Mass, Volume, Weight and Density

Section 3.3 Mass, Volume, Weight and Density

Objectives: Learning Objectives

In this section, learners will explore the concepts of mass, volume, weight, and density. By the end of this substrand, learners should be able to:

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Understand the concepts of mass, volume, weight, and density.

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Convert between different units of mass and volume.

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Explain the relationship between mass and weight and how it is affected by gravitational field strength.

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Subsection 3.3.1 Converting Units of Mass from One Form to Another

Activity 3.3.1.

Work in groups

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Draw a table similar to the one shown in Table 3.3.2 below.
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For each object in the list, follow the steps below to record its mass in all six units using the applet in Figure Figure 3.3.1.
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Figure 3.3.1. Conversion of Mass
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Select one object from the checklist in the upper left of the applet.
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Drag the slider to each of the following units in order. After selecting each unit, record the mass displayed for the object in your table:
1. Ounces
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2. Pounds
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3. Tons
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4. Milligrams
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5. Grams
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6. Kilograms
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De-select the desired object by clicking on it again.
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Select the next object and repeat steps \(3–5\) until all objects in the checklist have been measured and recorded in Table 3.3.2.
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After completing the table, answer the following questions based on your measurements:
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1. How many grams are in one kilogram? Use values from your table to support your answer.
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2. How many ounces are in one pound?
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3. Which object had the smallest mass in milligrams? How did its value compare in ounces?
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4. Describe one interesting pattern or relationship you observed when converting an object’s mass through all six units.
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Discuss and share your findings with the rest of the class.
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Table 3.3.2. Conversion Units of Mass

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Object	mass (ounces)	mass (pounds)	mass (tons)	mass (milligrams)	mass (grams)	mass (kilograms)
Pencil
cellphone
potatoes
milk jug
microwave
socks
pillow

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\(\textbf{Key TakeAway}\)

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Definition 3.3.3.

Mass is the measure of the amount of matter in an object.

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The SI unit of mass is the kilogram (kg).

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Table 3.3.4. Mass Conversion Table

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Unit	Abbreviation	Equivalent kilogram
\(1\) tonne (megagram)	t	\(1\,000\) kg
\(1\) megagram	Mg	\(1\,000\) kg
\(1\) Hectogram	Hg	\(0.1\) kg
\(1\) Decagram	Dg	\(0.01\) kg
\(1\) gram	g	\(0.001\) kg
\(1\) decigram	dg	\(0.0001\) kg
\(1\) centigram	cg	\(0.00001\) kg
\(1\) milligram	mg	\(0.000001\) kg
\(1\) microgram	μg	\(0.000000001\) kg

Example 3.3.5.

Convert the following masses into kilograms.

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\(\displaystyle 500 \text{ g} \)
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\(\displaystyle 2\,300 \,000 \, \, \mu\text{g}\)
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\(\displaystyle 480 \text{ dg} \)
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\(\displaystyle 3 \text{ t}\)
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Solution.

\(500 \text{ g} \)
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To convert g to kg, use the conversion factor
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\(1 \text{ kg} = 1\,000 \, \text{g} \)
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\begin{align*} 1 \text{ kg} \amp = 1\,000 \, \text{g} \\ 500 \text{ g} \amp = \, ? \text{ kg} \end{align*}

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\begin{align*} \frac{500 \text{ g} \times 1 \text{ kg}}{1\,000 \, \text{g}} \amp = 0.5 \text{ kg} \\ \amp = 0.5 \text{ kg} \end{align*}

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\(2\,300 \,000 \, \, \mu\text{g}\)
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To convert μg to kg, use the conversion factor
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\(1\,000\,000\,000 \, \mu\text{g} = 1 \text{ kg}\)
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\begin{align*} 1\,000\,000\,000 \, \mu\text{g} \amp = 1 \text{ kg} \\ 2\,300 \,000 \, \, \mu\text{g} \amp = \, ? \text{ kg} \end{align*}

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\begin{align*} \frac{2\,300 \,000 \, \, \mu\text{g} \times 1 \text{ kg}}{1\,000\,000\,000 \, \mu\text{g}} \amp = 0.0023 \text{ kg} \\ \amp = 0.0023 \text{ kg} \end{align*}

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\(480 \text{ dg} \)
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To convert dg to kg, use the conversion factor
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\(1 \, \text{dg} = 0.0001 \text{ kg} \)
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\begin{align*} 1 \text{ dg} \amp = 0.0001 \, \text{kg} \\ 480 \text{ dg} \amp = \, ? \text{ kg} \end{align*}

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\begin{align*} \frac{480 \text{ dg} \times 0.0001 \text{ kg}}{1 \, \text{dg}} \amp = 0.048 \text{ kg} \\ \amp = 0.048 \text{ kg} \end{align*}

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\(3 \text{ t}\)
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To convert tonnes to kilograms, multiply the number of tonnes by \(1000\text{.}\)
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\(1 \text{ t} = 1\,000 \text{ kg} \)
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Therefore, \(3\) tonnes \(= 3 × 1,000 = 3,000\) kilograms.
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\(3\) tonnes is equal to \(3,000\) kilograms.
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Example 3.3.6.

Convert the following masses into grams.

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\(\displaystyle 2300 \text{ mg}\)
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\(\displaystyle 70 \text{ Hg}\)
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\(\displaystyle 12 \text{ kg}\)
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\(\displaystyle 800 \text{ cg}\)
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Solution.

\(2300 \text{ mg}\)
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\(\begin{align} 1\,\text{mg} \amp= 0.000001\,\text{kg} \\ 1\,\text{kg} \amp= 1000\,\text{g} \\ 1\,\text{mg} \amp= (0.000001 \times 1000\,\text{g}) = 0.001\,\text{g} \\ \\ 1\,\text{mg} \amp= 0.001\,\text{g} \\ 2300\,\text{mg} \amp= \, ?\,\text{g} \\ \\ 2300\,\text{mg} \amp= \frac{2300\,\text{mg} \times 0.001\,\text{g}}{1\,\text{mg}} \\ \amp= 2.3\,\text{g} \end{align} \)
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\(70 \text{ Hg}\)
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\(\begin{align} 1 \text{ Hg} \amp= 0.1 \text{ kg} \\ 1 \text{ kg} \amp= 1000 \text{ g} \\ 1 \text{ Hg} \amp= (0.1 \times 1000 \text{ g}) = 100 \text{ g} \\ \\ 1 \text{ Hg} \amp= 100 \text{ g}\\ 70 \text{ Hg} \amp= \, ? \text{ g} \\ 70 \text{ Hg} \amp= \frac{70 \text{ Hg} \times 100 \text{ g}}{1 \text{ Hg}}\\ \amp=7000 \text{ g} \end{align}\)
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\(12 \text{ kg}\)
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\(1 \text{ kg} = 1000 \text{ g} \)
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\(\begin{align} 1\, \text{kg} \amp= 1000\, \text{g} \\ 12\, \text{kg} \amp= \, ?\, \text{g} \\ 12\, \text{kg} \amp= \frac{12\, \text{kg} \times 1000\, \text{g}}{1\, \text{kg}} \\ \amp= 12000\, \text{g} \end{align}\)
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\(800 \text{ cg}\)
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\(\begin{align} 1\, \text{g} \amp= 100\, \text{cg} \\ 1\, \text{cg} \amp= \frac{1\, \text{g}}{100} = 0.01\, \text{g} \\ \\ 800\, \text{cg} \amp= \, ?\, \text{g} \\ 800\, \text{cg} \amp= 800 \times 0.01\, \text{g} \\ \amp= 8\, \text{g} \end{align}\)
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Checkpoint 3.3.7.

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Subsection 3.3.2 Relating Mass and Weight

Activity 3.3.2.

Work in groups

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What you need:

Spring balance
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Objects with different mass (Stones, books, shoes)
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Triple Beam balance
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Measure the weight of each object using the spring balance.
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Measure the mass of each object on the triple beam balance.
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Record the mass and weight of each object in a table below.
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Calculate the weight of each object using the formula: \(\text{W} = \text{m} \times \text{g}\text{,}\) where \(g\) is the acceleration due to gravity (approximately \(10 \text{ N/kg}\)).
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Compare the weight calculated using the formula with the weight measured using the spring balance. Discuss any differences or similarities you observe.
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Discuss the relationship between mass and weight. How do they differ? How are they related?
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Share your findings with the rest of the class.
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Table 3.3.8. Mass and Weight Table

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Object	Mass
	In gram	In kg
1
2
3

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\(\textbf{Key TakeAway}\)

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Definition 3.3.9.

Weight is the measure of the force of gravity acting on an object.

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The SI unit of weight is the newton (N).

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Weight (W) \(=\) mass of an object (kg) \(\times\) gravitational force (N/kg)

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Remark 3.3.10.

Planets in the solar system have different gravitational forces. This means that the weight of an object will vary depending on the planet it is on, but its mass will remain constant.

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For example, the gravitational field strength on Earth is approximately \(10\) N/kg, which means that an object with a mass of \(4\) kg will weigh \(40\) N on Earth.

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Table 3.3.11. Difference Between Mass and Weight

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Mass	Weight
Measure of the amount of matter in an object	Measure of the force of gravity on an object
SI unit is the kilogram (kg)	SI unit is the newton (N)
Mass is constant regardless of location	Weight can change depending on the gravitational field 🔗

Example 3.3.12.

A table has a mass of \(35\) kg. What is its weight? (Take g \(= 10\) N/kg)

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Solution.

Weight \(=\) mass (kg) \(\times\) gravitational force (g)

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Weight \(= 35\) kg \(\times\) 10 N/kg \(= 350\) N

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Example 3.3.13.

The weight of a box is \(150\) N. What is its mass? (Take g \(= 10\) N/kg)

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Solution.

Weight \(=\) mass (kg) \(\times\) gravitational force (g)

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150 N \(=\) mass (kg) \(\times\) 10 N/kg

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mass \(= \frac{150 \text{ N}}{10 \text{ N/kg}} = 15 \text{ kg}\)

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Checkpoint 3.3.14.

TODO

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Subsection 3.3.3 Mass, Volume and Density

Activity 3.3.3.

What you need

An electronic weighing machine or a balance scale
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Water
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A large container
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Graduated cylinder, \(100\) ml
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Measure and record the mass of an empty graduated cylinder in grams.
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Pour \(150\) mL of water into the graduated cylinder.
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Weigh the graduated cylinder with \(150\) mL of water and record the total mass.
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Subtract the mass of the empty cylinder to find the mass of \(150\) mL of water. Record it in the table.
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Use the mass and volume of the water to calculate density. Record the density in g/cm\(^3\) in the table below.
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Density of water\(= \frac{ \text{mass of water}}{ \text{volume of water}}\)
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Carefully pour out some water until only \(100\) mL remains in the graduated cylinder.
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Repeat steps \(3\) to \(5\) for \(100\) mL of water.
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Pour out more water until only \(50\) mL remains. Repeat steps \(3\) to \(5\) again.
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Finally, pour out water until you have \(25\) mL remaining. Repeat the same process and record your results.
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Look at the values of density for the different volumes.
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- Are they almost the same?
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- What does that tell you about the density of water?
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What is the density of water in g/cm \(^3\text{?}\)
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Table 3.3.15. Calculating the density of different volumes of water

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Volume of water	\(150\) mL	\(100\) mL	\(50\) mL	\(25\) mL
Mass of empty graduated cylinder (g)
Mass of graduated cylinder + water (g)
Mass of water (g)
Density of water (g/cm\(^3\))

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\(\textbf{Key TakeAway}\)

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Definition 3.3.16.

Volume is the amount of space occupied by an object.

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The standard unit of volume is the cubic metre (m \(^3\)).

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The volume of a cube is calculated by multiplying the length of one side by itself three times, or cubing the length of one side.

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Remark 3.3.17.

Density is the mass of a substance per unit volume. It is calculated by dividing the mass of an object by its volume.

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Density \(= \frac{\text{mass}}{\text{volume}}\)

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Example 3.3.18.

Evans has \(370 \text{cm}^3 \) of milk in a container. If the density of milk is \(1.03 \text{g/cm}^3\text{,}\) what is the mass of the milk?

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Solution.

To find the mass of the milk, we can use the formula for density:

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\(\text{Density} = \frac{\text{mass}}{\text{volume}}\)

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Rearranging the formula to solve for mass gives us:

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\(\text{mass} = \text{Density} \times \text{volume}\)

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Substituting in the values we have:

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\(\text{mass} = 1.03 \text{g/cm}^3 \times 370 \text{cm}^3\)

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\(\text{mass} = 381.1 \text{g}\)

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Example 3.3.19.

A cuboid has a length of \(5 \text{cm}\text{,}\) a width of \(3 \text{cm}\text{,}\) and a height of \(2 \text{cm}\text{.}\) It’s mass is \(60 \text{g}\text{.}\) What is the density of the cuboid in \(\text{kg/m}^3\text{?}\)

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Solution.

\(\text{Density} = \frac{\text{mass}}{\text{volume}}\)

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First, we need to find the volume of the cuboid as follows:

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\(\text{Volume} = \text{length} \times \text{width} \times \text{height}\)

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Substituting in the values we have:

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\(\text{Volume} = 5 \text{cm} \times 3 \text{cm} \times 2 \text{cm} = 30 \text{cm}^3\)

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Now we can substitute the mass and volume into the density formula:

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\(\text{Density} = \frac{60 \text{g}}{30 \text{cm}^3} = 2 \text{g/cm}^3\)

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To convert this to \(\text{kg/m}^3\text{,}\) we multiply by \(1000\) (since \(1 \text{g/cm}^3 = 1000 \text{kg/m}^3\)):

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\(\text{Density} = 2 \text{g/cm}^3 \times 1000 = 2000 \text{kg/m}^3\)

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Example 3.3.20.

A cylinder has a radius of \(8 \text{cm}\) and a height of \(14 \text{cm}\text{.}\) It’s mass is \(85 \text{g}\text{.}\) What is the density of the cylinder?

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