Problem 47 - Algebra and Trigonometry with Analytic Geometry

Question

Rotation of compact discs (CDs) 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. (c) Find the angular speed (in rpm) of the drive motor when it is reading a track 3 centimeters from the center of the CD. (d) Find a function \(S\) that gives the drive motor speed in 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

(a) 400π rad/min; (b) ~119.4 cm/sec; (c) ~760 rpm; (d) Inversely proportional function, S(r) = 2280/r, validates results.

Step 1: Convert RPM to Radians Per Minute

The angular speed in radians per minute can be found using the conversion factor where \[ 1 \text{ rpm} = \frac{2\pi}{60} \text{ rad/min}. \]Given that the CD rotates at 200 rpm when reading a track 5.7 cm from the center, the angular speed is\[ \omega = 200 \times \frac{2\pi}{1} = 400\pi \text{ rad/min}. \]

Step 2: Calculate Linear Speed

The linear speed at a distance from the center can be calculated as \[ v = \omega \times r, \]where \( \omega \) is the angular speed in rad/min, and \( r \) is the radius. Here, \( \omega = 400\pi \text{ rad/min} \) and \( r = 5.7 \text{ cm} \), so \[ v = 400\pi \times 5.7 \text{ cm/min}. \]Convert cm/min to cm/sec: \[ v = \frac{400\pi \times 5.7}{60} \approx 119.4 \text{ cm/sec}. \]

Step 3: Find Angular Speed at Different Radius

Since the linear speed must remain constant, we equate the linear speed at 3 cm to that at 5.7 cm: \[ \omega_1 \times 3 = 400\pi \times 5.7 \Rightarrow \omega_1 = \frac{400\pi \times 5.7}{3} \approx 760 \text{ rpm}. \]

Step 4: Determine the Motor Speed Function

The speed of the motor as a function of radius, \( S(r) \), is given by keeping the linear speed constant:\[ \omega \times r = \text{constant}, \]specifically,\[ S(r) = \frac{400 \times 5.7}{r}, \]where \( S(r) = \omega \) in rpm and \( r \) is in cm. This is an example of inverse variation, as the radius decreases, the angular speed of the motor increases.

Step 5: Graph and Check Function

In the graph of \( S(r) = \frac{2280}{r} \) over the interval \( 2.3 \leq r \leq 5.9 \), you should find: when \( r = 3 \), \( S(3) = 760 \text{ rpm} \) and when \( r = 5.7 \), \( S(5.7) = 400 \text{ rpm}. \) The graph confirms the inverse relationship between \( S \) and \( r \).

Key Concepts

RPM to Radians Conversion

When working with rotations, converting revolutions per minute (RPM) to radians per minute is crucial. In this exercise, we start with a CD rotating at 200 RPM. To change this to radians per minute, we use the fact that one revolution corresponds to the angle of a full circle, which is \( 2\pi \) radians.
The conversion factor turns out to be \( 1 \text{ rpm} = \frac{2\pi}{60} \text{ rad/min} \). The calculation for our CD is simple – multiply 200 RPM by the conversion factor:

• The formula is \( \omega = 200 \times \frac{2\pi}{1} = 400\pi \text{ rad/min} \).

This result shows us the angular speed of the CD motor in a more versatile measurement unit, radians, which is key for further mathematical calculations.

Linear Speed Calculation

To find how fast a point on the CD moves, we need to calculate its linear speed. This is related to angular speed and the distance from the center. The relationship is defined by the formula \( v = \omega \times r \), where \( v \) is linear speed, \( \omega \) is angular speed, and \( r \) is the radius.
Inserting our values, \( \omega = 400\pi \text{ rad/min} \) and \( r = 5.7 \) cm, we find:

• \( v = 400\pi \times 5.7 \text{ cm/min} \).

To change this speed to centimeters per second, convert minutes to seconds by dividing by 60. Therefore, the linear speed becomes \( v \approx 119.4 \text{ cm/sec} \). This indicates the speed at which information is read from the CD track 5.7 centimeters from the center.

Inverse Variation

The exercise highlights an inverse variation relationship between motor speed in RPM and the track radius being read. This means that as the radius decreases, the angular speed must increase to keep the linear speed constant. Such an inverse relationship can be described in the formula:

• \( \omega \times r = \text{constant} \).

This implies:

• \( S(r) = \frac{400 \times 5.7}{r} \),

where \( S(r) \) is the RPM and \( r \) is the track radius in centimeters. The function \( S(r) \) gives us the motor speed necessary to maintain constant data reading speed and clearly illustrates that when \( r \) becomes smaller, \( S(r) \) becomes larger. This is a classic example of how inverse variations are applied in real-life scenarios.

Mathematical Functions

Understanding mathematical functions is essential to calculate how CD player motor speed varies with radius. In this problem, we define a function \( S \) such that \( S(r) = \frac{2280}{r} \). Understanding this function is valuable because it mathematically models the real process of maintaining constant data reading rates.
The function needs verification, which can be done by picking values for \( r \), such as 3 cm and 5.7 cm, and confirming the corresponding speeds. The calculated speeds are 760 RPM and 400 RPM, respectively. Graphing \( S(r) \) on the defined interval \( 2.3 \leq r \leq 5.9 \) allows us to visualize the inverse variance nature of the function and see how it behaves with changes in \( r \).
This exercise demonstrates how functions model physical phenomena and test understanding through problem-solving.