Problem 56 - General Chemistry: Principles and Modern Applications

Question

Calculate the mass of a cylinder of stainless steel \(\left(d=7.75 \mathrm{g} / \mathrm{cm}^{3}\right)\) with a height of \(18.35 \mathrm{cm}\) and a radius of \(1.88 \mathrm{cm}\).

Step-by-Step Solution

Answer

To find the mass, calculate the volume of the given cylinder first and then multiply it with the density to find the mass.

Step 1: Determine the Volume

The volume of a cylinder is given by the formula \(V = \pi \times r^2 \times h\). In this case, \(r = 1.88 cm\) and \(h = 18.35 cm\). Therefore we sub these values in to calculate the volume: \(V = 3.14 \times (1.88cm)^2 \times 18.35cm\).

Step 2: Calculate the Mass

Now, the mass of the cylinder can be calculated using the formula \(m = d \times V\). Here, \(d = 7.75g/cm^3\) and \(V\) is the value calculated in step 1. Substituting these values we can calculate the mass of the cylinder.

Key Concepts

Cylinder Volume Calculation

The volume of a cylinder is essential to determine when calculating the mass or capacity of a cylindrical object. To find the volume of a cylinder, you use the formula:

• \(V = \pi \times r^2 \times h\)

where:

• \(V\) is the volume,
• \(r\) is the radius of the cylinder's base, and
• \(h\) is the height of the cylinder.

This formula leverages the principles of geometry to encapsulate how much space the cylinder occupies. The radius is squared because you are calculating the area of the base before multiplying by the height, effectively extruding that area along the height dimension.
This method of calculation can be applied to any cylinder, irrespective of the material or size, making it a versatile tool in both mathematics and physics.

Understanding Density

Density is a measure of how much mass is contained in a given volume. It is a crucial concept in physics and chemistry because it helps to determine how substances will interact. The formula to calculate density is:

• \(d = \frac{m}{V}\)

where:

• \(d\) is the density,
• \(m\) is the mass, and
• \(V\) is the volume.

In the exercise, the density of stainless steel is given as \(7.75 \, \text{g/cm}^3\). This means that every cubic centimeter of stainless steel weighs 7.75 grams.
When you know the density of a material and the volume of an object made of that material, you can easily find its mass by rearranging the formula to \(m = d \times V\).
Understanding density helps explain why some materials float while others sink and why some objects are heavier despite being smaller in size.

Importance of Geometry Formulas

Geometry formulas are fundamental tools used to calculate various properties and dimensions of geometric shapes and objects. In the context of our exercise, using the correct formula allowed us to calculate the cylinder's volume accurately.
Geometry connects the dimensions, shapes, and spatial understanding of figures, playing a pivotal role in fields like engineering, architecture, and everyday problem-solving.

• Formulas specific to shapes: For cylinders, cones, spheres, and other 3D shapes, unique formulas help us find properties such as area, volume, and surface area.
• Application in problem-solving: By applying geometry formulas, complex problems become manageable, allowing for precise construction and understanding of objects.

Understanding geometry and its formulas not only helps in mathematical exercises but also enhances spatial reasoning and logical thinking, valuable skills in numerous real-world applications.