/*! 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 61 The penny-farthing is a bicycle ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 61

The penny-farthing is a bicycle that was popular between 1870 and \(1890 .\) As the drawing shows, this type of bicycle has a large front wheel and a small rear wheel. During a ride, the front wheel (radius \(=1.20 \mathrm{m})\) makes 276 revolutions. How many revolutions does the rear wheel (radius \(=0.340 \mathrm{m}\) ) make?

Short Answer

Expert verified
The rear wheel makes approximately 976 revolutions.

Step by step solution

01

Calculate the Distance Traveled by the Front Wheel

First, we need to find the circumference of the front wheel which is the distance the wheel covers in one complete revolution. This can be found using the formula for the circumference of a circle: \[ C = 2\pi r \]where \(r\) is the radius of the wheel. For the front wheel, \( r = 1.20 \, \text{m} \). Thus, the circumference is:\[ C = 2 \times \pi \times 1.20 = 2.4\pi \, \text{m} \]Since it makes 276 revolutions, the total distance traveled by the front wheel is:\[ \text{Distance} = 276 \times 2.4\pi \]
02

Calculate the Equivalent Revolutions for the Rear Wheel

Next, we'll find the number of revolutions the rear wheel makes while covering the same distance. We'll use the circumference of the rear wheel for this.First, calculate the circumference of the rear wheel (radius \(0.340 \, \text{m}\)) using the formula:\[ C = 2 \pi r \]Thus, the circumference is:\[ C = 2 \times \pi \times 0.340 = 0.68\pi \, \text{m} \]The number of revolutions \(N\) of the rear wheel needed to travel the same distance as the front wheel is given by:\[ N = \frac{\text{Total Distance}}{\text{Circumference of Rear Wheel}} = \frac{276 \times 2.4 \pi}{0.68 \pi} \]Simplifying, we have:\[ N = \frac{276 \times 2.4}{0.68} \approx 975.88 \]
03

Round the Number of Revolutions

Since the number of revolutions must be a whole number, we round 975.88 to the nearest whole number, which is 976.

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.

Radius of a Wheel
The radius is the distance from the center of a circle to any point on its edge. This measurement is crucial when determining how far a wheel can travel in one complete rotation. In the case of the penny-farthing bicycle, the radius helps us figure out different aspects such as circumference and distance traveled. For example, the larger front wheel of the penny-farthing has a radius of 1.20 meters. This significant size means it covers more ground with each revolution than the smaller rear wheel, which has a radius of 0.34 meters. Understanding the radius is important because it serves as the foundation for calculating the wheel's circumference and ultimately, the distance traveled. It's the starting point for our mathematics involving wheels.
Understanding Circumference
The circumference of a circle is the distance around its outer edge. Think of it as the wheel's perimeter. For a bicycle wheel, this means the distance the wheel covers in one complete rotation. To calculate the circumference, we use the formula:\[ C = 2\pi r \]where \(C\) is the circumference and \(r\) is the radius of the wheel. It tells us how far the wheel will travel with one turn.
  • Front wheel with radius 1.20 m has a circumference calculated as \(2.4\pi \) meters.
  • Rear wheel with a radius of 0.34 meters has a circumference of \(0.68\pi \) meters.
Knowing the circumference helps determine how many revolutions the wheel takes to cover a certain distance, making it a key concept in this type of mathematics problem.
Counting Revolutions
Revolutions refer to the number of times a wheel fully rotates around its axis. It is a measure of how far a bicycle travels based on how many times its wheels turn. In our problem with the penny-farthing bicycle, the front wheel completes 276 revolutions. By knowing the circumference of the wheel, we can calculate the total distance traveled as:\[ \text{Distance} = 276 \times 2.4\pi \].To make these calculations for the rear wheel, we need to ensure it covers the same distance. Therefore, we calculate the number of revolutions for the smaller rear wheel using:\[ N = \frac{276 \times 2.4\pi}{0.68\pi} \].This calculation shows us how many times the smaller wheel must turn to keep up with the larger wheel, helping us compare distances effectively.
Calculating Distance Traveled
Distance traveled refers to the total length the penny-farthing covers when riding. Understanding this concept helps us grasp how measurements such as wheel radius and circumference influence the overall journey.To find out the distance traveled by one wheel - say, the front wheel - we multiply the number of revolutions by the wheel's circumference. In this exercise, the front wheel travels \(276 \times 2.4\pi \) meters in total.To ensure the rear wheel travels the same distance, we calculate how many revolutions it needs based on its circumference, confirming:\[ \text{Revolutions} = \frac{\text{Total Distance}}{\text{Circumference of Rear Wheel}} \].Through this understanding, we can determine how efficient different wheel sizes are in covering distances, giving us powerful insight into the engineering feats of the penny-farthing bicycle.

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 drawing shows a golf ball passing through a windmill at a miniature golf course. The windmill has 8 blades and rotates at an angular speed of \(1.25 \mathrm{rad} / \mathrm{s}\). The opening between successive blades is equal to the width of a blade. A golf ball (diameter \(\left.4.50 \times 10^{-2} \mathrm{m}\right)\) has just reached the edge of one of the rotating blades (see the drawing). Ignoring the thickness of the blades, find the minimum linear speed with which the ball moves along the ground, such that the ball will not be hit by the next blade.

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 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?

The differential gear of a car axle allows the wheel on the left side of a car to rotate at a different angular speed than the wheel on the right side. A car is driving at a constant speed around a circular track on level ground, completing each lap in 19.5 s. The distance between the tires on the left and right sides of the car is \(1.60 \mathrm{m}\), and the radius of each wheel is \(0.350 \mathrm{m} .\) What is the difference between the angular speeds of the wheels on the left and right sides of the car?

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?

A ball of radius \(0.200 \mathrm{m}\) rolls with a constant linear speed of \(3.60 \mathrm{m} / \mathrm{s}\) along a horizontal table. The ball rolls off the edge and falls a vertical distance of \(2.10 \mathrm{m}\) before hitting the floor. What is the angular displacement of the ball while the ball is in the air?
See all solutions

Recommended explanations on Physics Textbooks

Scientific Method Physics

Read Explanation

Waves Physics

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Physical Quantities And Units

Read Explanation

Turning Points in Physics

Read Explanation

Electric Charge Field and Potential

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.