Problem 78 - College Physics

Question

A certain material has a density of \(9.0 \mathrm{~g} / \mathrm{cm}^{3} .\) It is formed into a solid rectangular brick with dimensions \(1.0 \mathrm{~cm} \times 2.0 \mathrm{~cm} \times 4.0 \mathrm{~cm} .\) (a) What is its mass in kilograms? (b) If you wanted to make a cube of this same material containing twice the mass of this brick, what would be the length of one side of the cube?

Step-by-Step Solution

Answer

The mass of the brick is 0.072 kg; the cube's side is approximately 2.52 cm.

Step 1: Calculate the Volume of the Brick

To find the mass, we first need the volume of the brick. Use the formula for volume of a rectangular prism: \[V = ext{length} \times ext{width} \times ext{height}\]Substitute the given dimensions:\[V = 1.0 \, \text{cm} \times 2.0 \, \text{cm} \times 4.0 \, \text{cm} = 8.0 \, \text{cm}^3\]

Step 2: Calculate the Mass of the Brick

Now, use the relationship between mass, volume, and density:\[\text{Mass} = \text{Density} \times \text{Volume}\]For this brick:\[\text{Mass} = 9.0 \, \text{g/cm}^3 \times 8.0 \, \text{cm}^3 = 72.0 \, \text{g}\]Convert grams to kilograms:\[72.0 \, \text{g} = \frac{72.0}{1000} \, \text{kg} = 0.072 \, \text{kg}\]

Step 3: Find the Mass of the Cube

The cube needs to have twice the mass of the brick. Therefore, the mass of the cube will be:\[\text{Mass of cube} = 2 \times 0.072 \, \text{kg} = 0.144 \, \text{kg}\]Convert this mass to grams for easier calculations:\[0.144 \, \text{kg} = 144.0 \, \text{g}\]

Step 4: Calculate the Volume of the Cube

Using the known density, calculate the volume of the cube:\[\text{Volume of cube} = \frac{\text{Mass of cube}}{\text{Density}} = \frac{144.0 \, \text{g}}{9.0 \, \text{g/cm}^3} = 16.0 \, \text{cm}^3\]

Step 5: Determine the Side Length of the Cube

The volume of a cube is given by:\[V = s^3\]Where \(s\) is the side length. Set \(s^3 = 16.0 \, \text{cm}^3\) and solve for \(s\):\[s = \sqrt[3]{16.0} \, \text{cm} \approx 2.52 \, \text{cm}\]

Key Concepts

Volume calculation

The concept of volume is essential when dealing with any three-dimensional object, such as our rectangular brick. Volume measures how much space an object occupies.

To calculate the volume of a rectangular prism (the shape of the brick), use the simple formula: \[ V = ext{length} \times \text{width} \times \text{height} \] This is like finding how many tiny cubes fit inside the object.

For the brick in the problem, with dimensions 1.0 cm, 2.0 cm, and 4.0 cm, substitute these values into the formula: \[ V = 1.0 \text{ cm} \times 2.0 \text{ cm} \times 4.0 \text{ cm} = 8.0 \text{ cm}^3 \] So, the brick occupies 8 cubic centimeters of space.

Breaking it down helps us see that volume tells us the total number of unit cubes needed to fill the brick entirely. This concept applies not only to rectangular prisms but to any solid object.

Mass calculation

Mass is a measure of how much matter an object contains. To find mass, we need to consider both the volume and density of the material. The relationship is straightforward: \[ \text{Mass} = \text{Density} \times \text{Volume} \] This tells us mass depends on both how dense the material is and how much space it takes up (its volume).

In our example, the brick's density is 9.0 g/cm³, and its volume is 8.0 cm³. So: \[ \text{Mass} = 9.0 \text{ g/cm}^3 \times 8.0 \text{ cm}^3 = 72.0 \text{ g} \]

Density acts like a bridge between volume and mass. It lets us calculate how "heavy" a given volume of material will be.

Metric conversions

Metric conversions are crucial when working with measurements because they help translate between different units. The metric system is especially user-friendly due to its base-10 design.

For mass, the basic conversion needed is between grams and kilograms. Remember, 1 kilogram equals 1000 grams. To convert grams to kilograms, divide the number by 1000.

In our example: \[ 72.0 \text{ g} = \frac{72.0}{1000} \text{ kg} = 0.072 \text{ kg} \]

Doing these conversions ensures your calculations make sense in real-world contexts, where different systems and units may be used. Always check units to avoid mistakes.

Rectangular prism

A rectangular prism is a 3D shape with six faces, each a rectangle. It looks like a box, and most everyday objects can be thought of as rectangular prisms.

In this exercise, the brick is a rectangular prism with given dimensions, making it easy to calculate volume using the length, width, and height. These types of shapes are straightforward to work with in geometry.

The volume formula applied to rectangular prisms is a powerful tool. It allows for quick and efficient calculation of the space inside.

Understanding the properties of a rectangular prism helps solve problems relating to space, such as packing, building, or filling. Whether in science labs or packing boxes, rectangular prisms are everywhere!