volume conversions ensures accurate calculations and prevents costly mistakes.

Mass Conversion

Mass conversion often involves converting between different systems of measurement, such as grams and kilograms. In our case, when dealing with the density of gold, the mass is originally provided in grams.
To convert grams into kilograms, remember that 1 kilogram equals 1,000 grams. This relationship is pivotal because it affects how we perceive and handle measurements at larger scales. Simply divide the mass in grams by 1,000 to convert it to kilograms.
For instance, when converting the gold's density from 19.3 g/cm³ to kg/m³, dividing by 1,000 adjusts the mass properly to suit different scientific contexts.

• This conversion ensures that when results scale up, like in mining, metallurgy, or crafting industries, they remain precise and comprehensible across various measurement systems.

Understanding mass conversion plays a significant role in international trade and scientific research. It's all about maintaining uniformity and avoiding miscommunications across borders and disciplines.