Log out Go to App
Go to App

Learning Materials

Features

Discover

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Mars orbits the Sun at a mean distance of 228 million \(\mathrm{km},\) in a period of 687 days. The Earth orbits at a mean distance of 149.6 million \(\mathrm{km},\) in a period of 365.26 days. a) Suppose Earth and Mars are positioned such that Earth lies on a straight line between Mars and the Sun. Exactly 365.26 days later, when the Earth has completed one orbit, what is the angle between the Earth-Sun line and the Mars-Sun line? b) The initial situation in part (a) is a closest approach of Mars to Earth. What is the time, in days, between two closest approaches? Assume constant speed and circular orbits for both Mars and Earth. c) Another way of expressing the answer to part (b) is in terms of the angle between the lines drawn through the Sun, Earth, and Mars in the two closest approach situations. What is that angle?

A CD starts from rest and speeds up to the operating angular frequency of the CD player. Compare the angular velocity and acceleration of a point on the edge of the CD to those of a point halfway between the center and the edge of the CD. Do the same for the linear velocity and acceleration.

A monster truck has tires with a diameter of \(1.10 \mathrm{~m}\) and is traveling at \(35.8 \mathrm{~m} / \mathrm{s}\). After the brakes are applied, the truck slows uniformly and is brought to rest after the tires rotate through 40.2 turns. a) What is the initial angular speed of the tires? b) What is the angular acceleration of the tires? c) What distance does the truck travel before coming to rest?

A hook above a stage is rated to support \(150 .\) lb. A 3 -lb rope is attached to the hook, and a 147 -lb actor is going to attempt to swing across the stage on the rope. Will the hook hold the actor up during the swing?

A \(80.0-\mathrm{kg}\) pilot in an aircraft moving at a constant speed of \(500 . \mathrm{m} / \mathrm{s}\) pulls out of a vertical dive along an arc of a circle of radius \(4000 . \mathrm{m}\).
See all solutions

Recommended explanations on Physics Textbooks

Modern Physics

Read Explanation

Kinematics Physics

Read Explanation

Physical Quantities And Units

Read Explanation

Scientific Method Physics

Read Explanation

Fundamentals of Physics

Read Explanation

Rotational Dynamics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free