Problem 48 - Basic Technical Mathematics with Calculus

Question

As a ball bearing rolls along a straight track, it makes 11.0 revolutions while traveling a distance of \(109 \mathrm{mm}\). Find its radius.

Step-by-Step Solution

Answer

The radius of the ball bearing is approximately 1.58 mm.

Step 1: Understanding the Problem

We need to find the radius of the ball bearing given it makes 11 revolutions and covers a distance of 109 mm. We know that one complete revolution covers a distance equal to the circumference of the circle described by the ball's path.

Step 2: Writing the Equation

The distance traveled by the ball bearing in terms of revolutions is given by the equation: \\[\text{Distance} = \text{Number of Revolutions} \times \text{Circumference of the ball bearing}\] \where the Circumference is \(2\pi r\) for a circle.

Step 3: Substituting Values

Given that the Number of Revolutions is 11, and the total Distance traveled is 109 mm, we substitute these values into the equation: \\[109 = 11 \times 2\pi r\]

Step 4: Solving for the Radius

To find the radius \(r\), solve the equation for \(r\): \\(109 = 11 \times 2\pi r\) \Divide both sides by \(22\pi\): \\[r = \frac{109}{22\pi}\]

Step 5: Calculating the Radius

Using a calculator, compute \(\frac{109}{22\times\pi}\) to find the value of \(r\). The approximate value is \(r \approx 1.58\) mm.

Key Concepts

Understanding Geometry Through Circles

In geometry, understanding the basic properties of circles is fundamental. A circle is a shape where all points are equidistant from a central point, known as the radius. This distance from the center to any point on the circle is crucial in solving problems like the one in our exercise. The circumference, or the perimeter of a circle, is calculated by the formula \(C = 2\pi r\). This formula tells us how far the circle's edge extends around the center. Knowing how to manipulate this formula can help in finding other unknowns, such as the radius, when the circumference is known.
When approaching problems like the described ball bearing, visualize the circle created by the rolling ball. Understanding how many times the ball makes a complete circle, or revolution, helps unravel the connection between linear and circular measurements.

Applying Calculus Concepts

Calculus, a branch of mathematics dealing with rates of change and the accumulation of quantities, can enhance understanding in problems involving rolling objects. While the original problem doesn't explicitly require calculus, understanding the relationship between distance and change can offer deeper insights.
In this context, calculus can be used to study the motion and path of rolling objects. As the ball bearing rolls, its motion can be described using derivative and integral concepts. If the speed were variable, derivatives could reveal the relationship between speed, time, and distance. Conversely, integrals help in calculating the total distance by accumulating tiny pieces of motion over time. For this straightforward problem, calculus offers a backdrop that enriches comprehension but isn't needed for the solution. Instead, it underpins the elegant interplay between geometry and motion in real-world applications.

Mathematical Calculations for Problem Solving

Mathematical calculations are at the heart of problem solving in exercises like this. The key is to translate a word problem into mathematical expressions that can be solved step by step.
1. **Set up the Equation:** Recognize what is known and what needs to be found. Write the basic formula for distance involving the circumference and revolutions as \( \text{Distance} = \text{Revolutions} \times \text{Circumference} \). This allows linkage of the rolling distance to the circle's properties.2. **Substitution:** Insert given values into the formula. Here, the problem gives total distance and number of revolutions, translating into the equation \(109 = 11 \times 2\pi r\).3. **Solve:** Rearrange the equation to isolate the unknown variable, in this case, the radius \( r \). Execute this calculation to solve for \( r \), ensuring use of correct units throughout.The effective combination of substitution, manipulation, and accurate computation proves invaluable in deriving solutions. It's this process that transforms a seemingly complex question into a series of manageable mathematical steps.