Problem 114 - 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. $$r=3.2 \mathrm{ft}, \omega=\frac{\pi \mathrm{rad}}{4 \mathrm{sec}}, t=3 \mathrm{min}$$

Step-by-Step Solution

Answer

The point travels approximately 452.389 ft.

Step 1: Convert Time to Appropriate Units

First, we need to convert the given time from minutes to seconds because the angular speed is provided per second. Since 1 minute is equal to 60 seconds, we have:\[ t = 3 \text{ minutes} = 3 \times 60 = 180 \text{ seconds} \]

Step 2: Calculate Angular Distance

Next, determine the angular distance the point travels, which is the product of angular speed and time:\[ \theta = \omega \times t = \frac{\pi}{4} \text{ rad/sec} \times 180 \text{ sec} = 45\pi \text{ radians} \]

Step 3: Calculate Linear Distance

The linear distance a point travels along a circle, also known as arc length, is the product of the circle’s radius and the angular distance traveled in radians. Using the formula \( s = r \times \theta \), we get:\[ s = 3.2 \text{ ft} \times 45\pi \]This simplifies to:\[ s = 144\pi \text{ ft} \approx 452.389 \text{ ft} \] (rounded to three significant digits)

Key Concepts

Angular Speed

Angular speed refers to how fast an object rotates or revolves relative to another point—the center of a circle, in most cases. It's the rate at which an angle changes with respect to time. In this context, angular speed is represented by \( \omega \). Mathematically, it's expressed in radians per second, which tells us how many radians an object covers in one second.

• To calculate angular speed: \( \omega = \frac{\text{angle (radians)}}{\text{time (seconds)}} \)

For the given exercise, \( \omega = \frac{\pi}{4} \text{ rad/sec} \). This means every second the object moves \( \frac{\pi}{4} \) radians around the circle's center.Understanding this concept helps us determine how quickly a point along the circle traverses its circular path.

Arc Length

The arc length is the measure of the distance along the curved line forming part of a circle. In circular motion, arc length \( s \) measures how much path an object has covered along the circle's circumference. This is essentially the 'linear distance' a point moves along the circle.

• Calculated using: \( s = r \times \theta \)

Where:- \( r \) is the radius of the circle,- \( \theta \) is the angular distance in radians.In our step, \( s = 3.2 \text{ ft} \times 45\pi \), where 3.2 ft is the radius and 45\( \pi \) is the angular distance covered. Calculating gives approximately 452.389 feet, indicating how much the point travels along the circle.

Angular Distance

Angular distance is the measurement of the angle through which a point or line has rotated in a specified direction around a specific axis. It's different from physical distance, as it relates to the idea of the circle having been turned. Angular distance is measured in radians, giving a clear view of how far 'around the circle' a point has moved.

• Calculated with: \( \theta = \omega \times t \)

Given in our steps, \( \theta = \frac{\pi}{4} \text{ rad/sec} \times 180 \text{ sec} = 45\pi \text{ radians} \). This figure reflects the total angle covered in the time frame.Hence, the concept of angular distance gives the revolution depth or the angle between the starting and final positions of the point on the circle.

Unit Conversion

Unit conversion is often necessary to ensure all quantities are in compatible formats for calculations. In our exercise, time conversion is critical because it matches the unit of angular speed, which is per second.

• 1 minute equals 60 seconds.

The given time was initially in minutes, so it was converted to seconds: \( t = 3 \text{ minutes} \times 60 = 180 \text{ seconds} \). Doing this ensures the time variable in equations aligns with the unit of measurement for angular speed. Proper unit conversion can simplify problems and prevent errors, ensuring calculations are accurate and results make sense.