/*! 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 -cm radius. The disk rotates at a constant angular velocity of 1.4 rev/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 59.34 meters.

Step by step solution

01

Understand the Problem

We need to find out how far a spot on the outer edge of the rotating disk of the floor polisher moves during a given time. For this, we'll calculate the total distance covered by a point on the disk's circumference in that time period.
02

Calculate the Circumference of the Disk

The circumference of the disk, which is a circle, can be calculated using the formula: \( C = 2 \pi r \). Given that the radius \( r \) is 15 cm, the circumference \( C \) is \( C = 2 \times \pi \times 15 \).
03

Convert Radius to Meters

Convert the radius from centimeters to meters to maintain consistency of units in calculations: \( 15 \text{ cm} = 0.15 \text{ m} \).
04

Calculate the Angular Distance in Revolutions

The polisher rotates at an angular velocity of 1.4 revolutions per second \( \text{rev/s} \). Over 45 seconds, it rotates \( 45 \times 1.4 = 63 \) revolutions.
05

Calculate Total Distance Moved

The total distance moved is the number of revolutions multiplied by the circumference: \( \text{Distance} = 63 \times (2 \pi \times 0.15) \text{ m} \).
06

Final Calculation

Calculate the final value of the distance: \( \text{Distance} = 63 \times 0.30\pi \approx 59.34 \text{ meters} \).

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 something rotates. It is the rate at which an object changes its angle, typically measured in revolutions per second (rev/s) or radians per second (rad/s). For our floor polisher, the angular velocity is given as 1.4 rev/s.
This means that every second, the disk completes 1.4 full circles. Understanding angular velocity helps us know how quick each complete rotation is and is essential for determining how much distance is covered over time.
  • Angular velocity tells us the speed of rotation.
  • Measured in rev/s, radians/s, or degrees/s.
  • 1.4 rev/s implies 1.4 complete turns every second.
Circumference Calculation
To find out how far a point on the edge of a rotating disk moves, we need to calculate its circumference. The circumference of a circle is the distance around it. It's calculated with the formula: \[ C = 2 \pi r \] where \( r \) is the radius of the circle.
For the polisher, the radius of the disk is 15 cm. When converted to meters (which we'll discuss further below), it's 0.15 m. So the circumference becomes: \[ C = 2 \pi \times 0.15 = 0.3\pi \] m.
  • Formula: \( C = 2 \pi r \)
  • Essential to find the travel distance of a point on the edge.
  • Radius for calculations should be in consistent units.
Unit Conversion
Unit conversion is crucial for consistency in calculations. Our exercise involves measurements in both centimeters and meters. Since we aim for the final distance in meters, all measurements must be in the same unit.
In this case, the radius is initially given as 15 cm. To convert to meters (the standard unit for length in physics), we use the conversion factor: \[ 1 ext{ cm} = 0.01 ext{ m} \].
Applying this, the radius becomes 0.15 m. This step ensures that the circumference calculation and the final distance are all in meters.
  • Be consistent with units to avoid calculation errors.
  • Convert cm to m by multiplying by 0.01.
  • Larger units make numbers easier to manage in physics.
Distance Calculation
After identifying the circumference and understanding the angular velocity, you can calculate the total distance a point on the disk edge moves. The trick is to combine the total revolutions completed over the time period with the circumference.
First, calculate total revolutions: \( 1.4 ext{ rev/s} \times 45 ext{ s} = 63 ext{ revolutions} \).
Next, multiply the number of revolutions by the circumference: \[ ext{Total Distance} = 63 \times (2\pi \times 0.15) = 63 \times 0.3\pi \].Carrying out the calculations finally, this comes out to approximately 59.34 meters. This represents the total path length traveled by a point on the peripheral of the disk in 45 seconds.
  • Total revolutions are time \( \times \) angular velocity.
  • Distance \( = \) revolutions \( \times \) circumference.
  • Final distance quantifies how far the spot traveled.

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

The sun has a mass of \(1.99 \times 10^{30} \mathrm{kg}\) and is moving in a circular orbit about the center of our galaxy, the Milky Way. The radius of the orbit is \(2.3 \times 10^{4}\) light-years ( 1 light-year \(\left.=9.5 \times 10^{15} \mathrm{m}\right)\), and the angular speed of the sun is \(1.1 \times 10^{-15} \mathrm{rad} / \mathrm{s} .\) (a) Determine the tangential speed of the sun. (b) What is the magnitude of the net force that acts on the sun to keep it moving around the center of the Milky Way?

A motorcycle accelerates uniformly from rest and reaches a linear speed of \(22.0 \mathrm{m} / \mathrm{s}\) in a time of \(9.00 \mathrm{s}\). The radius of each tire is \(0.280 \mathrm{m}\) What is the magnitude of the angular acceleration of each tire?

A baseball pitcher throws a baseball horizontally at a linear speed of \(42.5 \mathrm{m} / \mathrm{s}\) (about \(95 \mathrm{mi} / \mathrm{h}\) ). Before being caught, the baseball travels a horizontal distance of \(16.5 \mathrm{m}\) and rotates through an angle of \(49.0 \mathrm{rad}\). The baseball has a radius of \(3.67 \mathrm{cm}\) and is rotating about an axis as it travels, much like the earth does. What is the tangential speed of a point on the "equator" of the baseball?

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 car is traveling along a road, and its engine is turning over with an angular velocity of \(+220 \mathrm{rad} / \mathrm{s} .\) The driver steps on the accelerator, and in a time of \(10.0 \mathrm{s}\) the angular velocity increases to \(+280 \mathrm{rad} / \mathrm{s}\). (a) What would have been the angular displacement of the engine if its angular velocity had remained constant at the initial value of \(+220 \mathrm{rad} / \mathrm{s}\) during the entire \(10.0-\mathrm{s}\) interval? (b) What would have been the angular displacement if the angular velocity had been equal to its final value of \(+280 \mathrm{rad} / \mathrm{s}\) during the entire \(10.0-\mathrm{s}\) interval? (c) Determine the actual value of the angular displacement during the \(10.0-\) s interval.
See all solutions

Recommended explanations on Physics Textbooks

Electricity

Read Explanation

Electricity and Magnetism

Read Explanation

Linear Momentum

Read Explanation

Work Energy and Power

Read Explanation

Scientific Method Physics

Read Explanation

Fundamentals of Physics

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.