Problem 94 - Precalculus : Building Concepts and Connections

Question

A circular music box rotates at a constant rate while the music is playing. What is the linear speed of a fly that is perched on the music box at a point 2 inches from its center if it takes the music box 6 seconds to make one revolution? Express your answer in inches per second.

Step-by-Step Solution

Answer

\(\frac{2\pi}{3}\) inches per second.

Step 1: Calculate the angular speed

Angular speed, \(\omega\), is defined as the rate of change of angle with time. It is given by the formula \(\omega = \frac{2\pi}{T}\), where \(T\) is the time period for one revolution. In this case, it takes 6 seconds to make one revolution. So, \(\omega = \frac{2\pi}{6}\) rad/s.

Step 2: Calculate the linear speed

Now that we have the angular speed, we can calculate the linear speed. The linear speed, \(v\), is given by the formula \(v = r\omega\), where \(r\) is the radius and \(\omega\) is the angular speed. In this case, the radius is 2 inches. So, \(v = 2 \times \frac{2\pi}{6}\) inches/sec.

Step 3: Simplify the equation

By simplifying the equation \(v = 2 \times \frac{2\pi}{6}\), we get \(v = \frac{2\pi}{3}\) inches/sec.

Key Concepts

Angular Speed

Understanding angular speed is crucial when dealing with circular motion. Angular speed is a measure of how quickly an object rotates or revolves around a central point. It is usually denoted by the Greek letter \( \omega \) and is defined as the angle an object covers over a certain amount of time. Mathematically, it's expressed as \( \omega = \frac{\theta}{t} \) where \( \theta \) is the angle in radians and \( t \) is the time it takes to cover that angle.

For example, if a wheel completes a full 360-degree (or \( 2\pi \) radians) turn in 60 seconds, its angular speed would be \( \omega = \frac{2\pi \text{ radians}}{60 \text{ seconds}} \) rad/s. This angular speed remains constant for uniform circular motion. Diving into our music box example, where one revolution (\(\ 2\pi \) radians) takes 6 seconds, the angular speed \( \omega\) is calculated using the same formula, resulting in \(\omega = \frac{2\pi}{6}\) rad/s. This constant rate of angular rotation is what ultimately helps determine the linear speed of our perched fly.

Revolutions per Minute

Another common measure of rotational speed is revolutions per minute (RPM). This is a count of the number of full rotations completed in one minute. To convert between angular speed (in radians per second) and RPM, we use the relationship that one revolution is \( 2\pi \) radians and there are 60 seconds in a minute.

For instance, if an object's angular speed is \( \omega = 0.5 \) rad/s, to find the RPM, we calculate \(\frac{0.5 \text{ rad/s}}{2\pi \text{ rad/rev}} \times 60 \text{ s/min}\), thus obtaining its RPM. Converting the angular speed in our music box example to RPM can help us understand its rotational speed in a different and more commonly used unit, although it wasn't explicitly required for the linear speed calculation.

Radius and Linear Speed Relationship

The relationship between radius and linear speed is a foundational concept in circular motion. Linear speed (also known as tangential speed) is the actual speed of a point moving along the circular path, measured in linear units per time, such as inches per second or meters per second.

The linear speed, \( v \) is directly proportional to both the radius \( r \) of the path and the angular speed \( \omega \) of the object in motion, as expressed in the equation \( v = r\omega \) . This relationship shows that for a larger radius, an object will have a greater linear speed at the same angular speed.

In the context of our music box, with a radius of 2 inches and an angular speed \( \omega = \frac{2\pi}{6} \) rad/s, the linear speed \( v \) of the fly perched on the edge is given by \( v = 2 \times \frac{2\pi}{6} \) inches/sec. Simplifying this equation gives us a linear speed of \( \frac{2\pi}{3} \) inches/sec. It's clear how crucial the radius is in this scenario; a fly closer to the center, say at a radius of 1 inch, would travel at half the linear speed, exemplifying the direct proportionality in this relationship.