Problem 112 - A Graphical Approach to Precalculus with Limits

Question

The formula \(\omega=\frac{\theta}{t}\) can be rewritten as \(\theta=\) wt. Substituting wt for \(\theta\) changes \(s=r \theta\) to \(s=r \omega t\). Use the formula \(s=r \omega t\) to find the value of the missing variable. \(s=\frac{3 \pi}{4}\) kilometers, \(r=2\) kilometers, \(t=4\) seconds

Step-by-Step Solution

Answer

\(\omega = \frac{3\pi}{32}\) km/s.

Step 1: Understand the Given Formula

The given formula to use is \(s = r \omega t\), where \(s\) is the arc length, \(r\) is the radius, \(\omega\) is the angular velocity, and \(t\) is the time. We are given \(s = \frac{3\pi}{4}\) kilometers, \(r = 2\) kilometers, and \(t = 4\) seconds.

Step 2: Identify the Missing Variable

We need to find \(\omega\), the angular velocity, using the provided values in the formula.

Step 3: Substitute the Known Values

Substitute \(s = \frac{3\pi}{4}\), \(r = 2\), and \(t = 4\) into the formula \(s = r \omega t\). This gives us the equation: \(\frac{3\pi}{4} = 2 \cdot \omega \cdot 4\).

Step 4: Simplify the Equation

Start by simplifying the right side of the equation: \(2 \cdot \omega \cdot 4 = 8\omega\). Now, the equation is: \(\frac{3\pi}{4} = 8\omega\).

Step 5: Solve for \(\omega\)

To solve for \(\omega\), divide both sides of the equation by 8: \(\omega = \frac{3\pi}{4 \cdot 8}\). Simplify this to get \(\omega = \frac{3\pi}{32}\).

Key Concepts

Arc Length

Arc length is a concept that helps to measure the distance along the curved line of a circle segment, such as the portion of a wheel's circumference. In problems involving circles, the arc length represents a segment of the circle's perimeter.
When discussing motion in a circle, the arc length becomes a crucial element because it defines how far an object has moved if it is traveling along the circle's edge. This is distinctly different from a straight-line distance.

• The formula used is: \(s = r \theta\), where \(s\) is the arc length, \(r\) is the radius, and \(\theta\) is the central angle in radians.
• In the exercise, the given formula transforms to \(s = r \omega t\). This is because angular distance \(\theta\) can be expressed as \(\omega t\).

Understanding the concept of arc length is key to solving problems involving circular motion, such as tracking how far a wheel has traveled over a given time period.

Radius

The radius of a circle is the distance from its center to any point on the perimeter of the circle. It is fundamental to many computations involving circles.
The radius is not just important for evaluating arc length; it also factors into computing the area and the circumference of the circle.

• In our particular exercise, the radius was given as 2 kilometers, indicating the size of the circle involved in the problem.

By using the radius in conjunction with other variables like angular velocity and time, we can determine how the circle behaves, such as how much of it is covered over a specific interval of time.

Time

Time is a critical variable when dealing with problems of motion, especially circular motion. It helps in understanding how motion is spread out over a period. In the given problem, time is an essential factor in the formula \(s = r \omega t\).
The time variable allows us to calculate how an object moves consistently around a circle as it relates to angular velocity.

• Time is provided in seconds for this exercise to allow for straightforward substitution into our revised formula to solve for angular velocity.
• It is necessary to have consistent units for time to make meaningful physical interpretations.

Understanding how time interacts with arc length and radius underpins the behavior of rotating bodies and is central to solving real-world rotational motion problems.