Problem 112 - 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. $$r=2 \mathrm{mm}, \omega=6 \pi \frac{\mathrm{rad}}{\mathrm{sec}}, t=11 \mathrm{sec}$$

Step-by-Step Solution

Answer

The distance traveled is approximately 414 mm.

Step 1: Understand the Relationship Between Distance, Radius, and Angular Speed

The distance a point travels along a circle, known as the arc length, can be found using the formula: \[ s = heta imes r \]where \( s \) is the arc length, \( \theta \) is the angular displacement in radians, and \( r \) is the radius of the circle. The angular displacement \( \theta \) can also be determined from the angular speed \( \omega \) and time \( t \) using \( \theta = \omega \times t \).

Step 2: Calculate the Angular Displacement

Using the formula \( \theta = \omega \times t \), substitute the given values:\( \omega = 6\pi \frac{\text{rad}}{\text{sec}} \) and \( t = 11 \text{ sec} \).\[ \theta = 6\pi \times 11 = 66\pi \text{ radians} \]

Step 3: Calculate the Arc Length

Now use \( s = \theta \times r \) to find the arc length.Given \( \theta = 66\pi \text{ radians} \) and \( r = 2 \text{ mm} \):\[ s = 66\pi \times 2 = 132\pi \text{ mm} \]Convert this result to a numerical value:\[ s = 132 \times 3.14159 \approx 414 \text{ mm} \]

Step 4: Confirm the Answer to Three Significant Digits

Round \( 414 \text{ mm} \) to three significant digits:The answer remains \( 414 \text{ mm} \) since it already has the required precision.

Key Concepts

Understanding Angular Speed

Angular speed is a term used to describe how quickly an object rotates around a circular path. While linear speed measures distance over time in a straight line, angular speed measures the angle through which an object turns per unit of time. It is denoted by the Greek letter \( \omega \). A good analogy might be the hands of a clock, where the angular speed is the consistent rate at which they complete a rotation.
Typically, angular speed is expressed in radians per second \( \left( \frac{\text{rad}}{\text{sec}} \right) \). A full circle or rotation is equivalent to \( 2\pi \) radians. Thus, an object making one full revolution in one second would have an angular speed of \( 2\pi \frac{\text{rad}}{\text{sec}} \).

• Higher angular speed means quicker rotation.
• It plays a vital role in determining how fast a point moves along the circle's edge.
• Angular speed can be constant or variable depending on the conditions.

Exploring Angular Displacement

Angular displacement refers to the angle through which a point or line has been rotated in a specified direction and about a specified axis. In simpler terms, it is the angular "travel" of a point in circular motion. Angular displacement is used to find how far a point on a circle has traveled and is denoted by \( \theta \).
The relationship between angular displacement, angular speed, and time is given by the formula: \[ \theta = \omega \times t \] where \( \omega \) is the angular speed, and \( t \) is time. This straightforward multiplier tells us the total angular distance covered over the time \( t \).

• Measured in radians, it allows us to calculate the arc length of a circle.
• Key for solving problems linking angular motion to linear movement on a circle.
• Angular displacement ties directly into the arc length when multiplied by the radius.

The Role of Radius of Circle

The radius of a circle is a fundamental component in understanding circular motion. It is the distance from the circle's center to any point along its edge. In the context of this exercise, knowing the radius is crucial because it determines the path length or arc length a point will cover when it moves around the circle.
The formula connecting arc length \( s \), angular displacement \( \theta \), and radius \( r \) is:\[ s = \theta \times r \] We can see that the arc length increases with a larger radius, assuming the same angle is traveled.

• The radius is always positive, as it is a measure of distance.
• A key factor in calculating how far a point on the edge will travel for a given angle \( \theta \).
• The radius, when combined with angular values, provides a bridge to linear calculations.