Brass is a copper-zinc alloy. What is the mass in grams of a...

Brass is a copper-zinc alloy. What is the mass in grams of a brass cylinder having a length of 1.62 in. and a diameter of 0.514 in. if the composition of the brass is 67.0$\%$ copper and 33.0$\%$ zinc by mass? The density of copper is 8.92 $\mathrm{g} / \mathrm{cm}^{3}$ and the density of zinc is 7.14 $\mathrm{g} / \mathrm{cm}^{3} .$ Assume that the density of the brass varies linearly with= composition.

Key Concepts

Unit Conversion

Unit conversion is the process of converting measurements from one set of units to another. In this context, converting the dimensions from inches to centimeters is essential since the density values are given in grams per cubic centimeter. Accurate conversion is crucial to ensure that the calculated volume is in the same unit system as that used for the density of the materials.

Volume Calculation

Volume calculation involves determining the space occupied by a three-dimensional object. For a cylinder, this is done using the formula V = ?r²h, where r is the radius and h is the height (or length). This concept is important as it provides the geometric basis for calculating the mass once the density is known.

Density of Alloys

The density of an alloy, such as brass, is a property that depends on the densities of the constituent metals and their relative proportions. When the density is said to vary linearly with composition, it implies that the overall density of the alloy can be obtained as a weighted average of the densities of its individual components based on their mass fractions.

Linear Interpolation

Linear interpolation is a mathematical method used to estimate an unknown value that falls between two known values. In the case of brass, where the density changes linearly with copper and zinc composition, the density of brass can be determined by interpolating between the density of copper and the density of zinc according to the given mass percentages.

Transcript

00:01 We're told that we have a brass cylinder and brass is a copper zinc alloy.

00:06 So each has its own components and its each own density.

00:10 Now this brass cylinder has a length, which we can also call height, of 1 .62 inches.

00:17 And it has a diameter, that is, 0 .514 inches.

00:23 We also know that this is 67 % copper and 33 % zinc by mass.

00:29 We have the densities of both, and we want to find.

00:31 Find the mass in grams of this cylinder.

00:36 So before we can do this, because the densities are providing grams per cubic centimeter, we should probably make these inches as centimeters instead.

00:45 So let's convert them.

00:47 First up, it's helpful to know that for every inch, there is in fact 2 .54 centimeters.

00:56 So solving first, we get there's 4 .11 centimeters is the height.

01:05 Then let's do the diameter, so we'll do 0 .514 inches times 2 .54 centimeters in every inch.

01:17 And we get that the diameter is 1 .31 centimeters.

01:25 Next up, let's try to find the volume of this cylinder, which we can do with the following formula.

01:31 The volume of a right cylinder is pi r squared height.

01:36 We don't have r, but we do have diameter.

01:39 And diameter is twice the radius.

01:42 So let's divide that by 2 and find that the radius is 0 .653 centimeters.

01:51 Let's use that plugged into our volume formula.

01:55 Pi times 0 .653 centimeters squared times 4 .11 centimeters...