Question

A steel cylinder has a length of 2.16 in, a radius of 0.22 in, and a mass of \(41 \mathrm{~g}\). What is the density of the steel in \(\mathrm{g} / \mathrm{cm}^{3} ?\)

Short answer

The density of the steel is approx. \(7.85 \mathrm{g} / \mathrm{cm}^{3}\).

Step-by-step solution

Calculate the Volume of the Cylinder

To find the density, we first need to calculate the volume of the cylinder using the formula for the volume of a cylinder, which is \( V = \pi r^{2} h \), where \(r\) is the radius and \(h\) is the height (or length) of the cylinder. The radius is 0.22 inches and the length is 2.16 inches.

Convert the Volume from Cubic Inches to Cubic Centimeters

Since the density needs to be in \(\mathrm{g}/\mathrm{cm}^{3}\), we must convert the volume from cubic inches to cubic centimeters, knowing that 1 inch equals 2.54 cm. The volume in cubic centimeters is obtained by cubing the conversion factor (2.54) and multiplying it by the volume in cubic inches.

Calculate the Density of the Cylinder

Density is defined as mass per unit volume. Once the volume in cubic centimeters is obtained, the density in \(\mathrm{g}/\mathrm{cm}^{3}\) is calculated by dividing the mass of the cylinder (41 g) by the volume in cubic centimeters.

Key concepts

Volume of a Cylinder

The volume of a cylinder is a measure of how much space is inside the cylindrical object. To find this, we use a simple mathematical formula: \( V = \pi r^{2} h \). In this formula, \( r \) represents the radius — the distance from the center to the edge of the base of the cylinder, and \( h \) stands for the height or length of the cylinder.

For a steel cylinder with a radius of 0.22 inches and height of 2.16 inches, plugging the values into the formula gives us its volume. This is an essential step, as knowing the volume allows us to determine the density of the material, which is another way of understanding how tightly packed the substance is within that volume.

Applying the Formula

To calculate, you simply square the radius (multiply it by itself), multiply that result by \(\pi\), and then multiply it all by the height. It's important to perform each operation carefully to ensure accuracy in the volume, and consequently, the density calculation.

Unit Conversion

Unit conversion is crucial when working with measurements because different systems of measurement are used around the world. For density calculations, it's often necessary to convert units to get the correct density units, such as converting from grams per cubic inch to grams per cubic centimeter.

To do this, we use a conversion factor. In this example, since 1 inch is equal to 2.54 centimeters, we can find the volume in cubic centimeters by multiplying the volume in cubic inches by \(2.54^3\), which is the cubed conversion factor from inches to centimeters.

• Remember to cube the conversion factor (\(2.54 \) becomes \(2.54^3\)) when converting cubic units.
• Write down your units at each step to avoid confusion.
• Ensure that your final units match the units required by the problem you are solving.

It's a simple multiplicative process, but accuracy is key - minor mistakes in unit conversion can lead to significantly incorrect results.

Mass per Unit Volume

Density, or mass per unit volume, is expressed as the amount of mass present in a given volume. It's a vital concept because it tells us how compact or how sparse the material is. The formula to calculate density is \(\text{density} = \frac{\text{mass}}{\text{volume}}\).

For the steel cylinder with a mass of 41 grams and a volume we've calculated and converted to cubic centimeters, finding the density involves dividing the mass by the volume in cubic centimeters. This final step gives us the density in the desired units of \(\mathrm{g}/\mathrm{cm}^3\).

Why Does It Matter?

Density has many practical applications, like identifying substances, determining buoyancy, and in industrial processes where material selection is crucial. An understanding of density, combined with its calculation, is a fundamental skill in sciences like chemistry and physics, as well as in various engineering fields.