Problem 47 - Precalculus: Functions and Graphs

Question

Potation of compact disces (COs) The drive motor of a particular CD player is controlled to rotate at a speed of \(200 \mathrm{rpm}\) when reading a track 5.7 centimeters from the center of the CD. The speed of the drive motor must vary so that the reading of the data occurs at a constant rate. (a) Find the angular speed (in radians per minute) of the drive motor when it is reading a track 5.7 centimeters from the center of the CD.(b) Find the linear speed (in \(\mathrm{cm} / \mathrm{sec}\) ) of a point on the CD that is 5.7 centimeters from the center of the CD. Find the angular speed (in rpm) of the drive motor when it is reading a track 3 centimeters from the center of the CD. Find a function \(S\) that gives the drive motor speed in \(\mathrm{rpm}\) for any radius \(r\) in centimeters, where \(2.3 \leq r \leq 5.9\) What type of variation exists between the drive motor speed and the radius of the track being read? Check your answer by graphing \(S\) and finding the speeds for \(r=3\) and \(r=5.7\)

Step-by-Step Solution

Answer

The motor speed is 400π rad/min at 5.7 cm, v is 119.3 cm/sec. At 3 cm, speed is 380 rpm. Motor speed varies inversely with track radius: S(r)=3579/r.

Step 1: Calculate the Angular Speed for 5.7 cm

The given motor speed is 200 revolutions per minute (rpm). To convert this to radians per minute, use the relation \(1 \text{ revolution} = 2\pi \text{ radians}\). Thus, the angular speed \( \omega \) in radians per minute is given by: \[ \omega = 200 \times 2\pi = 400\pi \text{ radians per minute}. \]

Step 2: Calculate the Linear Speed for 5.7 cm

The linear speed \(v\) can be found from the angular speed using the formula \(v = r\omega\), where \(r\) is the radius in centimeters and \(\omega\) is the angular speed in radians per minute. First, convert the angular speed to radians per second: \[ \omega = \frac{400\pi}{60} \text{ radians per second}. \] Then, calculate the linear speed: \[ v = 5.7 \times \frac{400\pi}{60} \approx 119.3 \text{ cm/sec}. \]

Step 3: Calculate the Angular Speed for 3 cm

Since the linear speed of data read needs to be constant, use \(v = r \omega\) where \(v\) is constant. Given that the initial condition for \(r = 5.7\) cm is: \[ 119.3 = 5.7 \times \omega, \] solve for \(\omega\) when \(r = 3\) cm using: \[ 119.3 = 3 \times \omega, \] which yields \[ \omega = \frac{119.3}{3} \approx 39.77 \text{ radians per second}. \] Convert this back to rpm: \[ \omega = \frac{39.77 \times 60}{2\pi} \approx 380 \text{ rpm}. \]

Step 4: Find the Function for Motor Speed

Given that linear speed \(v\) is constant, \(v = r \omega = 119.3 \text{ cm/sec}\), express \(\omega\) as a function of \(r\): \[ \omega = \frac{119.3}{r}. \] Convert this to rpm: \[ \omega = \frac{119.3 \times 60}{2\pi r}. \] The function \( S(r) \) for motor speed in rpm as a function of radius \(r\) is: \[ S(r) = \frac{3579}{r}. \]

Step 5: Graph and Verify the Function

Graph the function \( S(r) = \frac{3579}{r} \) for \( r \) in the range \( 2.3 \leq r \leq 5.9 \). Check that for \( r = 3 \), \( S(3) \approx 380 \text{ rpm} \), and for \( r = 5.7 \), \( S(5.7) \approx 200 \text{ rpm} \). This confirms that the motor speed inversely varies with the radius of the CD track being read.

Key Concepts

Linear Speed

When discussing linear speed, we relate closely to the motion along a straight line. In the context of a rotating CD, linear speed refers to how fast a point on the outer edge of the rotating disc is moving.
The linear speed can be calculated by understanding the movement of the point in terms of distance per unit of time.
On a circular path, like with a CD, the linear speed is intimately connected to angular speed (how quickly the CD spins around its center) and radius (distance from the center to where you're measuring on the edge). Here's the formula for linear speed:

• Linear Speed, \[ v = r \omega \]
• \( v \) represents linear speed in cm/sec.
• \( r \) is the radius in centimeters.
• \( \omega \) is the angular speed in radians/sec.

In simpler words, if the CD spins faster or you're measuring farther from the center, the linear speed increases.
Understanding this concept helps to calculate how fast a specific point on the CD's surface travels as it rotates.

Radial Measure

Radial measure is a crucial part of understanding how objects move in circular paths. It often refers to measuring angles in terms of radians rather than degrees.
In the case of circular motion, one complete revolution equals \(2\pi\) radians.
For a CD or any circular object, radial measure helps transform our understanding of rotations into a form that easily ties to real-world data. Here's why this matters in angular measurements:

• Converting revolutions to radians allows us to work with the same units across formulas.
• It provides a direct connection to the circle's radius, making calculations seamless.

Imagine the CD as a clock. If the minute hand completes one full circle, it covers \(2\pi\) radians, providing a standardized way to refer to angular movement. Understanding radial measure enables students to seamlessly convert different types of measurements and better analyze rotational dynamics.

Function of Radius

The function of the radius in the context of a CD player's drive motor speed is a fascinating study in how variables are interdependent.
The radius here refers to the distance from the center of the CD to the point where data is being read. In this scenario, as the radius changes, the motor speed needs to adjust accordingly to maintain a constant linear speed. Let's consider the relationship between radius and speed:

• The speed variation function is given by \[ S(r) = \frac{3579}{r} \]
• This means the motor speed in rpm varies inversely proportional to the radius.

This inversely proportional relationship indicates that larger radii result in slower rotational speeds to keep the linear speed constant across different tracks. Graphing this function shows how motor speed declines as you measure farther from the center.
It elegantly demonstrates how changes in one variable (radius) necessitate adjustments in another (speed), underscoring the importance of understanding functional relationships in physics.