Question

In an engineering reference book, you find that the density of iron is \(0.284 \mathrm{lb} / \mathrm{in.}^{3} .\) What is the density in \(\mathrm{g} / \mathrm{cm}^{3} ?\)

Short answer

The density of iron is approximately \(7.86 \: g/cm^3\).

Step-by-step solution

Write down the given density of iron

The given density of iron is \(0.284 \: lb/in^3\).

First convert pounds to grams

Knowing that \(1 \: lb = 453.592 \: g\), the density of iron can be converted to grams per cubic inch. By multiplying the given density with the conversion factor \(453.592 \: g/lb\), we obtain \(0.284 \: lb/in^3 \times 453.592 \: g/lb = 128.8 \: g/in^3\).

Then convert cubic inches to cubic centimeters

Knowing that \(1 \: in^3 = 16.387 \: cm^3\), the density of iron can be converted to grams per cubic centimeter by dividing the density obtained in Step 2 by this conversion factor. So, \(128.8 \: g/in^3 / 16.387 \: cm^3/in^3 = 7.86 \: g/cm^3\).

Check units and round off

Check that the units have been correctly converted and the figure has been calculated with the right number of significant figures. Finally, round off the result to two decimal places.

Key concepts

Units Conversion

Converting units is an essential skill in science and engineering that allows us to express measurements in different systems. To perform a conversion, you need a conversion factor that relates the two units.
For example, to convert pounds to grams, we use the conversion factor of 1 pound = 453.592 grams. Similarly, to convert cubic inches to cubic centimeters, we use 1 cubic inch = 16.387 cubic centimeters.

• Multiply when converting to a larger unit.
• Divide when converting to a smaller unit.

This helps in seamlessly transitioning between different measurement systems, ensuring accuracy in calculations.

Significant Figures

Significant figures are crucial in scientific calculations because they indicate the precision of a measured or calculated quantity. They represent all the known digits in a number, plus one estimated digit.
For example, if a measurement is given as 0.284 lb/in³, it has three significant figures. In calculations, it's important to maintain significant figures to reflect the precision of the data:

• Non-zero digits are always significant.
• Any zeros between significant figures are significant.
• Leading zeros are not significant.

Ensure your final answer is rounded to the correct number of significant figures to maintain accuracy.

Cubic Centimeters

Cubic centimeters (cc or cm³) are a unit of volume used in the metric system. It measures the space occupied by an object, equivalent to a cube with each side of one centimeter.
In many scientific calculations, such as measuring the density of a substance, converting to cubic centimeters provides a standard metric for volume.

• 1 cm³ is equal to 0.061 cubic inches.
• 1 liter = 1000 cm³.

This unit is particularly common in fields like chemistry and medicine, where precise volume measurements are crucial.

Pounds to Grams

Converting pounds to grams is essential when working within the metric system. Understanding this conversion is important for accurate scientific calculations.
Typically, 1 pound is equivalent to 453.592 grams. This conversion factor allows us to switch from a larger imperial unit to a smaller metric unit. Using this, we ensure that mass measurements are consistent with other metric units.

• This conversion is a multiplicative process.
• Precision is key, so use the conversion factor with appropriate significant figures.

Mastering this helps you work effectively across different measurement systems.