Problem 113 - Precalculus

Question

Find the distance a point travels along a circle over a time \(t,\) given the angular speed \(\omega\) and radius \(r\) of the circle. Round your answers to three significant digits. $$r=5.2 \text { in., } \omega=\frac{\pi \mathrm{rad}}{15 \mathrm{sec}}, t=10 \mathrm{min}$$

Step-by-Step Solution

Answer

The point travels 653 inches along the circle.

Step 1: Convert Time to Consistent Units

Given time duration is in minutes, and we need to have it in seconds to match the angular speed's unit. To convert \(t = 10\) minutes to seconds: \[ t = 10 \text{ min} \times 60 \frac{\text{seconds}}{\text{minute}} = 600 \text{ seconds} \]

Step 2: Calculate Angular Distance

The angular distance \( \theta \) can be calculated using the formula \( \theta = \omega \times t \).Substitute the given \( \omega = \frac{\pi \text{ rad}}{15 \text{ sec}} \) and \( t = 600 \text{ sec} \):\[ \theta = \frac{\pi}{15} \times 600 = 40\pi \text{ radians} \]

Step 3: Apply Arc Length Formula

The distance travelled along the circle (arc length \( s \)) is found using the formula: \( s = r \cdot \theta \).Substitute \( r = 5.2 \text{ in.} \) and \( \theta = 40\pi \): \[ s = 5.2 \cdot 40\pi \approx 5.2 \times 125.664 = 653.4512 \text{ in.} \]

Step 4: Round the Arc Length

Round the calculated arc length to three significant digits: \[ 653.4512 \text{ in.} \approx 653 \text{ in.} \]

Step 5: Conclusion

The distance the point travels along the circle is 653 inches after rounding to three significant digits.

Key Concepts

Angular Speed

Angular speed is a measure of how quickly an object moves around a circular path. It tells us how much of the circle the object covers over a specific period of time. Angular speed is often denoted by \( \omega \) and is usually measured in radians per second (rad/s). This unit reflects how radians, a unit of angle, are traversed within a second.

To understand angular speed, picture a spinning wheel. If it completes one full revolution in a second, its angular speed is equal to its circumference divided over the time taken. Mathematically, if one complete cycle around a circle is \( 2\pi \) radians, the angular speed \( \omega \) can be expressed as \( \omega = \frac{\text{Total radians}}{\text{Time in seconds}} \). This gives us a versatile measure of circular motion.

Knowing the angular speed is crucial when calculating arc lengths, as it directly influences how much arc a point on the circle travels over time.

Radius of Circle

The radius of a circle is the distance from the center of the circle to any point on its circumference. The radius, denoted by \( r \), acts as a critical factor in various circular motion calculations, such as determining the arc length or angular velocity.

• The radius is always a straight line; it stretches from the center to the edge.
• It's a constant measure for any particular circle and doesn't change, regardless of the circle's rotation or movement.

Having a clear understanding of the radius is pivotal because it helps us calculate the distance travelled by a point along the arc of the circle—a concept important in determining arc lengths. In the context of our problem, this radius was used in the equation \( s = r \cdot \theta \) to help compute the arc length, demonstrating its importance in linking angle movement and actual travel distance along the circle's path.

Unit Conversion

Unit conversion is a fundamental skill in mathematics and physics, allowing us to maintain consistency within computations. When dealing with problems that involve multiple measurements, like time and angular speed, converting them into compatible units is crucial for obtaining the correct solution.

In the exercise, time was initially provided in minutes, but the angular speed was in terms of seconds (rad/s). Thus, we first needed to convert time from minutes to seconds:

• 1 minute = 60 seconds.
• Therefore, \( 10 \text{ minutes} \times 60 \frac{\text{seconds}}{\text{minute}} = 600 \text{ seconds}\).

This conversion was essential to ensure both time and angular speed shared the same unit framework, allowing for consistent calculations. Unit conversion acts as a bridge, making diverse measurements compatible with each other for accurate mathematical reasoning and problem-solving.