Density: Videos & Practice Problems

Fluids, including liquids and gases, can change shape to fit their containers, characterized by density (ρ), defined as mass (m) divided by volume (v). Common densities include freshwater (1000 kg/m³) and air (1.2 kg/m³). Specific gravity (SG) relates a material's density to freshwater's density, expressed as SG = ρ_material / 1000. Understanding these concepts aids in solving problems involving buoyancy and weight, where weight (W) is calculated using W = mg, with g as the acceleration due to gravity.

Intro to Density

Fluids, encompassing both liquids and gases, exhibit unique properties that distinguish them from solids. Unlike solids, which maintain a fixed shape, fluids can adapt their shape to fit the container they occupy. A fundamental concept in understanding fluids is density, which is defined as the mass of a substance divided by its volume. The Greek letter ρ (rho) represents density, and the formula can be expressed as:

$$\rho = \frac{m}{V}$$

Here, m denotes mass, measured in kilograms (kg), and V represents volume, typically expressed in cubic meters (m³). Volume is calculated by multiplying three dimensions: length, width, and height, which can also be referred to as depth. Thus, the formula for volume can be summarized as:

$$V = \text{length} \times \text{width} \times \text{height}$$

In practical applications, if the density of a material and its dimensions are known, one can easily calculate the mass using the rearranged formula:

$$m = \rho \times V$$

It is important to note that objects made from the same material will have the same density, regardless of their size. For instance, a small block of wood and a large block of wood will both have the same density, even though their masses and volumes differ. This consistency in density is crucial when analyzing mixtures of liquids, as liquids with higher densities will settle at the bottom of a container, while those with lower densities will rise to the top.

To illustrate these concepts, consider a scenario where we need to calculate the total weight of air in a large warehouse modeled as a rectangular prism with dimensions of 100 meters in length, 100 meters in width, and 10 meters in height. The density of air is approximately 1.225 kg/m³, and the acceleration due to gravity is taken as 10 m/s². First, we calculate the volume of the warehouse:

$$V = 100 \, \text{m} \times 100 \, \text{m} \times 10 \, \text{m} = 100,000 \, \text{m}³$$

Next, we find the mass of the air using the density:

$$m = \rho \times V = 1.225 \, \text{kg/m}³ \times 100,000 \, \text{m}³ = 122,500 \, \text{kg}$$

Finally, to determine the weight of the air, we apply the weight formula:

$$W = m \times g = 122,500 \, \text{kg} \times 10 \, \text{m/s}² = 1,225,000 \, \text{N}$$

This calculation reveals that the weight of the air in the warehouse is 1,225,000 newtons, highlighting the significant mass that can be contained within a large volume. Understanding these principles of fluid mechanics is essential for various applications in science and engineering.

Density Values & Conversions

Understanding density is crucial for solving various problems related to materials such as water, saltwater, blood, air, oil, and wood. Density is defined as mass per unit volume, and it is commonly expressed in several units. For instance, freshwater has a density of 1000 kilograms per cubic meter (kg/m³), which means that a volume of 1 cubic meter of freshwater has a mass of 1000 kilograms. This can also be expressed as 1 kilogram per liter (kg/L) or 1 gram per cubic centimeter (g/cm³). The conversion between these units is essential for calculations, especially when dealing with different substances.

For example, saltwater has a density of approximately 1030 kg/m³, which can be converted to 1.03 kg/L or 1.03 g/cm³. Similarly, whole blood has a density of about 1060 kg/m³. The conversions rely on two key factors: 1 cm³ = 1 mL and 1 m³ = 1000 L. These relationships allow for seamless transitions between different units of measurement.

When considering air at sea level, its density is significantly lower at about 1.2 kg/m³, which is approximately 800 times less dense than water. This lower density explains why we can move freely through air. Other materials like oil and wood typically have densities around 800 kg/m³, making them less dense than freshwater.

To illustrate the application of density in problem-solving, consider a scenario where you have 500 mL of a liquid with a density of 2.2 g/cm³. To find the mass, you can use the formula for density, which states that density (ρ) equals mass (m) divided by volume (V): ρ = m/V. Rearranging this gives m = ρ × V. Here, the volume is 500 mL, which is equivalent to 500 cm³ (since 1 mL = 1 cm³).

Substituting the values, you get:

m = 2.2 g/cm³ × 500 cm³ = 1100 g

To convert grams to kilograms (the SI unit), divide by 1000, resulting in 1.1 kg. To find the weight (w), use the equation w = mg, where g is the acceleration due to gravity, approximately 10 m/s² for simplification. Thus, the weight is:

w = 1.1 kg × 10 m/s² = 11 N

This example demonstrates how to apply density concepts and unit conversions to solve real-world problems effectively.

Specific Gravity

Specific gravity (SG) is a term that relates to the density of a material, defined as the ratio of the density of that material to the density of fresh water. The formula for specific gravity can be expressed as:

SG = \(\frac{\text{Density of material}}{\text{Density of fresh water}}\)

Since the density of fresh water is a constant value of 1,000 kg/m³, the specific gravity can also be simplified to:

SG = \(\frac{\text{Density of material}}{1000}\)

This means that specific gravity is a dimensionless number, as the units cancel out when dividing two densities. For example, if the density of a material is 2,000 kg/m³, its specific gravity would be:

SG = \(\frac{2000}{1000} = 2\)

This indicates that the material is twice as dense as water. Conversely, if a material has a specific gravity of 0.7, its density can be calculated as:

Density = SG × 1000 = \(0.7 \times 1000 = 700 \, \text{kg/m}^3\)

Specific gravity does not directly relate to gravitational force; rather, it serves as a comparative measure of density relative to water. Understanding specific gravity is essential in various applications, such as determining whether an object will float or sink in water.

To illustrate the application of specific gravity in calculations, consider a wooden cube with a specific gravity of 0.8 and a weight of 16,000 Newtons. To find the volume of the cube, we first need to determine its mass using the relationship between weight (W), mass (m), and gravitational acceleration (g):

W = m × g

Rearranging this gives:

m = \(\frac{W}{g}\)

Assuming g = 10 m/s² for simplicity, we find:

m = \(\frac{16000}{10} = 1600 \, \text{kg}\)

Next, we calculate the density of the wooden cube using its specific gravity:

Density = SG × 1000 = \(0.8 \times 1000 = 800 \, \text{kg/m}^3\)

Now, we can find the volume (V) using the density formula:

Density = \(\frac{m}{V}\) ⟹ V = \(\frac{m}{\text{Density}}\)

Substituting the values gives:

V = \(\frac{1600}{800} = 2 \, \text{m}^3\)

This calculation demonstrates how specific gravity can be used to derive other important properties of materials, reinforcing its significance in scientific and engineering contexts.

A wooden door is 1 m wide, 2.5 m tall, 6 cm thick, and weighs 400 N. What is the density of the wood in g/cm³? (use g = 10 m/s²)

• A. 0.0267 g/cm³
• B. 0.267 g/cm³
• C. 267 g/cm³
• D. 2670 g/cm³

Answer: (B)

Suppose an 80 kg (176 lb) person has 5.5 L of blood (1,060 kg/m³ ) in their body. How much of this person's total mass consists of blood? What percentage of the person's total mass is blood?

• A. 0.0519 kg
• B. 0.0583 kg
• C. 5.19 kg
• D. 5.83 kg

Answer: (D)

You want to verify if a 70-g crown is in fact made of pure gold (19.32 g/cm³ ), so you lower it by a string into a deep bucket of water that is filled to the top. When the crown is completely submerged, you measure that 3.62 mL of water has overflown. Is the crown made of pure gold?

• A. Yes
• B. No

Answer: (A)

Here’s what students ask on this topic:

What is the formula for density and how is it used in calculations?

The formula for density (ρ) is defined as the mass (m) of an object divided by its volume (V). Mathematically, it is expressed as:

$\rho =\frac{m}{V}$

In calculations, if you know the mass and volume of an object, you can determine its density by dividing the mass by the volume. For example, if an object has a mass of 10 kg and a volume of 2 m³, its density would be:

$\rho =\frac{10}{2}=5 \mathrm{kg/m³}$

How do you convert between different units of density?

To convert between different units of density, you need to use appropriate conversion factors. Common conversions include:

• 1 kg/m³ = 0.001 g/cm³
• 1 g/cm³ = 1000 kg/m³
• 1 kg/m³ = 0.001 kg/L

For example, to convert 1000 kg/m³ to g/cm³, you multiply by 0.001:

$1000 \mathrm{kg/m³} \times 0.001 \mathrm{g/cm³} = 1 \mathrm{g/cm³}$

What is specific gravity and how is it calculated?

Specific gravity (SG) is a dimensionless quantity that compares the density of a material to the density of freshwater. It is calculated using the formula:

$\mathrm{SG}=\frac{\mathrm{\rho \_material}}{1000}$

For example, if a material has a density of 2000 kg/m³, its specific gravity would be:

$\mathrm{SG}=\frac{2000}{1000}=2$

This means the material is twice as dense as freshwater.

How does density affect the buoyancy of an object?

Density plays a crucial role in determining the buoyancy of an object. An object will float in a fluid if its density is less than the density of the fluid. Conversely, it will sink if its density is greater. This is because buoyant force, which acts upward, is equal to the weight of the fluid displaced by the object. Mathematically, buoyant force (F_b) is given by:

$\mathrm{F\_b}=\mathrm{\rho \_fluid} \mathrm{V\_displaced} g$

Where ρ_fluid is the density of the fluid, V_displaced is the volume of fluid displaced, and g is the acceleration due to gravity.

What are some common densities of materials and their units?

Common densities of materials include:

• Freshwater: 1000 kg/m³
• Saltwater: 1030 kg/m³
• Air: 1.2 kg/m³
• Oil: ~800 kg/m³
• Wood: ~600-800 kg/m³

These densities are typically expressed in kilograms per cubic meter (kg/m³), but can also be converted to grams per cubic centimeter (g/cm³) or kilograms per liter (kg/L) using appropriate conversion factors.