/*! 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 56 Suppose you are riding a station... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 56

Suppose you are riding a stationary exercise bicycle, and the electronic meter indicates that the wheel is rotating at 9.1 rad/s. The wheel has a radius of 0.45 m. If you ride the bike for 35 min, how far would you have gone if the bike could move?

Short Answer

Expert verified
The distance would be about 8.597 km.

Step by step solution

01

Understand the Problem

We need to determine the distance you would have traveled if the bike could move given the wheel's rotational speed and radius, over a specific time duration.
02

Calculate the Circumference of the Wheel

The circumference of the wheel (which is the distance traveled in one full rotation) is given by the formula \( C = 2\pi r \), where \( r \) is the radius of the wheel. Here, \( r = 0.45 \text{ m} \). Thus, \( C = 2\pi \times 0.45 \approx 2.826 \text{ m} \).
03

Calculate the Total Rotations

We determine the number of rotations the wheel makes in 35 minutes. The wheel rotates at 9.1 rad/s, converting this into revolutions per second: \(\text{revolutions/s} = \frac{9.1}{2\pi} \approx 1.448 \text{ revolutions/s} \). The total number of seconds in 35 minutes is \(35 \times 60 = 2100 \text{ s}\). So, total rotations = \(1.448 \times 2100 \approx 3040.8\text{ revolutions}\).
04

Calculate the Total Distance Traveled

Multiply the total number of rotations by the circumference to find the total distance: \(\text{Distance} = 3040.8 \times 2.826 \approx 8596.85 \text{ m}\).
05

Convert the Distance to Kilometers

Convert the distance from meters to kilometers by dividing by 1000: \( \frac{8596.85}{1000} \approx 8.597 \text{ km}\).

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.

Exercise Bicycle Physics
Exercise bicycles offer a great way to explore concepts in physics, particularly those of rotational motion. Unlike bicycles that move through space, exercise bicycles are stationary. But their wheels still spin when you pedal. The movement of these wheels involves rotational motion, similar to how a bicycle wheel moves forward when riding outdoors. For those learning physics, this provides an excellent opportunity to study how concepts like angular speed and circumference can be applied in calculating theoretical travel distances.
Exercise bicycles help us understand that even when an object is not changing its position, the rotational aspects of its wheels can be studied to determine the distance it would theoretically cover if it could actually move.
Angular Speed
Angular speed is a measurement that describes how quickly an object revolves around a point or axis. In the scenario of an exercise bicycle, angular speed represents how fast the wheel is spinning.
It's usually measured in radians per second (rad/s). One complete turn of the wheel corresponds to an angular displacement of 2蟺 radians. Converting radians per second to revolutions per second involves dividing by the full circle radian measure, 2蟺.
Angular speed is essential in determining how many revolutions a wheel will make over a certain period, which is a crucial part of calculating potential distance traveled.
Calculation of Distance
The calculation of distance traveled is primarily about figuring out how far you would go if the stationary exercise bicycle could move. Because the bike isn鈥檛 moving, we use rotational movement to calculate a theoretical travel distance.
Here's how to do it:
  • Calculate the circumference of the wheel.
  • Determine how many rotations the wheel makes in the given time frame.
  • Multiply the number of rotations by the circumference to get the total distance.
  • Convert the result from meters to kilometers for more familiar measurement.
This gives you a good understanding of how rotational motion translates into theoretical linear travel in the context of a non-moving exercise bike.
Circumference of a Circle
The circumference of a circle is the distance around it. For a bicycle wheel, this is the path it would cover in one complete rotation.
The circumference is calculated using the formula: \[ C = 2\pi r \]where \( C \) is the circumference and \( r \) is the radius of the circle.
Knowing the circumference allows us to determine how much distance is covered in one full revolution of the wheel. This step is crucial in understanding rotational motion and how to calculate the distance in physics problems related to circular motion.

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

ssm A gymnast is performing a floor routine. In a tumbling run she spins through the air, increasing her angular velocity from 3.00 to 5.00 rev/s while rotating through one-half of a revolution. How much time does this maneuver take?

At the local swimming hole, a favorite trick is to run horizontally off a cliff that is 8.3 \(\mathrm{m}\) above the water. One diver runs off the edge of the cliff, tucks into a 鈥渂all,鈥� and rotates on the way down with an average angular speed of 1.6 rev/s. Ignore air resistance and determine the number of revolutions she makes while on the way down.

A person lowers a bucket into a well by turning the hand crank, as the drawing illustrates. The crank handle moves with a constant tangential speed of 1.20 m/s on its circular path. The rope holding the bucket unwinds with- out slipping on the barrel of the crank. Find the linear speed with which the bucket moves down the well.

The earth spins on its axis once a day and orbits the sun once a year \(\left(365_{4}^{1} \text { days). Determine the average angular velocity (in rad/s) of the }\right.\) earth as it \((\text { a) spins on its axis and }(b) \text { orbits the sun. In each case, }\) take the positive direction for the angular displacement to be the direction of the earth's motion.

A car is traveling with a speed of \(20.0 \mathrm{~m} / \mathrm{s}\) along a straight horizontal road. The wheels have a radius of \(0.300 \mathrm{~m} .\) If the car speeds up with a linear acceleration of \(1.50 \mathrm{~m} / \mathrm{s}^{2}\) for \(8.00 \mathrm{~s},\) find the angular displacement of each wheel during this period.
See all solutions

Recommended explanations on Physics Textbooks

Waves Physics

Read Explanation

Famous Physicists

Read Explanation

Space Physics

Read Explanation

Electrostatics

Read Explanation

Medical Physics

Read Explanation

Oscillations

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.