Problem 40 - Calculus with Concepts in Calculus

Question

Through how many complete revolutions does a bicycle wheel with radius 1 foot turn when the bicycle travels 1 mile?

Step-by-Step Solution

Answer

The bicycle wheel turns approximately 840 complete revolutions.

Step 1: Understand the relationship between distance and wheel revolutions

The number of revolutions a wheel makes is determined by how many times its circumference fits into the travel distance. We must calculate the circumference of the wheel and use it to determine the number of revolutions.

Step 2: Calculate the circumference of the wheel

The formula to calculate the circumference (C) of a circle is \( C = 2 \pi r \). Here, the radius \( r \) is 1 foot, so the circumference is \( 2 \pi \times 1 = 2 \pi \) feet.

Step 3: Convert the distance traveled from miles to feet

1 mile equals 5280 feet. Therefore, the bicycle travels 5280 feet.

Step 4: Calculate the number of complete revolutions

To find the number of complete revolutions, divide the total distance traveled by the circumference of the wheel: \( \frac{5280}{2\pi} \). Simplifying this gives \( \frac{5280}{6.2832(approx)} \approx 840 \) revolutions.

Key Concepts

Circumference of a Circle

Understanding the circumference of a circle is crucial when dealing with circular motion concepts, like revolutions of a bicycle wheel. The circumference is the distance around the circle. To find it, use the formula:

• \( C = 2 \pi r \)

Here, \( \pi \approx 3.14159 \) is a constant that helps describe the relationship between a circle's diameter and its circumference. The variable \( r \) represents the radius, or half the diameter, of the circle. In this exercise, the bicycle wheel has a radius of 1 foot, so the circumference is \( 2 \pi \times 1 = 2 \pi \) feet.
This calculation tells us the distance covered in one full rotation of the wheel. To link this to the problem, knowing the circumference lets us figure out how many such rotations fit into a longer travel distance, like a mile.

Unit Conversion

Mastering unit conversion is vital when solving problems involving different measurement systems. In the problem provided, the bicycle travels a distance measured in miles, but we need this in feet to align with the wheel's measurement. This is where unit conversion helps:

• Converting miles to feet is straightforward because 1 mile equals 5280 feet.

Unit conversion is essential in scientific calculations to ensure consistency. By accurately converting miles to feet, we make sure our calculations with the wheel's circumference (measured in feet) are logical and correct.
Always double-check your conversions to avoid errors, which can lead to incorrect results.

Distance and Displacement in Circles

In circular motion, understanding the difference between distance and displacement is important. Distance refers to the total ground covered, while displacement is the straight line measurement from the starting point to the endpoint, which can be zero in circular paths.
For the bicycle wheel problem, when we calculate the number of revolutions, we focus on the distance traveled by the wheel:

• The total distance the bicycle travels equals the number of wheel circumferences it covers.

In this example, the distance is the entire length of 1 mile, which is 5280 feet. However, displacement here isn't a factor since we're only dealing with how many repetitions the circular path of one circumference can occur in a straight travel distance. Thus, we determine revolutions by dividing the total travel distance by the wheel's circumference, showing us the complete cycles it undertakes.