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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: \( \frac{8596.85}{1000} \approx 8.597 \text{ km}\).

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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.

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Most popular questions from this chapter

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 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 string trimmer is a tool for cutting grass and weeds; it utilizes a length of nylon 鈥渟tring鈥� that rotates about an axis perpendicular to one end of the string. The string rotates at an angular speed of 47 rev/s, and its tip has a tangential speed of 54 m/s. What is the length of the rotating string?

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.

After \(10.0 \mathrm{s},\) a spinning roulette wheel at a casino has slowed down to an angular velocity of \(+1.88 \mathrm{rad} / \mathrm{s}\) . During this time, the wheel has an angular accelcration of \(-5.04 \mathrm{rad} / \mathrm{s}^{2}\) , Detcrmine the angular displacement of the wheel.

A wind turbine is initially spinning at a constant angular speed. As the wind鈥檚 strength gradually increases, the turbine experiences a constant angular acceleration of \(0.140 \mathrm{rad} / \mathrm{s}^{2}\). After making 2870 revolutions, its angular speed is \(137 \mathrm{rad} / \mathrm{s}\) (a) What is the initial angularvelocity of the turbine? (b) How much time elapses while the turbine is speeding up?
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