Question

A steel ball-bearing with a circumference of \(32.5 \mathrm{~mm}\) weighs \(4.20 \mathrm{~g}\). What is the density of the steel in \(\mathrm{g} / \mathrm{cm}^{3}\) ( \(\mathrm{V}\) of a sphere \(=\) \(\frac{4}{3} \pi r^{3} ;\) circumference of a circle \(\left.=2 \pi r\right) ?\)

Short answer

The density of the steel is approximately 7.81 g/cm^3.

Step-by-step solution

Find the radius

The circumference of the ball-bearing is given as 32.5 mm. Using the formula for the circumference of a circle, which is \(C = 2\pi r\), we can solve for the radius: \[r = \frac{C}{2\pi} = \frac{32.5 \text{ mm}}{2\pi} \] Calculate the radius.

Convert the radius to centimeters

Since we need the density in \( g/cm^3 \), the radius should be converted from millimeters to centimeters. 1 mm = 0.1 cm, so \[r = \frac{32.5}{2\pi} \times 0.1 \text{ cm}\].

Calculate the volume of the sphere

Using the formula for the volume of a sphere, \( V = \frac{4}{3} \pi r^3 \), we can find the volume in cubic centimeters. Substitute the radius from Step 2 into the volume formula: \[ V = \frac{4}{3} \pi \left( \frac{32.5}{2\pi} \times 0.1 \right)^3 \].

Calculate the density

Density is mass divided by volume. Given the mass is 4.20 g, calculate the density: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{4.20 \text{ g}}{V} \], using the volume found in Step 3.

Key concepts

Volume of a Sphere

To calculate the volume of a sphere, you need to use the formula: defined as \[ V = \frac{4}{3} \pi r^3 \] where \( V \) is the volume and \( r \) is the radius of the sphere. It's crucial to accurately find and convert the radius before using this formula. Given that you have the radius from the earlier steps (converted to centimetres if necessary), you can plug this value into the equation. This formula accounts for the three-dimensional space a sphere occupies. By cubing the radius, you ensure the volume is in terms of cubic units, making it easy to measure and compare with the mass to find density. Just ensure that all units are consistent.

Circumference to Radius Conversion

Understanding the relationship between a circle's circumference and its radius is key here. The formula for the circumference of a circle is: \[ C = 2\pi r \] where \( C \) is the circumference and \( r \) is the radius. To find the radius from a given circumference, rearrange the formula: \[ r = \frac{C}{2\pi} \]. This calculation provides the radius directly, which you will need for subsequent steps such as volume calculation. Consider an example from an exercise where the circumference provided is \(32.5 \text{ mm}\). By inserting this value into the radius formula, you get: \[ r = \frac{32.5}{2\pi} \]. Finally compute to get the radius. This approach lets you find the sphere's size directly, which is essential for further calculations like volume and density.

Unit Conversion

Unit conversion ensures that all measurements are in compatible units for calculations. For instance, if you have a measurement in millimetres but need the result in centimetres, you need to convert.For this process:

• 1 millimetre (mm) is equal to 0.1 centimetres (cm).

To convert a radius found in millimetres to centimetres, simply multiply by 0.1. For example, if the radius is \ \left(\frac{32.5}{2\pi} \right) \text{ mm}Convert to centimetres by: \[ r_{cm} = r_{mm} \times 0.1 \]In our case, it becomes \ \left(\frac{32.5}{2\pi} \times 0.1 \right) \text{ cm}This makes working with volumes and densities in \( \text{cm}^3 \) simpler and more accurate.Always ensure your final answers are presented in the correct units required by the problem.