Problem 107 - Precalculus

Question

Find the linear speed of a point traveling at a constant speed along the circumference of a circle with radius \(r\) and angular speed \(\omega\). $$\omega=\frac{2 \pi \mathrm{rad}}{3 \mathrm{sec}}, r=9 \mathrm{in.}$$

Step-by-Step Solution

Answer

The linear speed is approximately 18.85 in/sec.

Step 1: Determine Formula for Linear Speed

The linear speed, also known as tangential speed, of a point on the circumference can be found using the formula: \( v = r \cdot \omega \), where \( v \) is the linear speed, \( r \) is the radius of the circle, and \( \omega \) is the angular speed.

Step 2: Substitute Known Values

Substitute the given values into the formula: \( r = 9 \text{ in.} \) and \( \omega = \frac{2 \pi \text{ rad}}{3 \text{ sec}} \). This gives: \( v = 9 \cdot \frac{2 \pi}{3} \).

Step 3: Simplify the Expression

Simplify the expression for the linear speed: \( v = 9 \cdot \frac{2 \pi}{3} = \frac{18 \pi}{3} \), which simplifies to \( 6 \pi \).

Step 4: Calculate Final Result

Evaluate the expression to find the numerical value of the linear speed. Thus, \( v \approx 18.85 \) in/sec, using \( \pi \approx 3.14159 \).

Key Concepts

Angular Speed

Angular speed is a measure of how quickly an object travels around a circular path. It tells us how much angle is covered in a certain amount of time. We usually express angular speed in radians per second. In a circle, as a point moves, it covers an angle

• denoted as theta (\(\theta\))
• the time it takes is expressed in seconds

This speed is crucial in understanding rotational motion. For example, if we say an object has an angular speed \(\omega = \frac{2 \pi \text{ rad}}{3 \text{ sec}}\), it completes a full revolution (\(2\pi\) rad) in 3 seconds.
Connecting it to our exercise: When given this angular speed and a circle's radius, we can calculate the linear speed or tangential speed of a point on the circle's edge. Angular speed essentially helps us transition between rotational movement and straight-line speed.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its circumference. It is a key component in circular motion calculations because it directly affects the speed of any point traveling around the circle. The larger the radius, the farther a point must travel to complete one full rotation, potentially affecting its speed.

• Notation: Radius is usually denoted by the letter \(r\).
• Key role: Serves in the formula for linear speed: \(v = r \cdot \omega\).

For instance, in the exercise problem, we have \(r = 9 \text{ in.}\). By plugging this value into our formula with the given angular speed, the radius helps us determine the linear speed. Therefore, understanding the radius and its relationship with angular speed is crucial for calculations involving circular motions.

Tangential Speed

Tangential speed, also referred to as linear speed when specified in circular motion contexts, measures how fast a point on a circle's circumference is moving. Unlike angular speed, which is about the angle moved per second, tangential speed deals with actual distance covered per unit time. It's always oriented along the tangent to the circle at any point.

• Formula: Calculated as \(v = r \cdot \omega\).
• Relevance: Essential in fields requiring precise movement understanding, such as engineering and physics.

In practice, you find the tangential speed by multiplying the circle's radius by its angular speed. So, if a tire spins with an angular speed, the tangential speed tells us how swiftly the tire edge is covering ground. In our exercise, this calculated speed was about 18.85 inches per second, showing the point's fast linear movement around the circle's rim.