Problem 1 - College Physics

Question

You purchase a rectangular piece of metal that has dimensions 5.0 \(\mathrm{mm} \times 15.0 \mathrm{mm} \times 30.0 \mathrm{mm}\) and mass 0.0158 \(\mathrm{kg}\) . The seller tells you that the metal is gold. To check this, you compute the average density of the piece. What value do you get? Were you cheated?

Step-by-Step Solution

Answer

The metal's density is approximately 7,022.22 kg/m³; it's not gold. You were cheated.

Step 1: Understand the Problem

We are given a rectangular piece of metal with dimensions 5.0 mm, 15.0 mm, and 30.0 mm. The mass of this piece is 0.0158 kg. We need to find the volume of this piece and then calculate its density to determine if it matches the density of gold, which is approximately 19,320 kg/m³.

Step 2: Calculate the Volume of the Metal

First, convert the dimensions from millimeters to meters to work in standard SI units. - Length = 5.0 mm = 0.005 m- Width = 15.0 mm = 0.015 m- Height = 30.0 mm = 0.030 mThe volume \( V \) of a rectangular prism can be calculated as follows: \( V = ext{Length} \times ext{Width} \times ext{Height} = 0.005 \times 0.015 \times 0.030 \, ext{m}^3 \) \( V = 2.25 \times 10^{-6} \, ext{m}^3 \)

Step 3: Calculate the Density

Density \( \rho \) is calculated using the formula: \( \rho = \frac{\text{Mass}}{\text{Volume}} \) Substitute the given mass and the volume we calculated in the previous step: \( \rho = \frac{0.0158 \, \text{kg}}{2.25 \times 10^{-6} \, \text{m}^3} \) \( \rho \approx 7,022.22 \, \text{kg/m}^3 \)

Step 4: Compare with the Density of Gold

The calculated density of the metal is approximately 7,022.22 kg/m³. The known density of gold is about 19,320 kg/m³. Since the calculated density is much lower than that of gold, the metal cannot be pure gold.

Key Concepts

Volume Calculation

The volume of a three-dimensional object is essentially how much space it occupies. For objects with regular shapes, such as a rectangular prism, calculating volume is straightforward. Once you know the dimensions, you can use a formula to find it. In this exercise, we have a rectangular piece of metal, and we are looking to calculate its volume. To find the volume of a rectangular prism, simply multiply its length, width, and height together:\[V = \text{Length} \times \text{Width} \times \text{Height}\]Here, the dimensions are given in millimeters, which we converted to meters to keep them in SI units. The length is 5.0 mm, width 15.0 mm, and height 30.0 mm, which convert to 0.005 m, 0.015 m, and 0.030 m respectively. By multiplying these together, we find that\[V = 0.005 \times 0.015 \times 0.030 \, \text{m}^3 = 2.25 \times 10^{-6} \, \text{m}^3\]Always make sure that your units of measurement are consistent to avoid errors in calculation.

Rectangular Prism

A rectangular prism is a three-dimensional shape with six faces, all of which are rectangles. It's also referred to as a cuboid. The defining characteristic of a rectangular prism is that each of its angles is a right angle.
A real-world example of a rectangular prism can be a gemstone cut into that shape or a block of metal. Key properties:

• It has six faces, 12 edges, and 8 vertices.
• The opposite faces are parallel and identical in shape and size.
• Its volume can be found using the formula mentioned in the previous section.

Understanding the properties of a rectangular prism is essential for calculating its volume accurately, as it guides you on which dimensions need to be measured and multiplied together.

Density of Materials

The density of a material is a measure of how much mass it has relative to the volume it occupies. It is expressed in units of mass per unit volume, typically kilograms per cubic meter (kg/m³) in the SI system.The formula to calculate the density \( \rho \) is:\[\rho = \frac{\text{Mass}}{\text{Volume}}\]In our exercise, by substituting the mass of the metal piece, 0.0158 kg, and its volume, \(2.25 \times 10^{-6} \, \text{m}^3\), we determine the density as:\[\rho \approx 7,022.22 \, \text{kg/m}^3\]This value is much lower than the known density of gold, which stands at approximately 19,320 kg/m³. Thus, the metal is not pure gold. Remember, knowing the density of a material can be a helpful way to identify it, or to check its purity when compared against standard densities.

SI Units Conversion

SI units, or International System of Units, is a globally agreed-upon system for measuring various physical quantities such as length, mass, and volume. Consistency in using these units is crucial for scientific calculations. In this task, the dimensions of the metal were initially given in millimeters. Since we aim to calculate volume in cubic meters (m³) and density in kilograms per cubic meter (kg/m³), a conversion was necessary. Conversion steps:

• 1 mm is equivalent to 0.001 meters.
• This means 5.0 mm converts to 0.005 m, 15.0 mm to 0.015 m, and 30.0 mm to 0.030 m.

Accurate conversion ensures that calculations are correct and comprehensible. Improper handling of units can lead to significant errors in results. Always check your units before starting operations, and keep a conversion guide handy when dealing with unfamiliar measurements.