Question

A material will float on the surface of a liquid if the material has a density less than that of the liquid. Given that the density of water is approximately 1.0 g/mL under many conditions, will a block of material having a volume of \(1.2 \times 10^{4}\) in. \(^{3}\) and weighing 3.5 lb float or sink when placed in a reservoir of water?

Short answer

The block of material has a density of 0.00808 g/mL, which is less than the density of water (1.0 g/mL). Therefore, the block will float when placed in a reservoir of water.

Step-by-step solution

Convert Weight to Mass

We can convert the weight of the block (3.5 lb) to mass (in grams) by using the following relationship: 1 lb = 453.592 g Mass = Weight × Conversion factor = 3.5 lb × 453.592 g/lb ≈ 1587.572 g

Convert Volume from in\(^3\) to mL

Next, we need to convert the volume of the block from in\(^3\) to mL. The conversion factor is 1 in\(^3\) ≈ 16.3871 mL. Volume = Given Volume × Conversion factor = \(1.2 \times 10^{4}\) in\(^3\) × 16.3871 mL/in\(^3\) ≈ 196645.2 mL

Calculate the Density of the Block

Now that we have the mass in grams and the volume in mL, we can calculate the density of the material by dividing mass by volume. Density = Mass / Volume = 1587.572 g / 196645.2 mL ≈ 0.00808 g/mL

Compare the Density of the Block to Water

With the density of the material calculated, we compare it to the density of water (1.0 g/mL). Since the density of the material (0.00808 g/mL) is less than the density of water (1.0 g/mL), the block of material will float when placed in a reservoir of water.

Key concepts

Mass conversion

Mass conversion is when we change the unit of mass from one measurement system to another. In this context, we are converting the weight of a block given in pounds (lb) to grams (g), which is a common metric unit for mass.
The conversion is based on the fact that 1 lb is equal to 453.592 g. Therefore, to convert 3.5 lb to grams, you multiply 3.5 by 453.592. This gives a mass of approximately 1587.572 grams.

• 1 lb = 453.592 g
• 3.5 lb × 453.592 g/lb = 1587.572 g

This conversion is essential because it allows us to use consistent units (grams) across our calculations, especially for density, which is often expressed in g/mL.

Volume conversion

Volume conversion helps us express a given volume in a different comparable scale. Here, the block's volume is initially provided in cubic inches (in³), but we need it in milliliters (mL) for further calculations among metric units.
To convert cubic inches to mL, we use the conversion factor that 1 in³ is approximately equal to 16.3871 mL. By multiplying the given volume (1.2 x 10⁴ in³) by 16.3871, we find the volume in milliliters: approximately 196645.2 mL.

• 1 in³ ≈ 16.3871 mL
• 1.2 x 10⁴ in³ × 16.3871 mL/in³ ≈ 196645.2 mL

This conversion ensures that both the mass and volume of the object are in compatible units for determining density.

Density comparison

In order to compare densities, we need to calculate the density of the block and compare it with that of water. Density is calculated as mass divided by volume and is usually expressed in grams per milliliter (g/mL) or kilograms per liter (kg/L).
For the block, the mass is 1587.572 grams, and the volume is 196645.2 mL. Thus, its density is calculated as:\[ \text{Density} = \frac{1587.572 \text{ g}}{196645.2 \text{ mL}} \approx 0.00808 \text{ g/mL}\]When comparing this with the density of water (1.0 g/mL), we see a big difference.

• The block's density: 0.00808 g/mL
• Water's density: 1.0 g/mL

This simple comparison tells us that the block is much less dense than water.

Floating and sinking

Floating and sinking depend on the principle that an object will float in a liquid if its density is less than the density of the liquid. Conversely, an object will sink if its density is higher. This principle is a practical application of Archimedes' principle.
In the exercise, we discovered that the block has a density of 0.00808 g/mL, which is substantially lower than the density of water, which is 1.0 g/mL.

• Density of block: 0.00808 g/mL
• Density of water: 1.0 g/mL

Since the block’s density is less than that of water, the block will float. This consistent principle is what allows boats and other less dense objects to remain above the surface of the water.