/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 A floor polisher has a rotating ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 10

A floor polisher has a rotating disk that has a \(15-\mathrm{cm}\) radius. The disk rotates at a constant angular velocity of \(1.4 \mathrm{rev} / \mathrm{s}\) and is covered with a soft material that does the polishing. An operator holds the polisher in one place for \(45 \mathrm{s}\), in order to buff an especially scuffed area of the floor. How far (in meters) does a spot on the outer edge of the disk move during this time?

Short Answer

Expert verified
The spot moves approximately 59.35 meters.

Step by step solution

01

Determine the Distance per Revolution

First, calculate the circumference of the rotating disk. This is the distance a point at the outer edge moves in one complete revolution. The formula for the circumference of a circle is \(C = 2\pi r\) where \(r\) is the radius. Here, \(r = 15\,\mathrm{cm}\) or \(0.15\,\mathrm{m}\). Thus, the circumference \(C\) is \(2\pi \times 0.15 = 0.3\pi \; \mathrm{m}\).
02

Calculate Total Number of Revolutions

Next, determine how many complete revolutions the disk makes in the given time. We know the disk rotates at \(1.4\ \mathrm{rev/s}\), and the operator holds it in place for \(45\ \mathrm{s}\). Thus, the total number of revolutions is \(1.4 \times 45 = 63\ \mathrm{revolutions}\).
03

Compute Total Distance Traveled

Now, calculate the total distance traveled by the spot on the disk's edge. Multiply the circumference by the total number of revolutions. Thus, the total distance is \(0.3\pi \times 63 = 18.9\pi\ \mathrm{m}\).
04

Final Calculation

For a numerical answer, approximate \(\pi\) as \(3.14\). Therefore, the distance becomes approximately \(18.9 \times 3.14 = 59.346\ \mathrm{m}\).

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Velocity
Angular velocity is a measure of how fast an object rotates or revolves relative to another point, such as the center of a circle. It's often expressed in revolutions per second (rev/s) or radians per second. In our exercise, the disk of the floor polisher rotates at an angular velocity of \(1.4\ \mathrm{rev/s}\), indicating how many complete turns it makes each second.
  • Understanding angular velocity helps in figuring out the rotational speed of objects.
  • It indicates the number of rotations or revolutions occurring in a specified timeframe.
Angular velocity is crucial for calculating the number of times an object like a disk spins in a set period, which then helps to determine further motion-related parameters. Always remember, higher the angular velocity, faster the rotation.
Circumference
The circumference of a circle is the distance around its edge or perimeter. It's calculated using the formula \(C = 2\pi r\), where \(r\) is the radius. In this problem, the radius of the disk is given as \(15\ \mathrm{cm}\), or \(0.15\ \mathrm{m}\).
  • Circumference tells us how far a point travels if it makes one complete revolution along a circle's edge.
  • For the polishing disk, \(C = 2\pi \times 0.15 = 0.3\pi\ \mathrm{m}\).
  • This formula is fundamental in calculating the travel distance in rotational motion.
Knowing the circumference allows us to determine the distance a point on the disk's perimeter will travel in one full spin.
Distance Calculation
Distance calculation for a rotating object involves several key steps starting with understanding the object's path. For a point on the rotating disk, this path is defined by the circumference.
  • First, determine the distance per revolution using the circumference.
  • Multiply this distance by the number of revolutions to find the total distance.
  • In our case: total distance = \(0.3\pi \times 63 = 18.9\pi\ \mathrm{m}\).
This calculation reveals how far a point has traveled in a given amount of spinning, providing insights into movement in circular paths. Approximating \(\pi\) as \(3.14\), the total distance a spot moves on the outer edge of the polishing disk is roughly \(59.346\ \mathrm{m}\).
Revolutions
Revolutions are a unit for counting the complete turns an object makes around its axis. For the floor polisher, the disk completes several full revolutions while it is stationary timewise.
  • The disk's angular velocity, \(1.4\ \mathrm{rev/s}\), tells us how many revolutions occur every second.
  • Held stationary for \(45\ \mathrm{s}\), the total revolutions are \(1.4 \times 45 = 63\).
  • Each revolution contributes a full circumference of movement.
Calculating the number of revolutions provides the multiplier to use with the circumference when determining total traveled distance. This concept is essential in understanding rotational motion and distances covered by rotating objects.

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

A bicycle is rolling down a circular portion of a path; this portion of the path has a radius of \(9.00 \mathrm{m}\). As the drawing illustrates, the angular displacement of the bicycle is 0.960 rad. What is the angle (in radians) through which each bicycle wheel (radius \(=0.400 \mathrm{m}\) ) rotates?

The earth has a radius of \(6.38 \times 10^{6} \mathrm{m}\) and turns on its axis once every \(23.9 \mathrm{h}\). (a) What is the tangential speed (in \(\mathrm{m} / \mathrm{s}\) ) of a person living in Ecuador, a country that lies on the equator? (b) At what latitude (i.e., the angle \(\theta\) in the drawing ) is the tangential speed one-third that of a person living in Ecuador?

In 9.5 s a fisherman winds \(2.6 \mathrm{m}\) of fishing line onto a reel whose radius is \(3.0 \mathrm{cm}\) (assumed to be constant as an approximation). The line is being reeled in at a constant speed. Determine the angular speed of the reel.

A top is a toy that is made to spin on its pointed end by pulling on a string wrapped around the body of the top. The string has a length of \(64 \mathrm{cm}\) and is wound around the top at a spot where its radius is \(2.0 \mathrm{cm}\). The thickness of the string is negligible. The top is initially at rest. Someone pulls the free end of the string, thereby unwinding it and giving the top an angular acceleration of \(+12 \mathrm{rad} / \mathrm{s}^{2} .\) What is the final angular velocity of the top when the string is completely unwound?

A pitcher throws a curveball that reaches the catcher in 0.60 s. The ball curves because it is spinning at an average angular velocity of \(330 \mathrm{rev} / \mathrm{min}\) (assumed constant) on its way to the catcher's mitt. What is the angular displacement of the baseball (in radians) as it travels from the pitcher to the catcher?
See all solutions

Recommended explanations on Physics Textbooks

Force

Read Explanation

Astrophysics

Read Explanation

Dynamics

Read Explanation

Oscillations

Read Explanation

Physics of Motion

Read Explanation

Electromagnetism

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.