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Chapter 9: Problem 64

Determine the linear and angular speeds and accelerations of a speck of dirt located \(2.0 \mathrm{~cm}\) from the center of a CD rotating inside a CD player at 250 rpm.

Short Answer

Expert verified
Answer: The linear speed of the speck of dirt is approximately 0.5236 meters per second, and its linear acceleration is 0. The angular speed is approximately 26.18 radians per second, and the angular acceleration is 0.

Step by step solution

01

Convert rotational speed to angular speed

To find the angular speed of the speck of dirt on the CD, we first need to convert the given rotational speed (250 rpm) to angular speed in radians per second. To do this, we can use the conversion factor: 1 revolution = \(2\pi\) radians. Also, we need to convert minutes to seconds using the conversion factor: 1 minute = 60 seconds. The conversion from rpm to radians per second can be done as follows: Angular speed (in radians per second) = (Rotational speed (in rpm) * (2π radians per revolution)) / (60 seconds per minute)
02

Calculate the angular speed

Now, we can plug in the given values to calculate the angular speed: Angular speed = (250 * 2π) / 60 ≈ 26.18 radians per second
03

Calculate the linear speed

Next, we need to find the linear speed of the speck of dirt. Linear speed (v) can be calculated using the formula: v = rω, where r is the radius (distance from the center) and ω is the angular speed. Given that the speck of dirt is 2.0 cm from the center of the CD, we can convert the distance to meters (to keep units consistent) by dividing by 100. So, the radius is 0.02 meters. Now we can calculate the linear speed: Linear speed = rω = 0.02 * 26.18 ≈ 0.5236 meters per second
04

Calculate the angular acceleration

Since the CD is rotating at a constant speed (250 rpm), there is no angular acceleration acting on the speck of dirt. Therefore, the angular acceleration is 0.
05

Calculate the linear acceleration

Finally, we need to find the linear acceleration of the speck of dirt. Linear acceleration (a) can be calculated using the formula: a = rα, where r is the radius (distance from the center) and α is the angular acceleration. As we found in Step 4, the angular acceleration is 0. This means that the linear acceleration is also 0 since there is no change in the linear speed of the speck of dirt. To summarize, the linear speed of the speck of dirt is approximately 0.5236 meters per second and its linear acceleration is 0. The angular speed is approximately 26.18 radians per second and the angular acceleration is 0.

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