Problem 109 - 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 \(\omega\). $$\omega=\frac{\pi \mathrm{rad}}{20 \mathrm{sec}}, r=5 \mathrm{mm}$$

Step-by-Step Solution

Answer

The linear speed is \( \frac{\pi}{4} \text{ mm/sec} \).

Step 1: Understanding the Relationship Between Angular Speed and Linear Speed

The linear speed, \( v \), of a point traveling in a circle is related to its angular speed, \( \omega \), by the formula \( v = \omega \times r \). In this context, \( \omega \) is the angular speed, and \( r \) is the radius of the circle.

Step 2: Substituting the Given Values

Substitute the given values for \( \omega \) and \( r \) into the formula. Here, \( \omega = \frac{\pi \, \text{rad}}{20 \, \text{sec}} \) and \( r = 5 \, \text{mm} \).So, \( v = \left( \frac{\pi}{20} \right) \times 5 \).

Step 3: Calculating the Linear Speed

Calculate \( v \) using the substituted values. Simplifying will give:\[ v = \frac{\pi \times 5}{20} \]\[ v = \frac{5\pi}{20} \]\[ v = \frac{\pi}{4} \text{ mm/sec} \].

Key Concepts

Understanding Angular Speed

Angular speed (\( \omega \)) describes how fast an object travels around a circular path. It measures the rate of change of the angle (\( \theta \)) with respect to time. When an object is moving in a circle, it doesn't travel in a straight line, so angular speed helps in understanding its circular movement.
Angular speed is measured in angles per unit of time, common units include radians per second (rad/sec) or degrees per second (°/sec). For instance, if a wheel completes one full turn in one second, its angular speed would be \( 2\pi \, \text{rad/sec} \), because a full circle is \( 2\pi \) radians.
Key points about Angular Speed:

• Angular speed gives a sense of how quickly something is rotating or revolving.
• It's essential for calculations involving circular motion, providing a bridge to linear movement.

Explaining the Radius of a Circle

The radius (\( r \)) of a circle is a crucial measurement. It represents the distance from the center of the circle to any point along its circumference. This constant distance is a key component in many formulas, including those calculating the circle's area or circumference.
A circle's properties, such as its size and roundness, are directly influenced by its radius. In the exercise example, we have a circle with a radius of 5 mm, a fairly small circle.
Features of the Radius:

• The radius remains constant no matter where you measure from within the circle.
• It's half the diameter, important for converting or expressing other circle metrics.

Understanding the radius is vital, since it helps in linking the circular motion to linear concepts.

The Relationship Between Linear and Angular Speed

Linear speed (\( v \)) and angular speed (\( \omega \)) are interconnected by the formula \( v = \omega \times r \), which shows how rotation (angular movement) translates into movement along the circumference (linear distance).
This formula effectively merges circular and linear motion into a single framework, allowing calculations across different contexts of motion. Linear speed tells us how quickly a point on a circle's edge is moving, as if following a straight path along its circumference.
In the step-by-step solution, we calculated the linear speed from given values of angular speed and radius:

• Angular speed: \( \frac{\pi}{20} \) rad/sec
• Radius: 5 mm
• Calculation: \( v = \frac{\pi}{20} \times 5 = \frac{\pi}{4} \) mm/sec

This shows how each circle turn results in linear movement, demonstrating the relationship in a quantifiable way. Understanding this relationship is fundamental for tasks involving circular motions, such as wheels or orbits.