Problem 63 - Precalculus

Question

When being burned in a writable CD-R drive, the angular speed of a CD is often much faster than when playing audio, but the angular speed still varies to keep the linear speed constant where the disc is being written. When writing along the outer edge of the disc, the angular speed of one drive is about 4800 RPM (revolutions per minute). Find the linear speed if the CD has diameter of 120 millimeters.

Step-by-Step Solution

Answer

The linear speed is approximately 1809.56 meters per minute.

Step 1: Convert Diameter to Radius

First, we convert the diameter of the CD to a radius. The diameter is given as 120 millimeters. Radius is obtained by dividing the diameter by 2. Thus, the radius \( r \) of the CD is \( 120 \text{ mm} \div 2 = 60 \text{ mm} \).

Step 2: Convert Millimeters to Meters

To find the linear speed in meters per minute, we must first convert the radius from millimeters to meters, since 1 meter equals 1000 millimeters. Therefore, the radius in meters is \( 60 \text{ mm} \div 1000 = 0.06 \text{ m} \).

Step 3: Calculate the Angular Speed in Radians per Minute

The angular speed is given in revolutions per minute (RPM), which needs to be converted to radians per minute since linear speed is calculated using radians. One revolution equals \( 2\pi \) radians. So, \( 4800 \text{ RPM} \times 2\pi \text{ radians/revolution} = 9600\pi \text{ radians/minute} \).

Step 4: Calculate the Linear Speed

Linear speed \( v \) is given by the formula \( v = r \times \omega \), where \( r \) is the radius and \( \omega \) is the angular velocity in radians per unit time. Using our values, we have \( v = 0.06 \text{ m} \times 9600\pi \text{ radians/minute} \). Simplifying this gives \( v = 576\pi \text{ m/minute} \), and using \( \pi \approx 3.1416 \) gives \( v \approx 1809.56 \text{ m/minute} \).

Key Concepts

Diameter to Radius Conversion

In geometry, understanding the relationship between diameter and radius is crucial, particularly when dealing with circular objects. For any circle, the diameter is defined as the longest distance passing through the center, connecting two points on the circle. The radius is half that length, extending from the center to any point on the circle’s edge. To convert the diameter to the radius, simply divide by two. This makes calculations involving circles much more manageable. For instance, in the CD example where the diameter is 120 mm, dividing by 2 gives a radius of 60 mm.

• Formula: Radius = Diameter / 2
• Example: 120 mm / 2 = 60 mm

Unit Conversion

Unit conversion is essential in mathematical calculations to maintain consistency and accuracy, particularly in physics and engineering. When given measurements, converting them into a common unit system is a first critical step. In many cases, like with the CD, converting from millimeters to meters is necessary because standard velocity formulas use meters.

• 1 meter equals 1000 millimeters.
• To convert, divide millimeters by 1000.

For example, converting a radius of 60 mm to meters involves calculation: 60 mm divided by 1000, resulting in 0.06 m.

Angular Speed in Radians

Angular speed refers to the rate of rotation, typically expressed in terms of revolutions per minute (RPM) or radians per second/minute. In many physical applications, it's often necessary to convert RPM to radians per minute, especially for calculating linear speed. This is because the standard unit of angular measurement in physics is radians.

• 1 revolution = 2\( \pi \) radians.
• To convert RPM to radians/minute, multiply RPM by \( 2\pi \).

In our example, the CD spins at 4800 RPM, which can be converted to radians as follows: 4800 RPM \( \times 2\pi \) equals 9600\( \pi \) radians/minute. This conversion is fundamental for later linking angular movement to linear speed.

Geometry of Circles

The geometry of circles provides the foundation for solving various mathematical problems involving circular motion, such as determining linear speed. The relationship between angular speed and linear speed is crucial when interpreting circular movement.

• Linear speed \( v \) is tied to the radius \( r \) and angular speed \( \omega \).
• The formula \( v = r \times \omega \) captures this relationship.

This formula indicates that linear speed increases with larger radius or greater angular velocity, illustrating why converting units and measuring accurately is essential. In the CD example, calculating linear speed involves using the radius in meters and the angular speed in radians to find that it travels approximately 1809.56 m/minute.