Problem 7 - College Physics

Question

(a) What volume in liters is a cube \(20 \mathrm{~cm}\) on a side? (b) If the cube is filled with water, what is the mass of the water?

Step-by-Step Solution

Answer

(a) 8 liters, (b) 8 kg

Step 1: Calculate Volume of the Cube

To calculate the volume, we use the formula for the volume of a cube: \( V = a^3 \), where \( a \) is the side length. Here, \( a = 20 \) cm. Therefore, the volume \( V = 20^3 = 8000 \) cubic centimeters.

Step 2: Convert Cubic Centimeters to Liters

There are 1000 cubic centimeters in a liter. To convert 8000 cubic centimeters to liters, divide by 1000: \( 8000 \, \text{cm}^3 \div 1000 = 8 \text{ liters} \).

Step 3: Determine the Mass of Water

The density of water is approximately \( 1 \text{g/cm}^3 \), meaning 1000 grams per liter. Therefore, the mass of 8 liters of water is \( 8 \times 1000 = 8000 \text{ grams} \). To convert grams to kilograms, divide by 1000: \( 8000 \, \text{grams} \div 1000 = 8 \, \text{kg} \).

Key Concepts

Cubic Centimeters

Cubic centimeters are a unit of volume measurement often used for smaller objects and liquids. This unit helps to easily understand the space that an object might occupy or the capacity it can hold. The basic formula to find the volume of a cube is to use its side length. The formula is: \( V = a^3 \), where \( a \) is the length of one of the cube's sides. In this case, if the side of the cube is 20 cm, the volume in cubic centimeters would be calculated as follows:

• 20 cm \( \times \) 20 cm \( \times \) 20 cm = 8000 \( \text{cm}^3 \)

Knowing the volume in cubic centimeters is a first step in calculating further needs such as weight or liquid content.

Liters Conversion

Converting from cubic centimeters to liters is a common task, especially when dealing with liquids like water. Understanding the conversion rate makes it simple to move between these units.

• 1 liter = 1000 cubic centimeters

To convert cubic centimeters to liters, you need to divide the number of cubic centimeters by 1000. For example, if you have 8000 cubic centimeters:

• \( 8000 \, \text{cm}^3 \div 1000 = 8 \, \text{liters} \)

This straightforward conversion is useful in many practical scenarios, helping you quickly relate volume in typical household and scientific measurements.

Mass of Water

The mass of water can be calculated through its volume and density. Because water has a uniform density, converting volume to mass is straightforward.

• The density of water is approximately \(1 \, \text{g/cm}^3\) or 1000 grams per liter

Using this density, if you have 8 liters of water, you calculate the mass by multiplying the volume by water's density:

• 8 liters \( \times \) 1000 grams/liter = 8000 grams

To express this mass in kilograms, which is often more practical, divide by 1000:

• \( 8000 \, \text{grams} \div 1000 = 8 \, \text{kg} \)

Density of Water

Understanding the density of water is pivotal in many scientific and everyday calculations. Density is defined as the mass per unit volume, and for water, it is often cited as \(1 \, \text{g/cm}^3\). This means:

• 1 cubic centimeter of water weighs 1 gram
• 1 liter (which equals 1000 cubic centimeters) of water weighs 1000 grams

This consistency makes calculations with water straightforward and reliable. Whether determining how much a container can hold or converting between mass and volume, knowing the density simplifies the process. The typical property of water's density being 1 \(\text{g/cm}^3\) hails from its unique structure, which remains stable under most conditions encountered in daily life.