Problem 24 - Applied Physics

Question

An airplane circles an airport twice while \(5.00 \mathrm{mi}\) from the control tower. Find the length of the arc through which the plane travels.

Step-by-Step Solution

Answer

The airplane travels 20\(\pi\) miles.

Step 1: Understanding the Problem

The airplane travels in a circular path around the control tower at a constant radius of 5.00 miles. We need to calculate the length of the arc that the plane travels when making two full circles.

Step 2: Identify the Arc Formula

The length of an arc of a circle can be calculated using the formula \( L = heta imes r \), where \( L \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the angle in radians.

Step 3: Convert Revolutions to Radians

Since the plane circles the airport twice, the angle \( \theta \) becomes \( 2 \times 2\pi \) because one full revolution is \( 2\pi \) radians. Thus, \( \theta = 4\pi \) radians.

Step 4: Calculate Arc Length

Using the formula and the values \( r = 5.00 \) miles and \( \theta = 4\pi \) radians, calculate the arc length: \[ L = 4\pi \times 5.00 = 20\pi \; \text{miles} \]

Step 5: Final Answer

Thus, the length of the arc through which the plane travels is \( 20\pi \) miles.

Key Concepts

Circle Circumference

The circumference of a circle is like the perimeter, which is the whole boundary length around the circle. It tells you how far you would travel if you walked all the way around the circle once without shortcuts!
To calculate the circumference, we use the formula:

• \( C = 2\pi r \), where \( C \) represents the circumference.
• \( r \) is the radius, or the distance from the circle's center to its edge.

Let's use an example: If our circle has a radius of 5 miles, like the path the airplane follows, the circle's circumference would be:

• \( C = 2 \pi \times 5 = 10 \pi \; \text{miles}\)

Understanding the circumference helps relate it to the arc and angle in radians when we talk about parts of the circle.

Radian Measure

Radians offer a way to measure angles, different from degrees. While degrees circle uses 360 parts to denote a full revolution, radians use the length of the arc divided by the radius. It provides a natural and mathematical-friendly measure of angles.
One full circle, or revolution, is always \(2\pi\) radians. For example, if you open your compass to draw a circle and swing it around once, that's \(2\pi\) radians or 360 degrees.

• One-half circle: \(\pi\) radians
• One quarter-circle: \(\frac{\pi}{2}\) radians
• Three-quarters of a circle: \(\frac{3\pi}{2}\) radians

When you convert angles from radians to degrees (or vice versa), being comfortable with radians becomes essential, especially in formulas involving circles like the arc formula.

Revolution Conversion

Revolution conversion is about translating the number of revolutions of an object (around a circle) into radians. It's a key step in connecting real-world movements with mathematical calculations.
Remember:

• 1 revolution equals \(2\pi\) radians.

In our airplane example, if it circles the airport twice, we calculate the total radians as:

• 2 revolutions = \(2 \times 2\pi = 4\pi\) radians.

Why does this matter? Because radians are crucial for calculating the arc length, which tells us how much of the circle's circumference the airplane covers during its flights.

Arc Formula

The arc length formula is your go-to tool when you need to find out how much of the circle's path is covered for a particular angle, in radians. The beauty of the arc formula lies in its simplicity and flexibility.
The formula is given by:

• \( L = \theta \times r \)

Where:

• \( L \) is the arc length.
• \( \theta \) is the angle in radians.
• \( r \) is the radius of the circle.

In the context of the airplane circling the airport, we apply:

• \( \theta = 4\pi \) (from the revolution conversion)
• \( r = 5 \; \text{miles} \)
• \( L = 4\pi \times 5 = 20\pi \; \text{miles} \)

Arc length tells us precisely how far the airplane travels along its circular path, using the blend of real-world distance and abstract angular measure.