Math Problem Statement

a CD has a diameter of 110 millimeters. When playing audio, the angular speed varies to keep the linear speed constant where the disc is being read. When reading along the outer edge of the disc, the angular speed is about 220 RPM (revolutions per minute). Find the linear speed.

Solution

To find the linear speed when reading the outer edge of the CD, we can use the relationship between angular speed and linear speed. The formula for this is:

v=rωv = r \cdot \omega

Where:

  • vv is the linear speed.
  • rr is the radius of the CD.
  • ω\omega is the angular speed (in radians per second).

Step 1: Convert the diameter to radius.

Since the diameter of the CD is 110 mm, the radius is half of that:

r=110 mm2=55 mm=0.055 metersr = \frac{110 \text{ mm}}{2} = 55 \text{ mm} = 0.055 \text{ meters}

Step 2: Convert the angular speed from RPM to radians per second.

We are given the angular speed as 220 RPM (revolutions per minute). To convert RPM to radians per second:

ω=220 RPM×2π radians1 revolution×1 minute60 seconds\omega = 220 \text{ RPM} \times \frac{2\pi \text{ radians}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}}

ω=220×2π60\omega = 220 \times \frac{2\pi}{60}

Let’s calculate this.

ω=440π6023.04 radians/second\omega = \frac{440\pi}{60} \approx 23.04 \text{ radians/second}

Step 3: Calculate the linear speed.

Now, we can plug in the values into the linear speed formula:

v=rω=0.055 m23.04 rad/sv = r \cdot \omega = 0.055 \text{ m} \cdot 23.04 \text{ rad/s}

v1.267 meters per secondv \approx 1.267 \text{ meters per second}

Thus, the linear speed is approximately 1.27 meters per second.

Would you like further details or have any questions?


Here are five related questions to expand on the topic:

  1. How does the angular speed change when reading closer to the center of the CD?
  2. What would the linear speed be at the innermost part of the disc if the radius is much smaller?
  3. Why does the CD player keep the linear speed constant during playback?
  4. How do you convert linear speed back to angular speed if the radius and speed are known?
  5. How does the concept of angular velocity apply to other rotating objects like hard drives?

Tip: Always ensure units are consistent when using physical equations—convert millimeters to meters, RPM to radians per second, etc.

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Math Problem Analysis

Mathematical Concepts

Circular Motion
Angular Velocity
Linear Velocity

Formulas

v = r * ω
ω (radians per second) = RPM * (2π / 60)

Theorems

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Suitable Grade Level

Grades 10-12