Question

Calculate the density of a block of wood which weighs \(750 \mathrm{~kg}\) and has the dimensions \(25 \mathrm{~cm} \times 0.10 \mathrm{~m} \times 50.0 \mathrm{~m}\).

Short answer

The density of the block of wood can be calculated using the formula Density = Mass / Volume. After converting dimensions to meters and calculating the volume as \(1.25 \mathrm{~m^3}\), we can find the density: Density = \(\frac{750 \mathrm{~kg}}{1.25 \mathrm{~m^3}} = 600 \mathrm{~kg/m^3}\).

Step-by-step solution

Convert dimensions to consistent units

We need to convert the dimensions from cm and m to a consistent unit, preferably meters since the result would be in m³. The conversion factor for cm to m is 1 cm = 0.01 m. Dimension 1: \(25 \mathrm{~cm} \times 0.01 \mathrm{~m}/\mathrm{cm} = 0.25 \mathrm{~m}\) So, the first dimension is 0.25 meters.

Calculate the volume

Now that all dimensions are in meters, we can calculate the volume using the formula: Volume = Length × Width × Height Volume = \((0.25 \mathrm{~m}) \times (0.10 \mathrm{~m}) \times (50.0 \mathrm{~m})\) Volume = \(1.25 \mathrm{~m^3}\) So, the volume of the block of wood is 1.25 cubic meters.

Calculate the density

Now that we have both mass and volume, we can calculate the density using the formula: Density = Mass / Volume Density = \(\frac{750 \mathrm{~kg}}{1.25 \mathrm{~m^3}}\) Density = \(600 \mathrm{~kg/m^3}\) The density of the block of wood is 600 kg/m³.

Key concepts

Density Formula

Understanding how to calculate density is crucial in many areas of science and engineering. The density of a material is defined as its mass per unit volume. This is usually expressed using the formula:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
Using this formula requires knowing the mass of the object and its volume. Mass is typically measured in kilograms (kg) or grams (g), while volume may be measured in cubic meters (m³), liters (L), or cubic centimeters (cm³). The resulting density will have units that arise from the combination of these measurements, such as kilograms per cubic meter (kg/m³) or grams per cubic centimeter (g/cm³).
In the example provided, after determining the mass and volume of the block of wood, the formula was applied to find that the block’s density is 600 kg/m³. Calculating density is not only about plugging numbers into a formula; it involves understanding the physical meaning of density and its implications regarding the object's composition and buoyancy.

Unit Conversion

Unit conversion is an essential skill when working with different types of measurements. Inconsistency in units can lead to incorrect results, so ensuring that all measurements are in the same unit system (metric or imperial) is necessary for accurate calculations. In the given exercise, converting centimeters to meters was essential for calculating volume since the volume formula requires all dimensions to be in the same unit.
To convert centimeters to meters, you use the relation:
\[ 1 \text{ cm} = 0.01 \text{ m} \]
Multiplying the length in centimeters by 0.01 gives the length in meters. It’s important always to be mindful of the conversion factor since different units have different conversion factors. Simple mistakes in converting units can greatly alter the final calculation. Understanding and applying unit conversion correctly is vital in science, technology, and everyday life.

Mass and Volume Relationship

The relationship between mass and volume is at the heart of calculating density. Mass refers to the amount of matter in an object and is typically measured in kilograms or grams, while volume refers to the space it occupies, measured in cubic meters, liters, or cubic centimeters for solid objects.
When calculating the volume of an object from its dimensions, as done in the exercise, the formula used is:
\[ \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \]
Upon calculating the volume and knowing the mass, adjusting these measurements so that they correspond to each other is crucial—meaning they must be in compatible units to calculate density. The greater the mass for a given volume, the greater the density, which describes how tightly packed the material is within the object. Conversely, a large volume with little mass indicates a low density, which typically corresponds to lighter or more porous materials. This relationship is integral not only for finding density but also for understanding buoyancy, material properties, and design characteristics of different objects.