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Chapter 8: Problem 22

Interactive LearningWare 8.1 at reviews the approach that is necessary for solving problems such as this one. A motorcyclist is traveling along a road and accelerates for \(4.50 \mathrm{~s}\) to pass another cyclist. The angular acceleration of each wheel is \(+6.70 \mathrm{rad} / \mathrm{s}^{2}\) and, just after passing, the angular velocity of each is \(+74.5 \mathrm{rad} / \mathrm{s}\), where the plus signs indicate counterclockwise directions. What is the angular displacement of each wheel during this time?

Short Answer

Expert verified
The angular displacement is 267.41 rad.

Step by step solution

01

Understanding the Problem

We have a motorcyclist accelerating for 4.50 seconds with an angular acceleration of +6.70 rad/s虏, and we need to find the angular displacement given the final angular velocity is +74.5 rad/s.
02

Identify Known Variables

From the problem, we have:- Initial angular velocity (\( \omega_0 \)) is unknown.- Final angular velocity (\( \omega \)) is +74.5 rad/s.- Angular acceleration (\( \alpha \)) is +6.70 rad/s虏.- Time (\( t \)) is 4.50 s.- We need to find angular displacement (\( \theta \)).
03

Using Angular Motion Equation

Use the equation: \[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]We need to first determine \( \omega_0 \) using another equation since it's not given.
04

Find Initial Angular Velocity

Use the formula for final angular velocity:\[ \omega = \omega_0 + \alpha t \]Substitute the known values to find \( \omega_0 \):\[ 74.5 = \omega_0 + 6.70 \times 4.50 \]\[ \omega_0 = 74.5 - (6.70 \times 4.50) \]\[ \omega_0 = 44.35 \text{ rad/s}\]
05

Calculate Angular Displacement

Now, substitute \( \omega_0 \) back into the angular displacement equation:\[ \theta = 44.35 \times 4.50 + \frac{1}{2} \times 6.70 \times (4.50)^2 \]Calculate each part separately:\[ \theta_1 = 44.35 \times 4.50 = 199.575 \text{ rad} \]\[ \theta_2 = \frac{1}{2} \times 6.70 \times 20.25 = 67.8375 \text{ rad} \]Add the two parts:\[ \theta = 199.575 + 67.8375 = 267.4125 \text{ rad} \]
06

Final Answer

The angular displacement of each wheel during this time is 267.41 rad.

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Key Concepts

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

Understanding Angular Acceleration
Angular acceleration is a key concept in physics, especially when discussing rotational motion. It refers to how quickly the angular velocity of an object changes over time.
For instance, in our motorcycle exercise, the angular acceleration is given as\(+6.70 \, \mathrm{rad/s^2}\). This means that every second, the rate at which the wheels spin increases by \(6.70 \, \mathrm{rad/s}\).Here are some crucial points to remember about angular acceleration:
  • It can be positive or negative. Positive indicates an increase in angular velocity.
  • Measured in radians per second squared \(\left( \mathrm{rad/s^2} \right)\).
  • If zero, the angular velocity is constant.
Angular acceleration is essential for predicting how objects will move when they start or stop rotating, making it a vivid part of kinematic studies in rotational dynamics.
Exploring Angular Velocity
Angular velocity measures how fast an object rotates or spins.It's like the rotational equivalent of linear speed.It lets us know how many radians an object rotates per second.In the problem, the final angular velocity of the motorcyclist's wheel is \(+74.5 \, \mathrm{rad/s}\). A positive value indicates counterclockwise rotation.Key aspects of angular velocity:
  • Expressed in radians per second \(\left( \mathrm{rad/s} \right)\).
  • Determines the speed of rotation.
  • Can be increased or decreased through angular acceleration.
Understanding angular velocity is important in applications like robotics, where precise control of movement is crucial.It allows engineers and physicists to calculate how fast wheels or gears need to turn for a machine to operate at a desired speed.
Delving into Kinematics
Kinematics focuses on describing the motion of objects without concerning what causes this motion. For rotational motion, kinematics covers how angular displacement, angular velocity, and angular acceleration are related.Important formulas include:
  • \(\theta = \omega_0 t + \frac{1}{2} \alpha t^2\)
  • \(\omega = \omega_0 + \alpha t\)
These formulas help solve problems like the motorcyclist scenario, where we needed to find the angular displacement.By applying the kinematic equations, students can predict and describe motion even without seeing the physical outcome. This ability is useful in designing mechanical systems and understanding natural phenomena.
Analyzing Rotational Motion
Rotational motion refers to objects that move along a circular path. It鈥檚 distinct from linear motion, which occurs in a straight line. In our exercise, a motorcycle's wheels exhibit rotational motion as the bike travels along its path. Key insights into rotational motion include:
  • Motion occurs around a fixed axis, leading to circular paths.
  • Involves quantities like angular displacement, velocity, and acceleration.
  • Central to understanding the dynamics of many mechanical systems, ranging from wheels to planets.
Understanding rotational motion is pivotal in engineering, especially in designing things like gears, engines, and even amusement park rides. Recognizing how objects move in circular paths allows innovation and efficiency in various technological advancements.

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Most popular questions from this chapter

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 onethird that of a person living in Ecuador?

Two people start at the same place and walk around a circular lake in opposite directions. One has an angular speed of \(1.7 \times 10^{-3} \mathrm{rad} / \mathrm{s}\), while the other has an angular speed of \(3.4 \times 10^{-3} \mathrm{rad} / \mathrm{s}\). How long will it be before they meet?

ssm The drawing shows a device that can be used to measure the speed of a bullet. The device consists of two rotating disks, separated by a distance of \(d=0.850 \mathrm{~m}\), and rotating with an angular speed of \(95.0 \mathrm{rad} / \mathrm{s}\). The bullet first passes through the left disk and then through the right disk. It is found that the angular displacement between the two bullet holes is \(\theta=0.240 \mathrm{rad}\). From these data, determine the speed of the bullet.

Two identical dragsters, starting from rest, accelerate side-by- side along a straight track. The wheels on one of the cars roll without slipping, while the wheels on the other slip during part of the time. (a) For which car, he winner or the loser, do the wheels roll without slipping? Why? For the dragster whose wheels roll without slipping, is there (b) a relationship between its linear speed and the angular speed of its wheels, and (c) a relationship between the magnitude of its linear acceleration and the magnitude of the angular acceleration of its wheels? If a relationship exists in either case, what is it? Problem A dragster starts from rest and accelerates down the track. Each tire has a radius of \(0.320 \mathrm{~m}\) and rolls without slipping. At a distance of \(384 \mathrm{~m},\) the angular speed of the wheels is \(288 \mathrm{rad} / \mathrm{s}\). Determine, (a) the linear speed of the dragster and (b) the magnitude of the angular acceleration of its wheels.

Refer to Interactive Solution \(\underline{8.31}\) at to review a model for solving this problem. The take-up reel of a cassette tape has an average radius of \(1.4 \mathrm{~cm} .\) Find the length of tape (in meters) that passes around the reel in \(13 \mathrm{~s}\) when the reel rotates at an average angular speed of \(3.4 \mathrm{rad} / \mathrm{s}\)
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